CLT for random walks of commuting endomorphisms on compact abelian groups
aa r X i v : . [ m a t h . P R ] N ov CLT FOR RANDOM WALKS OF COMMUTINGENDOMORPHISMS ON COMPACT ABELIAN GROUPS
GUY COHEN AND JEAN-PIERRE CONZE
Abstract.
Let S be an abelian group of automorphisms of a probability space ( X, A , µ )with a finite system of generators ( A , ..., A d ). Let A ℓ denote A ℓ ...A ℓ d d , for ℓ =( ℓ , ..., ℓ d ). If ( Z k ) is a random walk on Z d , one can study the asymptotic distribu-tion of the sums P n − k =0 f ◦ A Z k ( ω ) and P ℓ ∈ Z d P ( Z n = ℓ ) A ℓ f , for a function f on X .In particular, given a random walk on commuting matrices in SL ( ρ, Z ) or in M ∗ ( ρ, Z )acting on the torus T ρ , ρ ≥
1, what is the asymptotic distribution of the associatedergodic sums along the random walk for a smooth function on T ρ after normalization?In this paper, we prove a central limit theorem when X is a compact abelian con-nected group G endowed with its Haar measure (e.g. a torus or a connected extensionof a torus), S a totally ergodic d -dimensional group of commuting algebraic automor-phisms of G and f a regular function on G . The proof is based on the cumulantmethod and on preliminary results on the spectral properties of the action of S , onrandom walks and on the variance of the associated ergodic sums. Contents
Introduction
Summation sequences
Summation sequences, kernels, examples
Spectral analysis, Lebesgue spectrum
Random walks
Random summation sequences (rws) defined by random walks
Auxiliary results on random walks ζ -regularity, variance Z d -actions by commuting endomorphisms on G Endomorphisms of a compact abelian group G The torus case: G = T ρ , examples of Z d -actions Date : 26 October 2014.2010
Mathematics Subject Classification.
Primary: 60F05, 28D05, 22D40, 60G50; Secondary: 47B15,37A25, 37A30.
Key words and phrases. quenched central limit theorem, Z d -action, random walk, self-intersectionsof a r.w., semigroup of endomorphisms, toral automorphism, mixing, S -unit, cumulant. Spectral densities and Fourier series for tori
CLT for summation sequences of endomorphisms on G Criterium for the CLT on a compact abelian connected group G Application to r.w. of commuting endomorphisms on G Random walks and quenched CLT
Powers of barycenter operators
Appendix I: self-intersections of a centered r.w. d=1: a.s. convergence of V n,p /V n, d=2: variance and SLLN for V n,p Appendix II: mixing, moments and cumulants
Introduction
Let S be a finitely generated abelian group and let ( T s , s ∈ S ) be a measure preservingaction of S on a probability space ( X, A , µ ). A probability distribution ( p s , s ∈ S ) on S defines a random walk (r.w.) ( Z n ) on S . We obtain a Markov chain on X (and arandom walk on the orbits of the action of S ) by defining the transition probabilityfrom x ∈ X to T s x as p s . The Markov operator P of the corresponding Markov chainis P f ( x ) = P s ∈S p s f ( T s x ). Limit theorems have been investigated for the associatedrandom process. For instance, conditions on f ∈ L ( µ ) are given in [11] for a “quenched”central limit theorem for the ergodic sums along the r.w. P n − k =0 f ( T Z k ( ω ) x ) when S is Z .“Quenched”, there, is understood for a.e. x ∈ X and in law with respect to ω . For otherlimit theorems in the quenched setting, see for instance [8] and references therein.Another point of view, in the study of random dynamical systems, is to prove limitinglaws for a.e. fixed ω (cf. [21]). This is our framework: for a fixed ω , we consider theasymptotic distribution of the above ergodic sums after normalization with respect to µ .More precisely, let ( T , ..., T d ) be a system of generators in S . Every T ∈ S can berepresented as T = T ℓ = T ℓ ...T ℓ d d , for ℓ = ( ℓ , ..., ℓ d ) ∈ N d . (The elements of Z d areunderlined to distinguish them from the scalars.) For a function f on X and T ∈ S , T f denotes the composition f ◦ T . The map f → T f defines an isometry on H = L ( µ ),the space of square integrable functions f on X such that µ ( f ) = 0.Given a r.w. W = ( Z n ) on Z d , we study the asymptotic distribution (for a fixed ω and with respect to the measure µ on X ) of P n − k =0 T Z k ( ω ) f, n → ∞ , after normalization.We consider also iterates of the Markov operator P introduced above (called belowbarycenter operator) and the asymptotic distribution of ( P n f ) n ≥ after normalization. LT FOR RANDOM WALKS OF COMMUTING ENDOMORPHISMS 3
Other sums for the random field (
T f, T ∈ S ), f ∈ L ( µ ), can be considered. Forinstance, if ( D n ) ⊂ N d is an increasing sequence of domains, a question is the as-ymptotic normality of | D n | − P ℓ ∈ D n T ℓ f and of the multidimensional “periodogram” | D n | − P ℓ ∈ D n e πi h ℓ,θ i T ℓ f, θ ∈ R d . All above questions can be formulated in the frame-work of what is called below “summation sequences” and their associated kernel.The specific model which is studied here is the following one: ( X, A , µ ) will be a compactabelian connected group G endowed with its Borel σ -algebra A and its Haar measure µ and S will be a semigroup of commuting algebraic endomorphisms of G . This extendsthe classical situation of a CLT for a single endomorphism (R. Fortet (1940) and M. Kac(1946) for G = T , V. Leonov (1960) for a general ergodic endomorphism of G ).The main result (Theorem 5.1) is a quenched CLT for ( P n − k =0 T Z k ( ω ) f ), when f is aregular function on G , ( T ℓ ) ℓ ∈ Z d a Z d -action on G by automorphisms and ( Z k ) a r.w. on Z d . After reduction of the r.w., we examine three different cases: a) moment of order 2,centering, dimension 1; b) moment of order 2, centering, dimension 2; c) transient case.This covers all cases when there is a moment of order 2. As usual, the transient case isthe easiest. The recurrent case requires to study the self-intersections of the r.w.The paper is organized as follows. In Sect. 1 we discuss methods of summation and recallsome facts on the spectral analysis of a finitely generated group of unitary operatorswith Lebesgue spectrum. The notion of “regular” summation sequence allows a unifiedtreatment of different cases, including ergodic sums over sets or sums along randomsequences generated by random walks. In Sect. 2 we prove auxiliary results on theregularity of quenched and barycenter summation sequences defined by a r.w. on Z d .These first two sections apply to a general Z d -action with Lebesgue spectrum.In Sect. 3, the model of multidimensional actions by endomorphisms or automorphismson a compact abelian group G is presented. We recall how to construct totally ergodictoral Z d -actions by automorphisms and we give explicit examples. The link between theregularity of a function f on G and its spectral density ϕ f is established, with a specifictreatment for tori for which the required regularity for f is weaker. A sufficient conditionfor the CLT on a compact abelian connected group G is given in Sect. 4.The results of the first sections are applied in Sect. 5. A quenched CLT for the ergodicsums of regular functions f along a r.w. on G by commuting automorphisms is shownwhen G is connected. For a transient r.w. the variance, which is related to a measure on T d with an absolutely continuous part, does not vanish. Another summation method,iteration of barycenters, yields a polynomial decay of the iterates for regular functionsand a CLT for the normalized sums, the nullity of the variance being characterized interms of coboundary. An appendix (Sect. 6) is devoted to the self-intersections propertiesof a r.w. In Sect. 7, we recall the cumulant method used in the proof of the CLT.To conclude this introduction, let us mention that in a previous work [5] we extendedto ergodic sums of multidimensional actions by endomorphisms the CLT proved by GUY COHEN AND JEAN-PIERRE CONZE
T. Fukuyama and B. Petit [14] for coprime integers acting on the circle. After com-pleting it, we were informed of the results of M. Levin [26] showing the CLT for ergodicsums over rectangles for actions by endomorphisms on tori. The proof of the CLT forsums over rectangles is based in both approaches, as well as in [14], on results on S -unitswhich imply mixing of all orders for connected groups (K. Schmidt and T. Ward [31]).The cumulant method used here is also based on this property of mixing of all orders.1. Summation sequences
Let us consider first the general framework of an abelian finitely generated group S (isomorphic to Z d ) of unitary operators on a Hilbert space H . There exists a systemof independent generators ( T , ..., T d ) in S and each element of S can be written in aunique way as T ℓ = T ℓ ...T ℓ d d , with ℓ = ( ℓ , ..., ℓ d ) ∈ Z d .Our main example (Sect. 3) will be given by groups of commuting automorphisms (orextensions of semigroups of commuting surjective endomorphisms) of a compact abeliangroup G . They act by composition on H = L ( G, µ ) (with µ the Haar measure of G )and yield examples of Z d -actions by unitary operators.Given f ∈ H , there are various choices of summation sequences for the random field( T ℓ f, ℓ ∈ Z d ). In the first part of this section, we discuss this point and the behavior ofthe kernels associated to summation sequences. The second part of the section is devotedto the spectral analysis of d -dimensional commuting actions with Lebesgue spectrum.1.1. Summation sequences, kernels, examples.Definition 1.1.
We call summation sequence a sequence ( R n ) n ≥ of functions from Z d to R + with 0 < P ℓ ∈ Z d R n ( ℓ ) < + ∞ , ∀ n ≥
1. Given S = { T ℓ , ℓ ∈ Z d } and f ∈ H , theassociated sums are P ℓ ∈ Z d R n ( ℓ ) T ℓ f .Let ˇ R n ( ℓ ) := R n ( − ℓ ). The normalized nonnegative kernel ˜ R n (with k ˜ R n k L ( T d ) = 1) is˜ R n ( t ) = | P ℓ ∈ Z d R n ( ℓ ) e πi h ℓ,t i | P ℓ ∈ Z d | R n ( ℓ ) | = X ℓ ∈ Z d ( R n ∗ ˇ R n )( ℓ )( R n ∗ ˇ R n )(0) e πi h ℓ,t i , t ∈ T d . (1) Definition 1.2.
We say that ( R n ) is ζ - regular , if ( ˜ R n ) n ≥ weakly converges to a proba-bility measure ζ on T d , i.e., R T d ˜ R n ϕ dt −→ n →∞ R T d ϕ dζ for every continuous ϕ on T d .The existence of the limit: L ( p ) = lim n →∞ Z ˜ R n ( t ) e − πi h p,t i dt = lim n ( R n ∗ ˇ R n )( p )( R n ∗ ˇ R n )(0) , for all p ∈ Z d , is equivalent to ζ -regularity with ˆ ζ ( p ) = L ( p ). Note that ˜ R n and ζ are even. Examples 1.3.
Summation over sets
A class of summation sequences is given by summation over sets. If ( D n ) n ≥ is a sequenceof finite subsets of Z d and R n = 1 D n , we get the “ergodic” sums P ℓ ∈ D n T ℓ f . The LT FOR RANDOM WALKS OF COMMUTING ENDOMORPHISMS 5 simplest choice for ( D n ) is an increasing family of d -dimensional rectangles. The sequence( R n ) = (1 D n ) is δ -regular if and only if ( D n ) satisfieslim n →∞ | D n | − | ( D n + p ) ∩ D n | = 1 , ∀ p ∈ Z d (Følner condition) . (2)a) (Rectangles) For N = ( N , ..., N d ) ∈ N d and D N := { ℓ ∈ N d : ℓ i ≤ N i , i = 1 , ..., d } ,the kernels are the d -dimensional Fej´er kernels K N ( t , ..., t d ) = K N ( t ) · · · K N d ( t d ) on T d , where K N ( t ) is the one-dimensional Fej´er kernel N ( sin πNt sin πt ) .b) A family of examples satisfying (2) can be obtained by taking a non-empty domain D ⊂ R d with “smooth” boundary and finite area and putting D n = λ n D ∩ Z d , where( λ n ) is an increasing sequence of real numbers tending to + ∞ .c) If ( D n ) is a (non Følner) sequence of domains in Z d such that lim n | ( D n + p ) ∩ D n || D n | =0 , ∀ p = 0, the associated kernel ( ˜ R n ) is ζ -regular, with ζ the uniform measure on T d . Examples 1.4.
Sequential summation
Let ( x k ) k ≥ be a sequence in Z d . Putting z n := P n − k =0 x k , n ≥
1, and R n ( ℓ ) = P n − k =0 z k = ℓ for ℓ ∈ Z d , we get a summation sequence of sequential type. Here, P ℓ ∈ Z d R n ( ℓ ) = n .The associated sums are P ℓ ∈ Z d R n ( ℓ ) T ℓ f = P n − k =0 T z k f . Let us define v n = v n, = { ≤ k ′ , k < n : z k = z k ′ } = X ℓ ∈ Z d R n ( ℓ ) , (3) v n,p := { ≤ k ′ , k < n : z k − z k ′ = p } = X ≤ k ′ ,k A reduction to the smallest lattice containing the support of the summation sequence isconvenient and can be done using Minkowski lemma.Below ( e , ..., e d ) is the canonical basis of R d . For r < d , E r (resp. E r ) is the latticeover Z (resp. the vector space) generated by ( e , ..., e r ) and F d − r the sublattice of Z d generated by ( e r +1 , ..., e d ).The set of non singular d × d -matrices with coefficients in Z is denoted by M ∗ ( d, Z ). GUY COHEN AND JEAN-PIERRE CONZE Lemma 1.5. a) Let L be a sublattice of Z d . If r ≥ is the dimension of the R -vectorspace L spanned by L , there exists C ∈ SL ( d, Z ) such that L = C E r .b) If r = d , there exists B ∈ M ∗ ( d, Z ) , such that L = B Z d and Card( Z d /L ) = | det( B ) | .Proof . a) The proof is by induction on r . By definition of L we can find a non zerovector v = ( v , ..., v d ) in L with integral coordinates such that gcd ( v , ..., v d ) = 1.Since a vector in Z d is extendable to a basis of Z d if and only if the gcd of its coordinatesis 1, we can construct an integral matrix C such that det C = 1 with v as first column.For w ∈ L , let ˜ w := h w, w i v − h v, w i v . The space L ∩ v ⊥ is generated by the lattice { ˜ w, w ∈ L } and has dimension r − 1. By the induction hypothesis, there is an integralmatrix C with det C = 1, such that C e r = e r and C E r − = C − ( L ∩ v ⊥ ). Taking C = C C , we have C E r − = C C E r − = L ∩ v ⊥ , Ce r = C C e r = C e r = v ∈ L .b) The existence of B is equivalent to the existence in L of a set { f , . . . , f d } of linearlyindependent vectors generating L . Let us construct such a set.The set { x : x = ( x , ..., x d ) ∈ L } is an ideal, hence the set of multiples of someinteger a . We can assume that a = 0 (otherwise we permute the coordinates). Let f = ( f , ..., f d ) ∈ L be such that f = a . The set { x − a − x f , x ∈ L } is a sublattice L of L with 0 as first coordinate of every element. It is of dimension d − R , hence,by induction hypothesis, generated by linearly independent vectors { f , . . . , f d − } in L .Clearly { f , . . . , f d − , f } is a set of linearly independent vectors generating L .Let R be a set of representatives of the classes of Z d modulo B Z d and let K be the unitcube in R d . Since if L is a cofinite lattice in Z d , two fundamental domains F , F in R d for L have the same measure: Leb( F ) = Leb( F ). Taking F = BK and F = S r ∈R ( K + r ),this implies | det( B ) | = Leb( BK ) = Leb (cid:16)S r ∈R ( K + r ) (cid:17) = Card( R ). (cid:3) Let us consider the lattice L in Z d generated by the elements ℓ ∈ Z d such that R n ( ℓ ) > n . For instance, for a sequential summation generated by ( x k ) k ≥ , the lattice L is the group generated in Z d by the vectors x k .By the previous lemma, after a change of basis given by a matrix in SL ( d, Z ) andreduction, we can assume without loss of generality that d is the genuine dimension , i.e.,that the vector space L generated by L over R has dimension d .1.2. Spectral analysis, Lebesgue spectrum. Recall that, if S is an abelian group of unitary operators, for every f ∈ H there is apositive finite measure ν f on T d , the spectral measure of f , with Fourier coefficientsˆ ν f ( ℓ ) = h T ℓ f, f i , ℓ ∈ Z d . When ν f is absolutely continuous, its density is denoted by ϕ f . In what follows, we assume that S (isomorphic to Z d ) has the Lebesgue spectrum propertyfor its action on H , i.e., there exists a closed subspace K such that { T ℓ K , ℓ ∈ Z d } is afamily of pairwise orthogonal subspaces spanning a dense subspace in H . If ( ψ j ) j ∈J isan orthonormal basis of K , { T ℓ ψ j , j ∈ J , ℓ ∈ Z d } is an orthonormal basis of H . LT FOR RANDOM WALKS OF COMMUTING ENDOMORPHISMS 7 In Sect. 3, for algebraic automorphisms of compact abelian groups, the characters willprovide a natural basis.Let H j be the closed subspace generated by ( T ℓ ψ j ) ℓ ∈ Z d and f j the orthogonal projectionof f on H j . We have ν f = P j ν f j . For j ∈ J , we denote by γ j an everywhere finitesquare integrable function on T d with Fourier coefficients a j, ℓ := h f, T ℓ ψ j i . A version ofthe density of the spectral measure corresponding to f j is | γ j | . By orthogonality of the subspaces H j , it follows that, for every f ∈ H , ν f has a density ϕ f in L ( dt ) which reads ϕ f ( t ) = P j ∈J | γ j ( t ) | . Changing the system of independent generators induces composition of the spectral den-sity by an automorphism acting on T d . After the reduction performed in 1.1.1 (Lemma1.5) we obtain an action with Lebesgue spectrum with, possibly, a smaller dimension.We have: R T d P j ∈J | γ j ( t ) | dt = P j ∈J P ℓ ∈ Z d | a j,ℓ | = R T d ϕ f ( t ) dt = k f k < ∞ . For t in T d , let M t f in K be defined, when the series converges in H , by: M t f := P j γ j ( t ) ψ j .Under the condition P j ∈J (cid:0) P ℓ | a j,ℓ | (cid:1) < + ∞ , M t f is defined for every t , the function t → k M t k is a continuous version of ϕ f . For a general f ∈ H , it is defined for t in a setof full measure in T d (cf. [4]). Variance for summation sequences Suppose that ( R n ) is a ζ -regular summation sequence. Let f be in L ( µ ) with a contin-uous spectral density ϕ f . By the spectral theorem, we have for θ ∈ T d :( X ℓ R n ( ℓ )) − k X ℓ ∈ Z d R n ( ℓ ) e πi h ℓ,θ i T ℓ f k = ( ˜ R n ∗ ϕ f )( θ ) −→ n →∞ ( ζ ∗ ϕ f )( θ ) . (6) Examples. 1) If ( D n ) is a Følner sequence of sets in Z d , then (for θ = 0) ζ = δ and we obtain the usual asymptotic variance σ ( f ) = ϕ f (0). More generally the usualasymptotic variance at θ for the “rotated” ergodic sums R θn f islim n | D n | − k X ℓ ∈ D n e πi h ℓ,θ i T ℓ f k = lim n ( ˜ R n ∗ ϕ f )( θ ) = ( δ ∗ ϕ f )( θ ) = ϕ f ( θ ) . When ( D n ) is a sequence of d -dimensional cubes, by the Fej´er-Lebesgue theorem, forevery f in H , since ϕ f ∈ L ( T d ), for a.e. θ , the variance at θ exists and is equal to ϕ f ( θ ).2) Let ( x k ) k ≥ be a sequence in Z d and z n = P n − k =0 x k . If ( z n ) is ζ -regular, then (cf. (5)):lim n v − n k n − X k =0 T z k f k = lim n Z ˜ R n ϕ f dt = ζ ( ϕ f ).2. Random walks Random summation sequences (rws) defined by random walks. For d ≥ J a set of indices, let Σ := { ℓ j , j ∈ J } be a set of vectors in Z d and( p j , j ∈ J ) a probability vector with p j > , ∀ j ∈ J . GUY COHEN AND JEAN-PIERRE CONZE Let ν denote the probability distribution with support Σ on Z d defined by ν ( ℓ ) = p j for ℓ = ℓ j , j ∈ J . The euclidian norm of ℓ ∈ Z d is denoted by | ℓ | .Let ( X k ) k ∈ Z be a sequence of i.i.d. Z d -valued random variables with common distribution ν , i.e., P ( X k = ℓ j ) = p j , j ∈ J . The associated random walk W = ( Z n ) in Z d (startingfrom 0) is defined by Z := 0, Z n := X + ... + X n − , n ≥ X (computed with a coefficient 2 π ) isΨ( t ) = E [e πi h X ,t i ] = X ℓ ∈ Z d P ( X = ℓ ) e πi h ℓ,t i = X j ∈ J p j e πi h ℓ j ,t i , t ∈ T d . (7) It satisfies: P ( Z n = k ) = Z T d Ψ n ( t ) e − πi h k,t i dt. (8)If (Ω , P ) denotes the product space (( Z d ) Z , ν ⊗ Z ) with coordinates ( X n ) n ∈ Z and τ theshift, then W can be viewed as the cocycle generated by the function ω → X ( ω ) underthe action of τ , i.e., Z n = P n − k =0 X ◦ τ k , n ≥ T , ..., T d be d commuting unitary operators on a Hilbert space H generating arepresentation of Z d in H with Lebesgue spectrum. Let ( Z n ) n ≥ = (( Z n , ..., Z dn )) n ≥ be ar.w. with values in Z d . We consider the random sequence of unitary operators ( T Z n ) n ≥ ,with T Z n = T Z n ...T Z dn d . For f ∈ H , the “quenched” process (for ω fixed) of the ergodicsums along the random walk is P n − k =0 T Z k ( ω ) f = P ℓ ∈ Z d R n ( ω, ℓ ) T ℓ f, with R n ( ω, ℓ ) = P n − k =0 Z k ( ω )= ℓ , n ≥ . (9)The sequence of “local times” ( R n ( ω, ℓ )) will be called a random walk sequential sum-mation (abbreviated in “r.w. summation” or in “rws”). It satisfies P ℓ R n ( ω, ℓ ) = n .We consider also the (Markovian) barycenter operator P : f → P j ∈ J p j T ℓ j f = E P [ T X ( . ) f ]and its powers: P n f = E P [ T Z n ( . ) f ]) = X ℓ ∈ Z d R n ( ℓ ) T ℓ f, with R n ( ℓ ) = P ( Z n = ℓ ) . (10)When the operators T j are given by measure preserving transformations of a probabilityspace ( X, A , µ ), we call quenched limit theorem a limit theorem for (9) w.r.t. µ . If thelimit is taken w.r.t. P × µ , it is called annealed limit theorem . Notations 2.1. The random versions of (3) and (4) are V n ( ω ) := { ≤ k ′ , k < n : Z k ( ω ) = Z k ′ ( ω ) } = X ℓ ∈ Z d R n ( ω, ℓ ) , (11) V n,p ( ω ) := { ≤ k ′ , k < n : Z k ( ω ) − Z k ′ ( ω ) = p } = X ≤ k ′ ,k We have: V n,p ( ω ) = n p =0 + n − X k =1 n − k − X j =0 (1 Z k ( τ j ω )= p + 1 Z k ( τ j ω )= − p );(13) E V n,p = n p =0 + n − X k =1 ( n − k ) [ P ( Z k = p ) + P ( Z k = − p )] . (14) Variance for quenched processes For a r.w. summation R n ( ω, ℓ ) = P n − k =0 Z k ( ω )= ℓ , let R ωn ( t ) and e R ωn ( t ) denote, respectively,the non normalized and normalized kernels (cf. (1)) R ωn ( t ) = | n − X k =0 e πi h Z k ( ω ) ,t i | = | X ℓ ∈ Z d R n ( ω, ℓ ) e πi h ℓ,t i | , e R ωn ( t ) = R ωn ( t ) / Z T d R ωn dt. (15)Recall that, if ϕ f is the spectral density of f ∈ H for the action of Z d , we have k n X k =1 T Z k ( ω ) f k = Z T d R ωn ( t ) ϕ f ( t ) dt by the spectral theorem (cf. (6)) and Z T d n R ωn ( t ) e − πi h p,t i dt = 1 n V n,p ( ω ) = 1 p =0 + n − X k =1 n n − k − X j =0 [1 Z k ( τ j ω )= p + 1 Z k ( τ j ω )= − p ] . (16) A rws defined by a r.w. ( Z n ) is said to be ζ -regular if ( ˜ R ωn ) n ≥ weakly converges to aprobability measure ζ (not depending on ω ) on T d for a.e. ω . This is equivalent to lim n →∞ V n,p ( ω ) V n ( ω ) = ˆ ζ ( p ), for a.e. ω and all p ∈ Z d (see Definition 1.2and Example 1.4). After auxiliary results on r.w.’s and self-intersections, we will showthat every rws is ζ -regular for some measure ζ .2.2. Auxiliary results on random walks. We recall now some general results on random walks in Z d (cf. [33]). Most of themare classical. Nevertheless, since we do not assume strict aperiodicity and in order toexplicit the constants in the limit theorems, we include reminders and some proofs. Westart with definitions and preliminary results.2.2.1. Reduced form of a random walk . Let L ( W ) (or simply L ) be the sublattice of Z d generated by the support Σ = { ℓ j , j ∈ J } of ν . From now on (without loss of generality, as shown by Lemma 1.5), we assume that L ( W ) is cofinite in Z d and we say that W is reduced (in other words “genuinely d -dimensional”, cf. [33]). Therefore the vector space L generated by L is R d and there is B ∈ M ∗ ( d, Z ) such that L = B Z d . The rank d will be sometimes denoted by d ( W ).Let D = D ( W ) be the sublattice of Z d generated by { ℓ j − ℓ j ′ , j, j ′ ∈ J } , D the vectorsubspace of R d generated by D and D ⊥ its orthogonal supplementary in R d . We denote dim( D ) by d ( W ). With the notations of 1.1.1, by Lemma 1.5, there is B ∈ M ∗ ( d, Z )such that D = B Z d if d ( W ) = d and D = B F d − if d ( W ) = d − Lemma 2.2. a) The quotient group L ( W ) /D ( W ) is cyclic (each ℓ j is a generator of thegroup). It is finite if and only if d ( W ) = d ( W ) .b) If W is reduced, then d ( W ) = d ( W ) or d ( W ) − .c) If W has a moment of order 1 and is centered, then d ( W ) = d ( W ) and D and L generate the same vector subspace.Proof . The point a) is clear, since ℓ j = ℓ mod D, ∀ j ∈ J . For b) , suppose D ⊥ = { } .There is v ∈ D ⊥ such that h v , ℓ i = 1. If v is in D ⊥ , then h v − h v, ℓ i v , ℓ j i = 0, whichimplies v = h v, ℓ i v , since L = R d . c) If v ∈ D ⊥ , then h v, ℓ j i = h v, ℓ i = P i p i h v, ℓ i = h v, P i p i ℓ i i = 0 , ∀ j ; hence v = 0. (cid:3) Recurrence/transience. Recall that a r.w. W = ( Z n ) is recurrent if and only if P ∞ n =1 P ( Z n = 0) = + ∞ . If P p j | ℓ j | < ∞ , then W is transient if P p j ℓ j = 0 and, if d = 1, recurrent if P p j ℓ j = 0. These last two results are special cases of a generalresult for cocycle over an ergodic dynamical system. A r.w. W is transient or recurrentaccording as ℜ e ( − Ψ ) is integrable on the d -dimensional unit cube or not ([33]).The local limit theorem (LLT) (cf. Theorem 2.6) implies that a (reduced) r.w. W withfinite variance is recurrent if and only if it is centered and d ( W ) = 1 or 2.For further references, let us introduce the following condition for a r.w. W : W is reduced, has a moment of order 2 and is centered. (17)If W (reduced) has a moment of order 2, either it is centered with d ( W ) ≤ d ( W ) = d , in the second d ( W ) = d or d − Annulators of L and D in T d . The values of t ∈ T d such that Ψ( t ) = 1 or | Ψ( t ) | = 1play an important role. They are characterized in the following lemma, where B and B are the matrices introduced at the beginning of 2.2.1. Recall that the annulator of asublattice L in Z d is the closed subgroup { t ∈ T d : e πi h r,t i = 1 , ∀ r ∈ L } . Notation 2.3. We denote by Γ (resp. Γ ) the annulator of L (resp. D ), by dγ (resp. dγ ) the Haar probability measure of the group Γ (resp. Γ ). Lemma 2.4. 1) We have Γ = { t ∈ T d : Ψ( t ) = 1 } = ( B t ) − Z d mod Z d . The group Γ isfinite and Card(Γ) = | det B | .2a) We have Γ = { t ∈ T d : | Ψ( t ) | = 1 } = f D ⊥ + ( B t ) − Z d mod Z d with f D ⊥ := D ⊥ / Z d ,which is either trivial or a 1-dimensional torus in T d . The quotient Γ / f D ⊥ is finite and a ( W ) := Card(Γ / f D ⊥ ) = | det B | .2b) If p ∈ D + nℓ , then the function F n,p ( t ) := Ψ( t ) n e − πi h p,t i is invariant by translationby elements of Γ . Its integral is 0 if p D + nℓ . We have | F n,p ( t ) | < , ∀ t Γ , and F n,p ( t ) = 1 , ∀ t ∈ Γ .Proof . We prove 2). The proof of 1) is analogous. We have | Ψ( t ) | ≤ | Ψ( t ) | = 1 if and only if e πi h ℓ j ,t i = e πi h ℓ ,t i , ∀ j ∈ J , i.e., if and only if t ∈ Γ . LT FOR RANDOM WALKS OF COMMUTING ENDOMORPHISMS 11 Recall that D = B Z d or B F d − . Let us treat the second case. By orthogonality, D ⊥ = ( B t ) − E . If t ∈ Γ , e πi h r,t i = 1 , ∀ r ∈ D , so that, for j = 2 , ..., d , e πi h B e j ,t i = 1,hence h e j , B t t i ∈ Z . It follows that the d − B t t are integers.Therefore, B t t ∈ E mod Z d , i.e., t ∈ ( B t ) − E + Γ = D ⊥ + Γ mod Z d , where Γ :=( B t ) − Z d mod Z d and f D ⊥ = D ⊥ modulo Z d is a closed 1-dimensional subtorus of T d .By Lemma 1.5.b, we have Card(Γ ) = Card( Z d /B t Z d ) = | det( B ) | . For a (reduced)centered r.w., Γ = Γ and Card(Γ ) = | det( B ) | .2b) Since F n,p ( t + t ) = e i h nℓ − p, t i F n,p ( t ) for t ∈ Γ , F n,p is invariant by all t ∈ Γ if p ∈ D + nℓ , and its integral is 0 if p D + nℓ . (cid:3) For a reduced r.w. W , there are ( r j , j ∈ J ) in Z d such that ℓ j = B r j and the latticegenerated by ( r j , j ∈ J ) is Z d . The r.w. is said to be aperiodic if L = Z d , stronglyaperiodic if D = Z d . If we replace W by the r.w. W ′ defined by the r.v. X ′ suchthat P ( X ′ = r j ) = p j , we obtain an aperiodic r.w. This allows in several proofs toassume aperiodicity without loss of generality when the r.w. is reduced (for example seeTheorem 2.10 in Sect. 6).Strong aperiodicity is equivalent to | Ψ( t ) | < t = 0 mod Z d (cf. [33] and Lemma2.4). It is also equivalent to: for every ℓ ∈ Z d , the additive group generated by Σ + ℓ is Z d . It implies d ( W ) = d ( W ) and a ( W ) = 1. Observe that, contrary to aperiodicity, itis not always possible to reduce proofs to the strictly aperiodic case.A r.w. is “deterministic” if P ( X = ℓ ) = 1 for some ℓ , so that | Ψ( t ) | ≡ Quadratic formLemma 2.5. Let W satisfy (17). Let Q be the quadratic form Q ( u ) = Var( h X , u i ) = P j ∈ J p j h ℓ j , u i and Λ the corresponding symmetric matrix. Then Q is definite positive.If Card( J ) = d + 1 , then det(Λ) = c Q Card( J ) j =1 p j , where c does not depend on the p j ’s.Proof . If Q ( u ) = 0, then h X , u i is a.e. constant, i.e., h ℓ i , u i = h ℓ j , u i for all i, j ∈ J ;hence u = 0, since (17) implies D = R d .Now, if d ′ = Card( J ) is finite, let J ′ be the set of indices { , ..., d ′ } . The quadraticform q ( u ) := P j ∈ J ′ p j u j − ( P j ∈ J ′ p j u j ) , u ∈ R d ′ − , is defined by the symmetric matrix A = D d ′ M , where D d ′ := diag ( p , ..., p d ′ ) and M := − p − p . − p d ′ − p − p . − p d ′ . . . . − p − p . − p d ′ .By subtracting line from line in the matrix M , we find det( M ) = 1 − P j ∈ J ′ p j andtherefore det( A ) = Q d ′ j =1 p j . The quadratic form q is positive definite since X j ∈ J ′ p j u j − ( X j ∈ J ′ p j u j ) − X ≤ j ′ U v is the vector with coordinates h ℓ j − ℓ , v i , j ∈ J ′ . The quadratic form Q can be written Q ( u ) = P j ∈ J ′ p j h ℓ j − ℓ , u i − ( P j ∈ J ′ p j h ℓ j − ℓ , u i ) = q ( U u ). Since D = R d , U is injective, hence an isomorphism ifand only if dim D = d = d ′ − d = d ′ − 1, the determinant of Λ = U t AU is c Q d +1 j =1 p j . The integer c does not dependon the probability vector ( p j ) j ∈ J . We have c = det( ˜ U ) where, for j = 1 , ..., d , the matrix˜ U representing U has as j -th row the d coordinates of the vector ℓ ( j +1) − ℓ . (cid:3) Local limit theorem. Let W = ( Z n ) be a reduced random walk in Z d associated to the distribution P ( X = ℓ j ) = p j , j ∈ J , with a finite second moment. Recall that a ( W ) and Λ are defined inLemmas 2.4 and 2.5. The local limit theorem (LLT) gives an equivalent of P ( Z n = k )when n tends to infinity: Theorem 2.6. Suppose W centered. If k D + nℓ , then P ( Z n = k ) = 0 . If k ∈ D + nℓ ,then we have with sup k | ε n ( k ) | = o (1) : (2 πn ) d P ( Z n = k ) = a ( W ) det(Λ) − e − n h Λ − k, k i + ε n ( k ) . (18) Remarks 2.7. 1) If ( Z n ) is strongly aperiodic centered with finite second moment, thenlim n →∞ [(2 πn ) d P ( Z n = k )] = det(Λ) − ([33], P.10). A version of the LLT in the centeredcase, extending a result of P´olya, was proved by van Kampen and Wintner (1939) in [18]without the strong aperiodicity assumption.2) By Lemma 2.5, if J is finite and Card( J ) = d + 1, then det(Λ) = c Q Card( J ) j =1 p j , where c is an integer ≥ p j ’s.3) It follows from (18) that, for every k , there is N ( k ) such that, for n ≥ N ( k ), P ( Z n = k ) > k ∈ D + nℓ . It implies also sup k ∈ Z d P ( Z n = k ) = O ( n − d ).4) The condition k ∈ D + nℓ reads nℓ = k mod D . Since here d ( W ) = d ( W ), L/D is a cyclic finite group by Lemma 2.2 and the set E ( k ) := { n ≥ k ∈ D + nℓ } for agiven k is an arithmetic progression.5) Let us mention a general version of the LLT which can be proved under the assumptionof finite second moment for a centered or non centered r.w.The quadratic form Q reads now: Q ( u ) = P j ∈ J p j h ℓ j , u i − ( P j ∈ J p j h ℓ j , u i ) . If Λ isthe self-adjoint operator such that Q ( u ) = h Λ u, u i , then D is invariant by Λ and therestriction of Q to D is definite positive. We denote by Λ the restriction of Λ to D . Theorem 2.8. For k ∈ Z d , let z n,k := k √ n −√ n E ( X ) = k √ n −√ n P j p j ℓ j . If k D + nℓ ,then P ( Z n = k ) = 0 . If k ∈ D + nℓ , then, with sup k | ε n ( k ) | = o (1) : (2 πn ) d W )2 P ( Z n = k ) = a ( W ) det(Λ ) − e − h Λ − z n,k , z n,k i + ε n ( k ) . (19) If the moment of order 3 is finite, then sup k | ε n ( k ) | = O ( n − ) . LT FOR RANDOM WALKS OF COMMUTING ENDOMORPHISMS 13 In the non centered case, when z n,k is not too large, for example when e −h Λ − z n,k , z n,k i ≥ k | ε n ( k ) | , Equation (19) gives an information on how the r.w. spreads around thedrift. With a finite third moment, it shows that, for any constant C , for n large, ther.w. takes with a positive probability the values k such that k ∈ D + nℓ which belongto a ball of radius C √ n centered at the drift n E ( X ).2.2.3. Upper bound for Φ n ( ω ) := sup ℓ ∈ Z d R n ( ω, ℓ ) = sup ℓ ∈ Z d P n − k =0 Z k ( ω )= ℓ . For the quenched CLT, we need some results on the local times and on the number ofself-intersections of a r.w. W . Proposition 2.9. (cf. [3] ) a) If the r.w. W has a moment of order 2, thenfor d = 1 : Φ n ( ω ) = o ( n + ε ); for d = 2 : Φ n ( ω ) = o ( n ε ) , ∀ ε > . (20) b) If W is transient with a moment of order η for some η > , then Φ n ( ω ) = o ( n ε ) , ∀ ε > .Proof . Let A rn := { ℓ ∈ Z d : | ℓ | ≤ n r } . If E ( k X k η ) < ∞ for η > 0, then for rη > P ∞ k =1 P ( | X k | > k r ) ≤ ( P ∞ k =1 k − rη ) E ( k X k η ) < ∞ . By Borel-Cantelli lemma, it follows | X k | ≤ k r a.s. for k big enough. Therefore there is N ( ω ) a.e. finite such that: | X + ... + X n − | ≤ n r +1 for n > N ( ω ). Hence sup ℓ ∈ Z d R mn ( ω, ℓ ) = sup ℓ ∈A r +1 n R mn ( ω, ℓ ) for n ≥ N ( ω ). a) We take r = 1. For all m ≥ C m , C ′ m independent of ℓ , we have: E [ R mn ( ., ℓ )] = E [ n − X k =0 Z k = ℓ ] m ≤ C m n m/ , for d = 1 , and ≤ C ′ m (Log n ) m , for d = 2 . (21)To show (21), it suffices to bound P ≤ k 0, there is r and N ( ω ) < + ∞ a.e. such thatsup ℓ ∈ Z d R mn ( ω, ℓ ) = sup ℓ ∈A r +1 n R mn ( ω, ℓ ) for n ≥ N ( ω ). In view of E [sup ℓ ∈A rn R mn ( ω, ℓ )] ≤ n d ( r +1) sup ℓ E [ R mn ( ω, ℓ )] ≤ C m n d ( r +1) M m , for every ε > m such that P ∞ n =1 E [ n − ε sup ℓ ∈A rn R n ( ω, ℓ )] m < ∞ . This implies b) . (cid:3) Self-intersections.1) Recurrent case The proof of the following result is postponed to Sect. 6. Theorem 2.10. Let W satisfy (17) and let p be in L ( W ) . If d ( W ) = 1 or 2, lim n V n,p ( ω ) V n ( ω ) =1 a.e. If d ( W ) = 2 , V n,p satisfies a SLLN: lim n V n,p ( ω ) / E V n,p = 1 a.e. 2) Transient case In the transient case (without moment assumptions), a general argument is available: Lemma 2.11. If (Ω , P , τ ) is an ergodic dynamical system and ( f k ) k ≥ a sequence offunctions in L (Ω , P ) such that P k ≥ k f k k r < ∞ , for some r > , then lim n n n − X k =1 n − k − X ℓ =0 f k ( τ ℓ ω ) = ∞ X k =1 Z f k d P , for a.e. ω. (22) Proof . We can assume f k ≥ 0. For the maximal function ˜ f k ( ω ) := sup n ≥ n P n − ℓ =0 f k ( τ ℓ ω ),by the ergodic maximal lemma, there is a finite constant C r such that k ˜ f k k r ≤ C r k f k k r .Therefore P ∞ k =1 ˜ f k ∈ L r (Ω , P ); hence P ∞ k =1 ˜ f k ( ω ) < + ∞ , for a.e. ω .Let ω be such that P ∞ k =1 ˜ f k ( ω ) < + ∞ . For ε > 0, there is L such that P k>L R f k d P ≤ P k>L k f k k r ≤ ε and P k>L ˜ f k ( ω ) ≤ ε ; hence, uniformly in n , n P nk = L +1 P n − k − ℓ =0 f k ( τ ℓ ω ) ≤ ε . By the ergodic theorem, we have lim n n P Lk =1 P n − k − ℓ =0 f k ( τ ℓ ω ) = P Lk =1 R f k d P .Therefore, for n big enough: | n P nk =1 P n − k − ℓ =0 f k ( τ ℓ ω ) − P ∞ k =1 R f k d P | ≤ ε . (cid:3) Recall that Ψ( t ) = E [ e πi h X ,t i ] , t ∈ T d . We use the notation: w ( t ) := 1 − | Ψ( t ) | | − Ψ( t ) | , c w = Z T d w dt, when w is integrable . (23)Outside the finite group Γ, w is well defined, nonnegative and w ( t ) = 0 only on Γ \ Γ.Hence it is positive for a.e. t , excepted when the r.w. is “deterministic”. Proposition 2.12. ( [33] ) Let W = ( Z n ) be a transient random walk in Z d .a) The function w is integrable on T d and there is a nonnegative constant K such that,if W is aperiodic, I ( p ) := 1 p =0 + ∞ X k =1 [ P ( Z k = p ) + P ( Z k = − p )] = Z T d cos 2 π h p, t i w ( t ) dt + K. (24) For a general reduced r.w. W, the sums I ( p ) are the Fourier coefficients of the measure wdt + Kdγ . LT FOR RANDOM WALKS OF COMMUTING ENDOMORPHISMS 15 b) If d > , then K = 0 ; if d = 1 and m ( W ) := P ℓ ∈ Z P ( X = ℓ ) | ℓ | < + ∞ , then K = | P ℓ ∈ Z P ( X = ℓ ) ℓ | − ; if d = 1 and m ( W ) = + ∞ , then K = 0 .Proof . For completeness we give a proof of a), following Spitzer ( § 9, P2 in [33]). For b)a more difficult result without assumption on the moments need to be used (see Spitzer § 24, P5, P6, P8, T2 in [33]).Let ( Z n ) be a transient, reduced, aperiodic r.w. Since P ∞ k =1 P ( Z k = p ) < + ∞ , we have ∞ X k =0 λ k P ( Z k = p ) = Z T d e πi h p,t i − λ Ψ( t ) dt, ∀ λ ∈ [0 , , ∞ X k =0 [ P ( Z k = p ) + P ( Z k = − p )] = 2 lim λ ↑ Z T d cos 2 π h p, t i ℜ e ( 11 − λ Ψ( t ) ) dt ;therefore, I ( p ) = lim λ ↑ R T d cos 2 π h p, t i w λ ( t ) dt , with w λ ( t ) := 1 − λ | Ψ( t ) | | − λ Ψ( t ) | .Taking p = 0 in the previous formula, we deduce, by Fatou’s lemma, the integrability of w on T d and with a nonnegative constant K the equality: I (0) = 1 + 2 ∞ X k =1 P ( Z k = 0) = lim λ ↑ Z T d w λ ( t ) dt = Z T d w ( t ) dt + K = c w + K, since w λ ( t ) ≥ 0, lim λ ↑ w λ ( t ) = −| Ψ( t ) | | − Ψ( t ) | = w ( t ), for t ∈ T d \{ } .By aperiodicity of W , Ψ( t ) = 1 for t ∈ U cη , where U cη is the complementary in T d of theball U η of radius η centered at 0, for η > 0. This implies sup t ∈ U cη sup λ< w λ ( t ) < + ∞ .Therefore, we get: lim λ ↑ Z U cη cos 2 π h p, t i w λ ( t ) dt = Z U cη cos 2 π h p, t i w ( t ) dt , hence: I ( p ) = Z U cη cos 2 π h p, t i w ( t ) dt + lim λ ↑ Z U η cos 2 π h p, t i w λ ( t ) dt, ∀ η > . (25)Let ε > 0. By positivity of w , we have, for η ( ε ) small enough:(1 − ε ) Z U η ( ε ) w λ ( t ) dt ≤ Z U η ( ε ) cos 2 π h p, t i w λ ( t ) dt ≤ (1 + ε ) Z U η ( ε ) w λ ( t ) dt ;hence, using (25): (1 − ε ) Z U η ( ε ) w λ ( t ) dt − Z U η ( ε ) cos 2 π h p, t i w ( t ) dt ≤ I ( p ) − Z T d cos 2 π h p, t i w ( t ) dt ≤ (1 + ε ) Z U η ( ε ) w λ ( t ) dt − Z U η ( ε ) cos 2 π h p, t i w ( t ) dt. For ε small enough, R U η ( ε ) cos 2 π h p, t i w ( t ) dt can be made arbitrary small, since w isintegrable, as well as ε sup λ< R U η w λ dt , since sup λ< R T d w λ dt < ∞ . Therefore the valueof I ( p ) − R T d cos 2 π h p, t i w ( t ) dt does not depend on p , hence is equal to K . This shows(24). The Riemann-Lebesgue Lemma implies K = lim | p |→ + ∞ P ∞ k =1 P ( Z k = ± p ). In the aperiodic case, by (24) the sums I ( p ) are the Fourier coefficients of the measure wdt + Kδ . In the general case, by replacing ℓ j by r j with ℓ j = Br j and t by B t t (cf.Subsection 2.2), we get an aperiodic r.w. and we apply the previous result. The pushforward of the measure δ by B t is the measure dγ , which shows a). (cid:3) ζ -regularity, variance. Regularity of summation sequences defined by r.w. We have V n,p = 0, for p L ( W ). If W satisfies (17), by the LLT (Theorem 2.6), for p ∈ L ( W ), with constants C i independent of p : E V n,p ∼ C n for d = 1 , E V n,p ∼ C n Log n for d = 2 , E V n,p ∼ C d n for d > . (26)It follows lim n E V n,p / E V n, = 1 , ∀ p ∈ L ( W ). Let us check (26) for d = 2. Let C be theconstant a ( W ) det(Λ) − (2 π ) − . Recall that L/D is a finite cyclic group (each ℓ j is agenerator of the cyclic group L/D , cf. Lemma 2.2).By (18), for every ε > 0, there is K ε such that: | N X n =1 P ( Z n = p ) − C N X n =1 nℓ = p mod D n |≤ C N X n =1 nℓ = p mod D | e − n h Λ − p, p i − ε n ( p ) | n ≤ C ( K ε + ε Log N ) . Since (Log N ) − P Nn =1 1 nℓ p mod D n → (Card L/D ) − we have:lim N [(Log N ) − N X n =1 P ( Z n = p )] = C lim N [(Log N ) − N X n =1 nℓ = p mod D n ] = C Card L/D . (27)From (27) and (14), we have in dimension 2, for every p ∈ L : E V N,p N Log N = 1 p =0 + P N − k =1 (1 − kN ) [ P ( Z k = p ) + P ( Z k = − p )]Log N → C Card L/D . Recall that ζ -regularity of a rws ( R n ( ℓ, ω )) n ≥ is equivalent to lim n V n,p ( ω ) V n, ( ω ) = ˆ ζ ( p ) , ∀ p ∈ Z d , for a.e. ω . Theorem 2.13. 1) If the rws is defined by a r.w. W which satisfies (17) with d ( W ) ≤ (hence centered and recurrent), then it is ζ -regular with ζ = dγ (= δ if W is strictlyaperiodic).- For d ( W ) = 1 , the normalization V n, ( ω ) depends on ω .- For d ( W ) = 2 , the normalization is C n Log n , with C = π − a ( W ) det(Λ) − . LT FOR RANDOM WALKS OF COMMUTING ENDOMORPHISMS 17 2) If the rws is defined by a transient r.w., then it is ζ -regular and the normalization is C n , with C = 1 + 2 P ∞ k =1 P ( Z k = 0) .If d = 1 , then C = c w + K and dζ ( t ) = ( c w + K ) − ( w ( t ) dt + dγ ( t )) , where c w is definedin (23) and K is the constant given by Proposition 2.12 ( K = 0 if m ( W ) is finite). If d ≥ , then C = c w and dζ ( t ) = c − w w ( t ) dt .Proof . 1) The recurrent case follows from Theorem 2.10.As V n ( ω ) / E V n → 1, we choose the constant C such that lim n E V n /Cn Log n = 1, i.e., bythe LLT: C = 2 lim n ( P n − k =1 P ( Z k = 0) / Log n ) = π − a ( W ) det(Λ) − .2) In the transient case, the Fourier coefficients of the kernel n R ωn given by (16) convergeby Lemma 2.11 to the finite sum of the series 1 p =0 + P ∞ k =1 [ P ( Z k = p ) + P ( Z k = − p )],which are the Fourier coefficients of the measure wdt + Kdγ according to Proposition2.12. (cid:3) Remarks 2.14. 1) With Condition (17), if Card( J ) = d + 1, then det(Λ) = c Q p j ,where c is an integer ≥ ℓ j ’s (cf. Lemma 2.5). In the strictlyaperiodic case, we obtain C = π − ( c Q p j ) − .2) For d = 1, the asymptotic variance w.r.t. P × µ is the same as the quenched variance.3) For a reduced r.w., ζ is a discrete measure only in the centered case when d ≤ 2, andin the non centered case when d = 1 and the r.w. is deterministic.4) As an example, let us consider a r.w. W on Z defined by P ( X k = ℓ j ) = p j , with ℓ = 1and 0 < p < 1. Suppose that W is transient. Let S be a map with Lebesgue spectrum.The random sequence ( S Z k ( ω ) ) can be viewed as a random commutative perturbation ofthe iterates of S . If ϕ f is continuous, the asymptotic variance of √ n k P n − k =0 f ( S Z k ( ω ) . ) k is R ϕ f dζ . By Theorem 2.13, the variance is = 0 if f 0. A CLT for the quenchedergodic sums can be shown when S is of hyperbolic type.2.3.2. Regularity of barycenter summations. Let W = ( Z n ) be a reduced r.w. in Z d and let ˜ W = ( ˜ Z n ) be the symmetrized r.w. If P is the barycenter operator defined as in (10) by P f = P j p j T ℓ j f , we have P n f = X ℓ ∈ Z d R n ( ℓ ) T ℓ f, with R n ( ℓ ) = P ( Z n = ℓ )and by (1) ( R n ∗ ˇ R n )( ℓ ) = P ( ˜ Z n = ℓ ). We have d ( ˜ W ) = d ( W ) − δ = d ( W ) with δ = 0 or1. The behaviour of P n is given by the LLT applied to ˜ W . When δ = 0, Γ is finite andits probability Haar measure dγ is a discrete measure. If δ = 1, dγ is the product of adiscrete measure by the uniform measure on a circle. The kernel ˜ R n is P ℓ P ( ˜ Z n = ℓ ) P ( ˜ Z n =0) e πi h ℓ, t i and we have k P n f k = Z X | E ( f ( A Z n ( . ) x )) | dµ ( x ) = X ℓ ∈ Z d P ( ˜ Z n = ℓ ) ˆ ϕ f ( ℓ ) . Proposition 2.15. The kernel ( ˜ R n ) (with a normalization factor P ℓ R n ( ℓ ) of order n d − δ ) weakly converges to the measure dγ .Proof . The characteristic function of ˜ X is | Ψ( t ) | = | E ( e πi h X ,t i ) | . The LLT given byTheorem 2.8 ( ˜ W is centered but non necessarily reduced) implies P ( ˜ Z n = ℓ ) = (2 πn ) − d 0( ˜ W )2 [ a det( ˜Λ ) − e − n h ˜Λ − ℓ, ℓ i + ε n ( ℓ )] . For the r.w. ( ˜ Z n ), the condition k ∈ D + nℓ reduces to k ∈ D , since 0 belongs to thesupport of the symmetrized distribution. Therefore, for the symmetric r.w., we obtain:lim n R | Ψ( t ) | n e − πi h p,t i dt R | Ψ( t ) | n dt = lim n P ( ˜ Z n = p ) P ( ˜ Z n = 0) → D ( p ) , ∀ p ∈ Z d . (28)Remark that when W is strictly aperiodic, one can show that (cid:0) | Ψ( t ) | n / R | Ψ( t ) | n dt (cid:1) n ≥ is an approximation of identity. This gives a direct proof of (28).Since 1 D is the Fourier transform of dγ viewed as a measure on the torus T d , the weakconvergence of ( ˜ R n ) to dγ follows from (28). (cid:3) Z d -actions by commuting endomorphisms on G Endomorphisms of a compact abelian group G . Let G be a compact abelian group with Haar measure µ . The group of characters of G is denoted by ˆ G or H and the set of non trivial characters by ˆ G ∗ or H ∗ . The Fouriercoefficients of a function f in L ( G, µ ) are c f ( χ ) := R G χ f dµ , χ ∈ ˆ G .Every surjective endomorphism B of G defines a measure preserving transformation on( G, µ ) and a dual injective endomorphism on ˆ G . For simplicity, we use the same notationfor the actions on G and on ˆ G . If f is function on G , Bf stands for f ◦ B .We consider a semigroup S of surjective commuting endomorphisms of G , for examplethe semigroup generated by commuting matrices on a torus. It will be useful to extendit to a group of automorphisms acting on a (possibly) bigger group ˜ G (a solenoidal groupwhen G is a torus). Let us recall briefly the construction. Lemma 3.1. There is a smallest compact abelian group ˜ G (connected, if G is connected)such that G is a factor of ˜ G and S is embedded in a group ˜ S of automorphisms of ˜ G .Proof . On the set { ( χ, A ) , χ ∈ H, A ∈ S} we consider the law ( χ, A ) + ( χ ′ , A ′ ) =( A ′ χ + Aχ ′ , AA ′ )). Let ˜ H be the quotient by the equivalence relation R defined by( χ, A ) R ( χ ′ , A ′ ) if and only if A ′ χ = Aχ ′ .The transitivity of the relation R follows from the injectivity of the dual action of each A ∈ S . The map χ ∈ ˆ G → ( χ, Id ) / R is injective. The elements A ∈ S act on ˜ H by( χ, B ) / R → ( Aχ, B ) / R . The equivalence classes are stable by this action. We canidentify S and its image. For A ∈ S , the automorphism ( χ, B ) / R → ( χ, AB ) / R is theinverse of ( χ, B ) / R → ( Aχ, B ) / R . LT FOR RANDOM WALKS OF COMMUTING ENDOMORPHISMS 19 Let ˜ G be the compact abelian group (with Haar measure ˜ µ ) dual of the group ˜ H endowedwith the discrete topology. The group H = ˆ G is isomorphic to a subgroup of ˜ H and G is a factor of ˜ G . We obtain an embedding of S in a group ˜ S of automorphisms of ˜ H and, by duality, in a group of automorphisms of ˜ G . If ˆ G is torsion free, then ˜ H is alsotorsion free and its dual ˜ G is a connected compact abelian group. (cid:3) If ( A , ..., A d ) are d algebraically independent automorphisms of ˜ G generating ˜ S , then˜ S = { A ℓ = A ℓ ...A ℓ d d , ℓ = ( ℓ , ..., ℓ d ) ∈ Z d } and ˜ S is isomorphic to Z d if it is torsion free.The corresponding Z d -action on ( ˜ G, ˜ µ ) is denoted A . Assumption 3.2. In what follows, we consider a set ( B j , j ∈ J ) of commuting surjectiveendomorphisms of G and the generated semigroup S . Denoting by ˜ S the extension of S toa group of automorphisms of ˜ G (if the B j ’s are invertible, then ˜ G is just G ), we assumethat ˜ S is torsion-free and non trivial, hence has a system of d ∈ [1 , + ∞ ] algebraicallyindependent generators A , ..., A d (not necessarily in S ). We suppose d finite. In otherwords we consider a set ( B j , j ∈ J ) of endomorphisms such that B j = A ℓ j , where A , ..., A d are d algebraically independent commuting automorphisms of ˜ G . We begin with some spectral results. The measure preserving Z d -action A is said to be totally ergodic if A ℓ on ( ˜ G, ˜ µ ) is ergodic for every ℓ ∈ Z d \{ } .One easily show that total ergodicity is equivalent to: A ℓ χ = χ for any non trivialcharacter χ and ℓ = 0 (free dual Z d -action on H ∗ ), to the Lebesgue spectrum propertyfor ˜ S acting on ( ˜ G, ˜ µ ), as well as to 2-mixing, i.e., lim k ℓ k→∞ µ ( C ∩ A − ℓ C ) = µ ( C ) µ ( C ),for all Borel sets C , C of G .For a semigroup S = { A ℓ , ℓ ∈ ( Z + ) d } , total ergodicity of the generated group is equiva-lent to the property (expressed on the dual of G ): A ℓ χ = A ℓ ′ χ for χ ∈ H ∗ and ℓ = ℓ ′ .Examples for G = T ρ are given in Subsection 3.2. What we call “totally ergodic” for ageneral compact abelian group G is also called “partially hyperbolic” for the torus. Spectral density ϕ f of a regular function f For f in L ( ˜ G ), we have ˆ ϕ f ( n ) = h f, A n f i = P χ ∈ ˜ H c f ( A n χ ) c f ( χ ) , n ∈ Z d . Observethat, using the projection Π from ˜ G to G , a function f on the group G can be lifted to˜ G and a character χ ∈ ˆ G viewed as a character on ˜ G via composition by Π. Putting˜ f ( x ) = f (Π x ), we have ˜ f = f ◦ Π = P χ ∈ ˆ G c f ( χ ) χ ◦ Π, so that the only non zero Fouriercoefficients of ˜ f correspond to characters on G .If f is a function defined on G , the Fourier analysis can be done for the Z d -action A inthe group ˜ G , but expressed in terms of the Fourier coefficients of f computed in G . Forthe spectral density of f , viewed as the spectral density of ˜ f for the Z d -action A on ˜ G ,one checks that ˆ ϕ f ( n ) = P χ ∈ ˆ G c f ( A n χ ) c f ( χ ) , n ∈ Z d , where c f ( A n χ ) = 0 if A n χ ˆ G .For the action by endomorphisms of compact abelian groups, the family ( ψ j ) (cf. nota-tions in Subsection 1.2) is ( χ j ). We have: a j,ℓ = h f, A ℓ χ j i = c f ( A ℓ χ j ) , γ j ( t ) = P ℓ ∈ Z d c f ( A ℓ χ j ) e πi h ℓ,t i , M t f = P j ∈ J γ j ( t ) χ j ,where J denotes a section of the action of ˜ S on ˜ H ∗ , i.e., a subset { χ j } j ∈ J ⊂ ˜ H ∗ = ˜ H \{ } such that every χ ∈ ˜ H ∗ can be written in a unique way as χ = A n ...A n d d χ j , with j ∈ J and ( n , ..., n d ) ∈ Z d .Therefore, M t f is defined for every t , if P j ∈ J (cid:0) P ℓ ∈ Z d | c f ( A ℓ χ j ) | ) < ∞ .We denote by AC ( G ) the class of real functions on G with absolutely convergent Fourierseries and µ ( f ) = 0, endowed with the norm: k f k c := P χ ∈ ˆ G | c f ( χ ) | < + ∞ . Proposition 3.3. If f is in AC ( G ) , then P ℓ ∈ Z d |h A ℓ f, f i| < ∞ and the spectral density ϕ f of f is continuous. For any subset E of ˆ G , if f ( x ) = P χ ∈E c f ( χ ) χ , then k ϕ f − f k ∞ ≤ k f − f k c . (29) Proof . (We work in ˜ G , but for simplicity we do not write ˜). Since every χ ∈ H ∗ can be written in an unique way as χ = A ℓ χ j , with j ∈ J and ℓ ∈ Z d , we have: P ℓ ∈ Z d | c f ( A ℓ χ j ) | ≤ P j ∈ J P ℓ ∈ Z d | c f ( A ℓ χ j ) | = P χ ∈ ˜ H ∗ | c f ( χ ) | = k f k c .Then, for every j ∈ J , the series defining γ j is uniformly converging, γ j is continuousand P j ∈ J k γ j k ∞ ≤ P j ∈ J P ℓ ∈ Z d | c f ( A ℓ χ j ) | = k f k c .The spectral density of f is ϕ f ( t ) = P j ∈ J | P ℓ ∈ Z d \{ } c f ( A ℓ χ j ) e πi h ℓ,t i | . The function P j ∈ J | γ j | is a continuous version of the spectral density ϕ f and k ϕ f k ∞ is bounded by X j ∈ J ( X ℓ ∈ Z d | c f ( A ℓ χ j ) | ) ≤ X j ∈ J ( X ℓ ∈ Z d | c f ( A ℓ χ j ) | )( X χ ∈ ˆ G | c f ( χ ) | ) ≤ ( X χ ∈ ˆ G | c f ( χ ) | ) = k f k c . Inequality (29) follows by replacing f by f − f . (cid:3) The torus case: G = T ρ , examples of Z d -actions. Every B in the semigroup M ∗ ( ρ, Z ) of non singular ρ × ρ matrices with coefficients in Z defines a surjective endomorphism of T ρ and a measure preserving transformation on( T ρ , µ ). It defines also a dual endomorphism of the group of characters H = c T ρ identifiedwith Z ρ (action by the transposed of B ). Since we compose commuting matrices, forsimplicity we do not write the transposition. When B is in the group GL ( ρ, Z ) of matriceswith coefficients in Z and determinant ± 1, it defines an automorphism of T ρ .For the torus, the construction in Lemma 3.1 reduces to the following. Let B j , j ∈ J ,be matrices in M ∗ ( ρ, Z ) and q j = | det( B j ) | . Suppose for simplicity J finite. Then ˜ G isthe compact group dual of the discrete group ˜ H := { Q j q ℓ j j k, k ∈ Z ρ , ℓ j ∈ Z − } , Z ρ is asubgroup of ˜ H and T ρ is a factor of ˜ G .It is well known that A ∈ M ∗ ( ρ, Z ) acts ergodically on ( T ρ , µ ) if and only if A has noeigenvalue root of unity. A Z d -action ( A ℓ , ℓ ∈ Z d ) on ( T ρ , µ ) is totally ergodic if and onlyif it is free on Z ρ \{ } , or equivalently if A ℓ has no eigenvalue root of unity if ℓ = 0. LT FOR RANDOM WALKS OF COMMUTING ENDOMORPHISMS 21 Lemma 3.4. Let M ∈ M ∗ ( ρ, Z ) be a matrix with irreducible (over Q ) characteristicpolynomial P . If { B j , j ∈ J } are d matrices in M ∗ ( ρ, Z ) commuting with M , theygenerate a commutative semigroup S of endomorphisms on T ρ which is totally ergodic ifand only if B ℓ = B ℓ ′ , for ℓ = ℓ ′ (where B ℓ := B ℓ ....B ℓ d d ) . The Z d -action extending S isthe product of a totally ergodic Z d ′ -action, with d ′ ≤ d by an action of finite order.Proof . Since P is irreducible, the eigenvalues of M are distinct. It follows that thematrices B j are simultaneously diagonalizable on C , hence are pairwise commuting.Now suppose that there are ℓ ∈ Z d \{ } and v ∈ Z ρ \ { } such that B ℓ v = v . Let E bethe subspace of R ρ generated by v and its images by M . The restriction of B ℓ to E isthe identity. E is M -invariant, the characteristic polynomial of the restriction of M to E has rational coefficients and factorizes P . By the assumption of irreducibility over Q ,this implies E = R ρ . Therefore B ℓ is the identity.Let K be the kernel of the homomorphism h : ℓ → B ℓ and ˜ h the quotient of h in Z d / K .The finitely generated group Z d / K is isomorphic to U ⊕ T , with U isomorphic to Z d ′ for d ′ ∈ [0 , d ], the restriction of ˜ h to U a totally ergodic Z d ′ -action and T a finite group. (cid:3) Examples of Z d -actions by automorphisms In general proving total ergodicity and computing explicit independent generators isdifficult. This may be easier with endomorphisms. For example, let ( q j , j ∈ J ) beintegers > x → q j x mod 1 be the corresponding endomorphisms acting on T . They generate a semigroup embedded in a group acting giving a totally ergodic Z d -action on an extension of T , where d ∈ [1 , + ∞ ] is the dimension of the vector spaceover Q generated by Log q j , j ∈ J . The construction extends to ρ × ρ matrices B j , j ∈ J ,such that | det( B j ) | > q j by | det( B j ) | , under the condition of Lemma3.4. Without irreducibility condition, the action of commuting ρ × ρ matrices B j , whenthe numbers Log | det( B j ) | are linearly independent over Q , extends to a totally ergodic Z d -action with d = Card( J ) on an extension of T ρ .On the contrary, for automorphisms of G = T ρ , it can be difficult to compute independentgenerators of the generated group for ρ > Z d -action by automorphisms on T ρ is related to the group of unitsin number fields (cf. [19]). To simplify let us consider a matrix M ∈ GL ( ρ, Z ) withan irreducible (over Q ) characteristic polynomial P . The elements of the centralizer of M in GL ( ρ, R ) are simultaneously diagonalizable. The centralizer of M in M ( ρ, Q ) canbe identified with the ring of polynomials in M with rational coefficients modulo theprincipal ideal generated by the polynomial P and hence with the field Q ( λ ), where λ isan eigenvalue of M , by the map p ( A ) → p ( λ ) with p ∈ Q [ X ].By Dirichlet’s theorem, if P has d real roots and d pairs of complex conjugate roots,there are d + d − K ( P ) associated to P . The centralizer C ( M ) of M in GL ( ρ, Z ) provides a totally ergodic Z d + d − -action by automorphisms on T ρ (up to a product by a finitecyclic group consisting of roots of unity). The computation of the number of real rootsof P gives the dimension d = d + d − Z d free action on T ρ generated by C ( M ).Nevertheless it can be difficult to compute elements with determinant ± C ( M ). Theexplicit computation of fundamental units (hence of independent generators) relies onan algorithm (see H. Cohen’s book [6]) which is in practice limited to low dimensions. Examples for T Let P ( x ) = − x + qx + n be a polynomial with coefficients in Z , irreducible over Q .Let M = n q be its companion matrix. Let λ be a root of P . If the field K ( P )is listed in a table giving the characteristics of the first cubic real fields (see [6], [34]), wefind a pair of fundamental units for the group of units in the ring of integers in K ( P ) ofthe form P ( λ ), P ( λ ), with P , P ∈ Z [ X ]. The matrices A = P ( M ) and A = P ( M )provide elements of C ( M ) giving a totally ergodic Z -action on T by automorphisms. 1) Explicit examples (from the table in [34] ) a) Let us consider the polynomial P ( x ) = − x + 12 x + 10 and its companion matrix M = . Let λ be a root of P . The table gives a pair of fundamental units: P ( λ ) = λ − λ − , P ( λ ) = − λ + λ + 11. Let A , A be the matrices A = P ( M ) = − − − − − 26 9 , A = P ( M ) = 11 1 − − − − . b) Consider now the polynomial P ( x ) = − x + 9 x + 2 and its companion matrix M ′ = . Let λ be a root of P . The table gives a pair of fundamental units for thealgebraic group associated to P : P ( λ ) = 85 λ − λ − , P ( λ ) = − λ + 4 λ + 161.Take A = P ( M ′ ) = − − 245 85170 706 − − − , A = P ( M ′ ) = 161 4 − − − − .In both cases a) and b), the matrices A and A are in GL (3 , Z ) and generate a totallyergodic actions of Z by automorphisms on T . 2) A simple example on T If P ( x ) = x + ax + bx + ax + 1, the polynomial P has two real roots: λ , λ − and twocomplex conjugate roots of modulus 1: λ , λ . LT FOR RANDOM WALKS OF COMMUTING ENDOMORPHISMS 23 Let σ j = λ j + λ j , j = 0 , 1. They are roots of Z − aZ + b − a − b + 8 > , a > , b > , a > b + 2 are satisfied (i.e., 2 < b < a − , a > 4, since 2 a − ≤ a + 2), then λ , λ − are solutions of λ − σ λ + 1 = 0,and λ , λ are solutions of λ − σ λ + 1 = 0, where σ = − a − √ a − b + 8 , σ = − a + 12 √ a − b + 8 . The polynomial P is not factorizable over Q . Indeed, suppose that P = P P with P , P with rational coefficients and degree ≥ 1. Since the roots of P are irrational, thedegrees of P and P are 2. Necessarily their roots are, say, λ , λ for P , λ , λ − for P .The sum λ + λ , root of Z − aZ + b − P arenot rational. Let us take A := − − a − b − a , B = A + I. From the irreducibilityover Q , it follows that, if there is a non zero fixed integral vector for A k B ℓ , where k, ℓ are in Z , then we have A k B ℓ = Id . This implies: λ k ( λ − ℓ = 1, hence, since we have | λ | = 1, it follows | λ + 1 | = 1, i.e. λ is also solution of z − z + 1 = 0, which is not true.An example is P ( x ) = x + 5 x + 7 x + 5 x + 1. If A is the companion matrix, then thematrices A and B = A + I generate a Z -totally ergodic action on T . 3) Construction by blocks Let M , M be two ergodic matrices respectively of dimension d and d . Let ( p i , q i ), i = 1 , 2, be two pairs of integers such that p q − p q = 0. On the torus T d + d we obtain a Z -totally ergodic action by taking A , A of the following form: A = (cid:18) M p M q (cid:19) , A = (cid:18) M p M q (cid:19) . Indeed, if there exists v = (cid:18) v v (cid:19) ∈ Z d + d \ { } invariant by A n A ℓ , then M np + ℓp v = v , M nq + ℓq v = v , which implies np + ℓp = 0, nq + ℓq = 0; hence n = ℓ = 0.This method gives explicit free Z -actions on T . The same method gives free Z -actionson T . As discuss above, it is more difficult to explicit examples of full dimension, i.e.,with 3 independent generators on T , or with 4 independent generators on T .3.3. Spectral densities and Fourier series for tori. The continuity of ϕ f for a general compact abelian group follows from the absoluteconvergence of the Fourier series of f , hence, for G = T ρ from the condition: | c f ( k ) | = O ( k k k − β ) , with β > ρ. (30)Condition (30) implies a rate of approximation of f by the partial sums of its Fourierseries, but for the torus a weaker regularity condition on f can be used. First, let usrecall a result on the approximation of functions by trigonometric polynomials. For f ∈ L ( T ρ ), the Fourier partial sums of f over squares of sides N are denoted by s N ( f ). The integral modulus of continuity of f is defined as ω ( δ, f ) = sup | τ |≤ δ,..., | τ ρ |≤ δ k f ( . + τ , · · · , . + τ ρ ) − f k L ( T ρ ) . Lemma 3.5. The following condition on the modulus of continuity ∃ α > d and C ( f ) < + ∞ such that ω ( δ, f ) ≤ C ( f ) (ln 1 δ ) − α , ∀ δ > , (31) implies, for a constant R ( f ) : k f − s N ( f ) k ≤ R ( f ) (ln N ) − α , with α > d. (32) Proof . Let K N ,...,N ρ be the ρ -dimensional Fej´er kernel, for N , ..., N ρ ≥ 1, and let J N ,...,N ρ ( t , · · · , t ρ ) = K N ,...,N ρ ( t , · · · , t ρ ) / k K N ,...,N ρ k L ( T ρ ) be the ρ -dimensional Jack-son’s kernel. Using the moment inequalities R t k J N ( t ) dt = O ( N − k ) , ∀ N ≥ , k =0 , , 2, satisfied by the 1-dimensional Jackson’s kernel, we obtain that there exists a posi-tive constant C ρ such that, for every f ∈ L ( T ρ ), k J N,...,N ∗ f − f k ≤ C ρ ω ( N , f ) , ∀ N ≥ k f − s N ( f ) k ≤ C ρ ω ( N , f ) , ∀ f ∈ L ( T ρ ) , ∀ N ≥ 1, since k f − s N ( f ) k ≤k f − P k for every trigonometric polynomial P in ρ variables of degree at most N ×· · ·× N .Hence (32) follows from (31). (cid:3) The required regularity in the next theorem is weaker than in Proposition 3.3. The proofis like that of the analogous result in [25]. It uses the lemma below due to D. Damjanovi´cand A. Katok [9], extended by M. Levin [26] to endomorphisms. Lemma 3.6. [9] If ( A n , n ∈ Z d ) is a totally ergodic Z d -action on T ρ by automorphisms,there are τ > and C > , such that for all ( n, k ) ∈ Z d × ( Z ρ \{ } ) for which A n k ∈ Z ρ . k A n k k ≥ Ce τ k n k k k k − ρ . (33) Theorem 3.7. Let ℓ → A ℓ be a totally ergodic d -dimensional action by commutingendomorphisms on T ρ . Let f be in L ( T ρ ) satisfying the regularity condition (31) ormore generally (32). Let f ( x ) := P n ∈E c n ( f ) e πi h n,x i , where E is a subset of Z ρ . Thenthere are finite constants B ( f ) , C ( f ) depending only on R ( f ) such that |h A ℓ f , f i| ≤ B ( f ) k f k k ℓ k − α , ∀ ℓ = 0 , (34) the spectral density is continuous, P ℓ ∈ Z d |h A ℓ f, f i| < ∞ and k ϕ f − f k ∞ ≤ C ( f ) k f − f k .Proof . (Recall the convention c A ℓ k ( f ) = 0 if A ℓ k Z ρ .) It suffices to prove the result for f since, by setting c f ( n ) = 0 outside E , we obtain (34) with the same constant B ( f ) asshown by the proof. Let λ, b, h be such that 1 < λ < e τ , < b < λ ρ , h := λb − ρ > τ is given by Lemma 3.6). We have for ℓ ∈ Z d : h A ℓ f, f i = X k ∈ Z ρ c k ( f ) c A ℓ k ( f ) = X k k k
From Inequality (33) of Lemma 3.6, we deduce that, if k k k < b k ℓ k , then k A ℓ k k ≥ Dλ k ℓ k k k k − ρ ≥ Dλ k ℓ k b − ρ k ℓ k = Dh k ℓ k , ℓ = 0. It follows, for the sum (1): | ( A ) | ≤ ( X k k k Dh k ℓ k | c m ( f ) | . By Parseval inequality and (32), there is a finite constant B ( f ) such that, for ℓ = 0:( X k m k >Dh k ℓ k | c m ( f ) | ) ≤ k f − s [ Dh k ℓ k ] ,..., [ Dh k ℓ | ] ( f ) k ≤ R ( f )(ln[ Dh k ℓ k ]) α ≤ B ( f ) k ℓ k − α . (36)From the previous inequalities, it follows: | ( A ) | ≤ B ( f ) k f k k ℓ k − α , ∀ k ℓ k 6 = 0. Analo-gously, we obtain | P k k k≥ b k ℓ k c k ( f ) c A ℓ k ( f ) | ≤ B ( f ) k f k k ℓ k − α , ℓ = 0 for ( B ) in (35).Taking B ( f ) = B ( f )+ B ( f ), (34) follows from (35) and implies the last statements. (cid:3) CLT for summation sequences of endomorphisms on G Criterium for the CLT on a compact abelian connected group G . For Z d -dynamical systems satisfying the K -property, a martingale-type property canbe used to obtain a CLT. (For martingale methods applied to d -dimensional randomfields, see for example [16], [35].) For Z d -action by automorphisms, the K -property isequivalent to mixing of all orders (cf. [31]) for zero-dimensional compact abelian groupsbut does not hold for instance for the model that we consider on tori. In the absence of K -property, we will use for abelian semigroups of endomorphisms of connected compactabelian groups the method of mixing of all orders applied by Leonov to a single ergodicautomorphism. Mixing actions by endomorphisms ( G connected) The proof of the CLT given by Leonov in [24] for a single ergodic endomorphism A of acompact abelian group G is based on the computation of the moments of the ergodic sums S n f when f is a trigonometric polynomial. It uses the fact that A is mixing of all orders,which follows from the K -property for the Z -action of a single ergodic automorphism([29]). For Z d -actions by automorphisms on compact abelian groups, mixing of all ordersis not always true (cf. [22], [32]), but it is satisfied for actions on connected compactabelian groups (Theorem 4.1 below) and the method of moments can be used.In 1992, W. Philip in [28] and K. Schmidt and T. Ward in [31] applied results on the num-ber of solutions of S -units equations (see ([30, 12]) to endomorphisms or automorphismsof compact abelian groups. Theorem 4.1. ( [31, Corollary 3.3] ) Every 2-mixing Z d -action by automorphisms on acompact connected abelian group G is mixing of all orders. With the notations of Lemma 3.1, if S is a totally ergodic semigroup of endomorphisms ona compact connected abelian group G , then its extension ˜ S to a group of automorphismsof ˜ G is mixing of all orders by Theorem 4.1. From now on, we consider a totally ergodic Z d -action ℓ → A ℓ by commuting automor-phisms on G (or on the extension ˜ G , but we will not write ˜ ) which is mixing of all orders(an assumption satisfied when G is connected by Theorem 4.1). Let ( R n ) n ≥ be a summation sequence on Z d . We use the notations and the results ofthe appendix (Sect. 7) on cumulants. For f ∈ L ( G ), we put σ n ( f ) := k P ℓ R n ( ℓ ) A ℓ f k and assume σ n ( f ) = 0, for n big. Lemma 4.2. For a trigonometric polynomial f with zero mean, the condition X ℓ r Y k =1 R n ( ℓ + j k ) = o ( σ rn ( f )) , ∀{ j , ..., j r } ∈ Z d , ∀ r ≥ , implies (37) σ n ( f ) − X ℓ R n ( ℓ ) A ℓ f distrib −→ n →∞ N (0 , . (38) Proof . Let ( χ k , k ∈ Λ) be a finite set of characters on G , χ the trivial character. If f = P k ∈ Λ c k ( f ) χ k , the moments of the process ( f ( A n . )) n ∈ Z d are m f ( n , ..., n r ) = Z f ( A n x ) ...f ( A n r x ) dx = X k ,...,k r ∈ Λ c k ...c k r A n χ k ...A nr χ kr = χ . For r fixed, the function ( k , ..., k r ) → m f ( k , ..., k r ) takes a finite number of values, since m f is a sum with coefficients 0 or 1 of the products c k ...c k r with k j in a finite set. Thecumulants of given order according to Identity (56) take also a finite number of values.Therefore, since mixing of all orders implies lim max i,j k ℓ i − ℓ j k→∞ s f ( ℓ , ..., ℓ r ) = 0 (cf. Notation(61)) by Lemma 7.6, there is M r such that s f ( ℓ , ..., ℓ r ) = 0 if max i,j k ℓ i − ℓ j k > M r .We apply Theorem 7.2 (cf. appendix). Let us check (60), i.e., X ( ℓ ,...,ℓ r ) ∈ ( Z d ) r c ( X ℓ , ..., X ℓ r ) R n ( ℓ ) ...R n ( ℓ r ) = o ( k Y n k r ) , ∀ r ≥ . Using (58), we obtain | X ℓ ,...,ℓ r s f ( ℓ , ..., ℓ r ) R n ( ℓ ) ...R n ( ℓ r ) | = | X max i,j k ℓ i − ℓ j k≤ M r s f ( ℓ , ℓ , ..., ℓ r ) R n ( ℓ ) ...R n ( ℓ r ) |≤ X ℓ X k j k ,..., k j r k≤ M r , j =0 | s f ( ℓ, ℓ + j , ..., ℓ + j r ) | r Y k =1 R n ( ℓ + j k )= X ℓ X k j k ,..., k j r k≤ M r , j =0 | s f ( j , j , ..., j r ) | r Y k =1 R n ( ℓ + j k ) . LT FOR RANDOM WALKS OF COMMUTING ENDOMORPHISMS 27 The right hand side is less than C P ℓ P k j k ,..., k j r k≤ M r , j =0 Q rk =1 R n ( ℓ + j k ). Therefore(37) implies (60). (cid:3) Theorem 4.3. Let ( R n ) n ≥ be a summation sequence on Z d which is ζ -regular (cf.Definition 1.2). Let f be a function in AC ( G ) with spectral density ϕ f . The condition (cid:0) sup ℓ R n ( ℓ ) (cid:1) r − X ℓ R n ( ℓ ) = o (( X ℓ ∈ Z d R n ( ℓ ) ) r/ ) , for every r ≥ , (39) implies (with the convention that the limiting distribution is δ if σ ( f ) = 0 ) ( X ℓ ∈ Z d R n ( ℓ ) ) − X ℓ ∈ Z d R n ( ℓ ) f ( A ℓ . ) distr −→ n →∞ N (0 , ζ ( ϕ f )) . (40) Proof . We use (29) and the ζ -regularity of ( R n ): for g in AC ( G ),( X ℓ ∈ Z d R n ( ℓ ) ) − k X ℓ ∈ Z d R n ( ℓ ) A ℓ g k = Z T d ˜ R n ϕ g dt → n →∞ ζ ( ϕ g ) . Let ( E s ) s ≥ be an increasing sequence of finite sets in ˆ G with union ˆ G \ { } and let f s ( x ) := P χ ∈E s c f ( χ ) χ be the trigonometric polynomial obtained by restriction of theFourier series of f to E s . Let us consider the processes defined respectively by U sn := ( X ℓ ∈ Z d R n ( ℓ ) ) − X ℓ ∈ Z d R n ( ℓ ) f s ( A ℓ . ) , U n := ( X ℓ ∈ Z d R n ( ℓ ) ) − X ℓ ∈ Z d R n ( ℓ ) f ( A ℓ . ) . We can suppose ζ ( ϕ f ) > 0, since otherwise the limiting distribution is δ . By Proposition3.3 we have ζ ( ϕ f − f s ) ≤ k f − f s k c . It follows ζ ( ϕ f s ) = 0 for s big enough. We can applyLemma 4.2 to the trigonometric polynomials f s , since σ n ( f s ) ∼ ( P ℓ ∈ Z d R n ( ℓ ) ) ζ ( ϕ f s )with ζ ( ϕ f s ) > U sn distr −→ n →∞ N (0 , ζ ( ϕ f s )) for every s . Moreover, sincelim n Z | U sn − U n | dµ = lim n Z T d ˜ R n ϕ f − f s dt = ζ ( ϕ f − f s ) ≤ k f − f s k c , we have lim sup n µ [ | U sn − U n | > ε ] ≤ ε − lim sup n R | U sn − U n | dµ → s →∞ ε > s lim sup n µ [ | U sn − U n | > ε ] = 0, ∀ ε > 0, is satisfied and theconclusion U n distr −→ n →∞ N (0 , ζ ( ϕ f )) follows from Theorem 3.2 in [2]. (cid:3) With the notations of Sect. 1, Theorem 4.3 implies for sequential summations: Corollary 4.4. If ( x n ) is a sequence in Z d such that z n = P n − k =0 x k is ζ -regular, thenthe convergence v − n P n − k =0 f ( A z k . ) distr −→ n →∞ N (0 , ζ ( ϕ f )) follows from the condition n (cid:0) sup ℓ n − X k =0 z k = ℓ (cid:1) r − = o ( v r/ n ) , ∀ r ≥ . (41)Before considering random walks, let us apply Theorem 4.3 to summation over sets: Corollary 4.5. Let ( D n ) n ≥ be a Følner sequence of sets in N d and let f be in AC ( G ) .We have σ ( f ) = lim n k P ℓ ∈ D n A ℓ f k / | D n | = ϕ f (0) and | D n | − X ℓ ∈ D n A ℓ f ( . ) distr −→ n →∞ N (0 , σ ( f )) . Proof . The sequence R n ( ℓ ) = 1 D n ( ℓ ) is ζ -regular, with ζ = δ . Suppose that ϕ f (0) = 0.We have σ n ( f ) ∼ | D n | ϕ f (0) and R n ( ℓ + j k ) = 0 or 1. Therefore Condition (39) holdsand the result follows from Theorem 4.3. (cid:3) Remarks 4.6. 1) The previous result is valid for the rotated sums: for f in AC ( G ),for every θ , σ θ ( f ) = ϕ f ( θ ) , | D n | − X ℓ ∈ D n e πi h ℓ,θ i f ( A ℓ . ) distr −→ n →∞ N (0 , σ θ ( f )) . (42)2) When G = T d , in view of Theorem 3.7, the conclusions of Theorem 4.3 and Corol-lary 4.5 are valid under the weaker regularity assumption (32), in particular under alogarithmic H¨olderian regularity (condition (31)).3) As mentioned in the introduction, the result of Corollary 4.5 for the sums over d -dimensional rectangles and regular functions was obtained by M. Levin ([26]). A CLT for the rotated sums for a.e. θ without regularity assumption When ( D n ) is a sequence of d -dimensional cubes in Z d , the following CLT for the rotatedsums holds for a.e. θ without regularity assumptions on f . Theorem 4.7. Let ( D n ) n ≥ be a sequence of cubes in Z d . For f in L ( G ) , we have fora.e. θ ∈ T d : σ θ ( f ) = ϕ f ( θ ) and | D n | − P ℓ ∈ D n e πi h ℓ,θ i A ℓ f ( . ) distr −→ n →∞ N (0 , σ θ ( f )) .Proof . As in [4], we use the relation lim n | D n | − k P ℓ ∈ D n e πi h ℓ,θ i T ℓ ( f − M θ f ) k = 0,which is satisfied, for any f ∈ L ( G ), for θ in a set of full measure (depending on f ). (cid:3) Application to r.w. of commuting endomorphisms on G Now we apply the previous sections to random walks of commuting endomorphisms orautomorphisms on a compact abelian group G .Let us consider a family ( B j , j ∈ J ) of commuting endomorphisms of G (extended ifnecessary to automorphisms of ˜ G ) and a probability vector ν = ( p j , j ∈ J ) such that p j > , ∀ j . These data define a random walk U n := Y ...Y n − where ( Y k ) k ≥ are i.i.d. r.v.with common distribution P ( Y k = B j ) = p j .If the B j ’s are given a priori, we can try to express U n via a r.w. on Z d for some d . Thismeans that we have to find algebraically independent generators A , ..., A d for some d ,in order to express the B j ’s as B j = A ℓ j , for ℓ j ∈ Z d . Another approach is to start froma totally ergodic Z d -action A on G (or ˜ G ) and from a r.w. W on Z d and to transfer W to a r.w. on the group of automorphisms of ˜ G , thus getting the B j ’s a posteriori. LT FOR RANDOM WALKS OF COMMUTING ENDOMORPHISMS 29 For example, when G is a torus, we can use the method described in Subsection 3.2,which gives an explicit construction of algebraically independent generators of a totallyergodic Z d -action by invertible matrices on a torus.With (Ω , P ) = (( Z d ) Z , ν ⊗ Z ) and ( X n ) the sequence of coordinates maps of Ω, we obtainon (Ω × ˜ G, P × ˜ µ ) a dynamical system ( ω, x ) → ( τ ω, A X ( ω ) x ). The iterates are ( ω, x ) → ( τ n ω, A X ( ω )+ ... + X n − ( ω ) x ) = ( τ n ω, A Z n ( ω ) x ), n ≥ U n = Y ...Y n − can be expressed as ( A Z n ), where ( Z n ) is the r.w. in Z d (which can be and will be assumed reduced) with distribution P ( X = ℓ j ) = p j .5.1. Random walks and quenched CLT. Recall that the measure dγ is defined in Definition 2.3, w ( t ) = −| Ψ( t ) | | − Ψ( t ) | and V n ( ω ) = { ≤ k ′ , k < n : Z k ( ω ) = Z k ′ ( ω ) } . Theorem 5.1. Let W be a reduced centered r.w. Let ℓ → A ℓ be a totally ergodic Z d -action by automorphisms on G . Let f be in AC ( G ) with spectral density ϕ f .I) Suppose W with a finite moment of order 2 and centered.a) If d = 1 , then, for a.e. ω , V n ( ω ) − n − X k =0 A Z k ( ω ) f ( . ) distr −→ n →∞ N (0 , γ ( ϕ f )) .b) If d = 2 , then, for a.e. ω , ( Cn Log n ) − n − X k =0 A Z k ( ω ) f ( . ) distr −→ n →∞ N (0 , γ ( ϕ f )) , with C = π − a ( W ) det(Λ) − .II) If W is transient with finite moment of order η for some positive η , then, for a.e. ω , ( Cn ) − n − X k =0 A Z k ( ω ) f ( . ) distr −→ n →∞ N (0 , ζ ( ϕ f )) . We have for ζ the following cases:if d = 1 , then C = c w + K and dζ ( t ) = ( c w + K ) − ( w ( t ) dt + dγ ( t )) , (cf. notations ofTheorem 2.13) ( K = 0 if m ( W ) is finite);if d ≥ , then C = c w and dζ ( t ) = c − w w ( t ) dt (the absolutely continuous part is nontrivial if and only if the random walk is non deterministic).Proof . Theorem 2.13 gives the ζ -regularity for the r.w. summation ( R n ( ω, ℓ )) n ≥ =( P n − k =0 Z k ( ω )= ℓ ) n ≥ and the expression of ζ . This is the first step (variance). It remainsto check Condition (41) of Corollary 4.4, which reads here: n (cid:0) sup ℓ n − X k =0 Z k ( ω )= ℓ (cid:1) r − = o ( V n ( ω ) r/ ) , for every r ≥ . (43)a) For the recurrent 1-dimensional case, since P ℓ R n ( ω, ℓ ) ≥ Cn / LogLog n for a.e. ω by (47), to have (43) it suffices that n (sup ℓ R n ( ω, ℓ )) r − = o (cid:0) ( n LogLog n ) r/ (cid:1) , ∀ r ≥ 3, i.e.,sup ℓ R n ( ω, ℓ ) = o ( n r − r − (LogLog n ) − r r − ) , ∀ r ≥ . The condition is satisfied, since the exponent r − r − is bigger than and sup ℓ R n ( ℓ ) ≤ n + ε .b) For the recurrent 2-dimensional case, for a.e. P ℓ R n ( ω, ℓ ) ∼ E P ℓ R n ( ., ℓ ) ∼ Cn Log n .We need: n (cid:0) sup ℓ R n ( ℓ ) (cid:1) r − = o (( n Log n ) r/ ) , ∀ r ≥ 3, i.e.,sup ℓ R n ( ℓ ) = o ( n r − r − (Log n ) r r − ) , ∀ r ≥ . The above condition is satisfied, since the exponent n r − r − increases from to when r varies from 3 to + ∞ and by Lemma 2.9 sup ℓ R n ( ℓ ) = o ( n ε ) , ∀ ε > > f ≡ (cid:3) Remark 5.2. If (Φ n ( ω, x )) n ≥ is a process depending on two variables x and ω suchthat, for a normalization by σ n independent of ω , R e itσ − n Φ n ( ω,x ) dµ ( x ) → e − t / , for a.e. ω , then the CLT holds for Φ n w.r.t. dµ × d P holds, since by the dominated convergencetheorem: R Ω (cid:0)R X e itσ − n Φ n ( ω,x ) dµ ( x )) d P ( ω ) → e − t / .It follows that, for a r.w. W which is transient or with finite variance, centering and d ( W ) = 2, the annealed version of the CLT in Theorem 5.1.b is satisfied. This resultcan be viewed as a “toral” version of Bolthausen theorem in [3].For W with finite variance, centering and d ( W ) = 1, the theorem of Kesten and Spitzerin [20] for a r.w. in random scenery gives an (annealed) convergence toward a distributionwhich is not the normal law. For the quenched process in their model or in the toralmodel of Theorem 5.1, convergence toward a normal law holds, but with a normalizationdepending on ω . Let us mention that, in the toral model, the annealed theorem analogousto the result of [20] for the recurrent 1-dimensional r.w. holds (S. Le Borgne, personalcommunication). Let us mention other results of quenched type like for instance in [17]. An example Let us give an explicit example on T . Consider the centered random walk on Z withdistribution ν supported on ℓ = (2 , ℓ = (1 , − ℓ = ( − , P ( X = ℓ j ) = , for j = 1 , , 3. With the notation of Subsection 2.2, here D is the sublatticegenerated by { (1 , , (4 , − } and we have D ⊥ = { } and a = 15.Let A and A be the commuting matrices computed in Subsection 3.2 (example 1.a).Let the matrices B j , j = 1 , , , be defined by B = A A = 29 23 − − − 67 23230 196 − ,B = A A − = − − 11 440 35 − − − 92 35 , B = A − A = 107 16 − − 70 23 16160 122 23 .The random walk on T such that with equal probability we move from x ∈ T to B j x , j = 1 , , 3, gives rise to a random process( U n ( ω ) x ) n ≥ on the torus such that for a regular LT FOR RANDOM WALKS OF COMMUTING ENDOMORPHISMS 31 function f the limiting distribution of (( n Log n ) − P n − k =0 f ( V k ( ω ) . )) is a normal law, withvariance σ f (0) = c ϕ f (0), where the constant c is given by the LLT.5.2. Powers of barycenter operators. We show now that the iterates of the barycenter operators satisfy the condition of The-orem 4.3. Let ( B j , j ∈ J ) be a set of endomorphisms satisfying Assumption 3.2, i.e.,such that B j = A ℓ j , ℓ j ∈ Z d , where A , ..., A d are d algebraically independent commutingautomorphisms of ˜ G . We suppose that the corresponding Z d -action A on ( ˜ G, ˜ µ ) is totallyergodic.Let P be the barycenter operator P f ( x ) := P j ∈ J p j f ( B j x ). The associated randomwalk ( Z n ) on Z d is defined by P ( X = ℓ j ) = p j , for j ∈ J . The lattice L ( ˜ W ) generatedby the support of the distribution of ˜ W coincides with D ( W ). The lattices D ( W )and D ( ˜ W ) are the same. We use the LLT (Theorem 2.8) for ˜ W (with the exponent d ( ˜ W ) = d ( W ) = d ( W ) or d ( W ) − ).We suppose that D ( W ) is not trivial, so that W is not deterministic and d = d ( W ) ≥ dγ was defined in Notation 2.3 (see also 2.3.2). Theorem 5.3. Let f be a function in AC ( G ) with spectral density ϕ f . Then, for aconstant C depending on the random walk, we have w.r.t. the Haar measure on G : C n d P n f distr −→ n →∞ N (0 , σ P ( f )) , with σ P ( f ) = Z ϕ f dγ . In particular, if B j = A j , j = 1 , ..., d , then (4 π ) d ( p ...p d ) n d P n f distr −→ n →∞ N (0 , σ P ( f )) , with σ P ( f ) = Z T ϕ f ( u, u, ..., u ) du. Proof . We apply Proposition 2.15. Here R n ( ℓ ) = P ( Z n = ℓ ), P ℓ ∈ Z d R n ( ℓ ) = 1 and byTheorem 2.6, sup ℓ R n ( ℓ ) = O ( n − d / ). If σ P ( f ) = 0, the variance σ n ( f ) is asymptoticallylike P ℓ ∈ Z d R n ( ℓ ) = P ( ˜ Z n = 0) ∼ C n − d / , for a constant C > (cid:0) sup ℓ R n ( ℓ ) (cid:1) r − = o ( n − rd / ) , ∀ r ≥ 3. It is satisfied, since n − ( r − d / = o ( n − rd / ) , ∀ r ≥ 3. We conclude by Theorem 4.3. (cid:3) Examples and remarks. 1) Let A , A be two commuting matrices with coefficients in M ( ρ, Z ) generating atotally ergodic Z -action on T ρ , ρ ≥ 3. Let P be the barycenter operator P f ( x ) := p f ( A x ) + p f ( A x ), p , p > , p + p = 1.The symmetrized r.w. ( ˜ Z n ) is 1-dimensional and the support of the distribution of˜ X is { (0 , , (1 , − , ( − , } . If ϕ f is continuous, then lim n →∞ √ p p πn k P n f k = R T ϕ f ( u, u ) du . If f satisfies the regularity condition (31) on T ρ , we have, with σ P ( f ) = R T ϕ f ( u, u ) du , (4 p p π ) n P n f distr −→ n →∞ N (0 , σ P ( f )). If f is in AC ( G ), then σ P ( f ) = 0 if and only if ϕ f ( u, u ) = 0, for every u ∈ T . Inparticular, if f is not a mixed coboundary (cf. Theorem 5.4), then σ P ( f ) = 0 and therate of convergence of k P n f k to 0 is the polynomial rate given by Theorem 5.3.2) Suppose now that A , A are automorphisms. We can take, for examples the matricescomputed in Subsection 3.2. Let P be defined by P f ( x ) := p f ( A x ) + p f ( A x ) + p f ( A − x ) + p f ( A − x ), p j > P j p j = 1.We have analogous result, excepted that here L ( ˜ W ) = D ( ˜ W ) = D ( W ) = Z (1 , ⊕ Z ( − , dγ is the barycenter of δ (0 , and δ ( , ) .3) Observe that the previous example for the quenched limit theorem used for its com-putation the tables of units in a field number. As remarked in Sect. 3, for barycenters,it is easier to give examples which are endomorphisms. The maps on the torus are notnecessarily invertible, but the analysis of the process uses a symmetrized r.w. on Z d .The difficulty is then to compute the dimension of the associated r.w. unless it is givenby primality conditions of the determinants. Let us give a simple example.Let P on L ( T ) be defined by: ( P f )( x ) = f (2 x ) + f (3 x ) + f (5 x ) + f (6 x ) + f (15 x ). The behavior of the iterates P n is given by the LLT applied to the sym-metrized r.w. ( ˜ Z n ) (strictly aperiodic in Z ) with distribution supported on ˜Σ = { (0 , , , ± (1 , , , ± (0 , , , ± (0 , , , ± (1 , − , , ± (1 , , − , ± (0 , , − , ± (1 , , − , ± (1 , − , − } . If f is regular on T , the process ( n − / P n f ( . )) n ≥ converges in distri-bution to a normal law (non degenerate f is not a mixed coboundary).4) For ν a discrete measure on the semigroup T of commuting endomorphisms of G , letus consider a barycenter of the form P f ( x ) = P T ∈T ν ( T ) f ( T x ). When there is a finitemoment of order 2 and d ( W ) < + ∞ , the decay of P n f is of order n − d ( W )4 when ϕ f iscontinuous and ϕ f (0) = 0. A question is to estimate the decay when ν has an infinitesupport and d ( W ) is infinite and to study the asymptotic distribution of the normalizediterates (if there is a normalization).For example if P f ( x ) = P q ∈P ν ( q ) f ( qx ), where P is the set of prime numbers and( ν ( q ) , q ∈ P ) a probability vector with ν ( q ) > q , what is the decay to0 of k P n f k , when f is H¨olderian on the circle?When d ( W ) is infinite, a partial result is that the decay is faster than Cn − r , for every r ≥ 1. This follows from the following observation:Let P and P be commuting contractions of L ( G ) such that k P n f k ≤ M n − r . Let α ∈ ]0 , , β = 1 − α . Then we have: k ( αP + βP ) n f ) k ≤ P nk =0 (cid:0) nk (cid:1) α k β n − k k P k f k . There is c < P k ≤ n α (cid:0) nk (cid:1) α k β n − k ≤ c n ; therefore: P nk =0 (cid:0) nk (cid:1) α k β n − k k P k f k ≤ M ( n α ) − r + c n ≤ M ′ n − r .5) The case of commutative or amenable actions strongly differs from the case of nonamenable actions for which a “spectral gap property” is available in certain cases ([15]),which implies a quenched CLT theorem (cf. ([7]). LT FOR RANDOM WALKS OF COMMUTING ENDOMORPHISMS 33 For actions by algebraic non commuting automorphisms B j , j = 1 , ..., d , on the torus,the existence of a spectral gap for P of the form P f ( x ) := P j ∈ J p j f ( B j x ) is related tothe fact that the generated group has no factor torus on which it is virtually abelian (cf.[1]). In general, a question is to split the action into a (virtually) abelian action on the L -space of a factor torus and a supplementary subspace with a spectral gap, in orderto get a full description of the quenched and the barycenter processes. Coboundary characterization For a Z d -action S with Lebesgue spectrum by automorphisms on T ρ and f on T ρ , let usgive a characterization for ϕ f (0) = 0 in terms of coboundaries.For the dual action of Z d on Z ρ , we construct the section J introduced before Propo-sition 3.3 in the following way. For a fixed ℓ , the set { A k ℓ, k ∈ Z d } is discrete andlim k k k→∞ k A k ℓ k = + ∞ . Therefore we can choose an element j in each class modulo theaction of S on Z ρ which achieves the minimum of the norm. By this choice, we have k j k ≤ k A k j k , ∀ j ∈ J , k ∈ Z d . (44) Theorem 5.4. If | c f ( k ) | = O ( k k k − β ) , with β > ρ , then ϕ f (0) = 0 if and only if f satisfies the following mixed coboundary condition: there are continuous functions u i , ≤ i ≤ d such that f = d X i =1 ( I − A i ) u i . (45) Proof . Let ε ∈ ]0 , β − ρ [. If δ = ( β − ρ − ε ) / ( β (1 + ρ )), we have δβρ − β (1 − δ ) = − ( ρ + ε ).There is a constant C such that k k k d e − δβτ k k k ≤ C , ∀ k ∈ Z d . According to (33), wehave | c f ( A k j ) | ≤ C k A k j k β ≤ Ce − βτ k k k k j k βρ ; hence e δβτ k k k | c f ( A k j ) | δ ≤ C k j k δβρ . (46)For every ℓ ∈ Z ρ \{ } , there is a unique ( k, j ) ∈ Z d × J such that A k j = ℓ ; hence byInequality (44): X j ∈ J X k ∈ Z d k k k d | c f ( A k j ) | = X j ∈ J X k ∈ Z d k k k d | c f ( A k j ) | δ | c f ( A k j ) | − δ ≤ C X j ∈ J X k ∈ Z d e δβτ k k k | c f ( A k j ) | δ | c f ( A k j ) | − δ ≤ C X j ∈ J X k ∈ Z d k j k δβρ | c f ( A k j ) | − δ . According to (46) and (44), the right hand side is less than C X j ∈ J X k ∈ Z d k A k j k δβρ | c f ( A k j ) | − δ = C X ℓ ∈ Z ρ \{ } k ℓ k δβρ | c f ( ℓ ) | − δ ≤ C X ℓ ∈ Z ρ \{ } k ℓ k δβρ − β (1 − δ ) ≤ C X ℓ ∈ Z ρ \{ } k ℓ k − ( ρ + ε ) < + ∞ . The sufficient condition for (45) given in [5] reads here: P j ∈ J P k ∈ Z d (1+ k k k d ) | c f ( A k j ) | < ∞ . This condition holds by the previous inequality. Since here the functions involved in the proof of the coboundary characterization are characters, hence continuous anduniformly bounded, the functions u i in (45) are continuous. (cid:3) Appendix I: self-intersections of a centered r.w. In this appendix, we prove Theorem 2.10. As already noticed, we can assume aperiodicityin the proofs. d=1: a.s. convergence of V n,p /V n, . We need the following lemmas. Lemma 6.1. If W is a 1-dimensional r.w. with finite variance and centered, then, fora.e. ω , there is C ( ω ) > such that V n ( ω ) ≥ C ( ω ) n (LogLog n ) − . (47) Proof . By the law of iterated logarithm, there is a constant c > ω ,the inequality | Z n ( ω ) | > c ( n LogLog n ) is satisfied only for finitely many values of n .This implies that, for a.e. ω , there is N ( ω ) such that | Z n ( ω ) | ≤ ( c n LogLog n ) , for n ≥ N ( ω ). It follows, for N ( ω ) ≤ k < n , | Z k ( ω ) | ≤ ( c k LogLog k ) ≤ ( c n LogLog n ) .Therefore, if R n ( ω ) is the set of points visited by the random walk up to time n , we haveCard( R n ( ω )) ≤ c ( ω ) n LogLog n ) , with an a.e. finite constant c ( ω ).We have n = P ℓ ∈ Z P n − k =0 Z k ( ω ) = ℓ ≤ ( P ℓ ∈R n ( ω ) ( P n − k =0 Z k ( ω ) = ℓ ) ) Card( R n ( ω )) byCauchy-Schwarz inequality; hence: V n ( ω ) ≥ n / Card( R n ( ω )), which implies (47). (cid:3) Lemma 6.2. For an aperiodic r.w. in dimension 1 with finite variance and centered,we have: sup n | n X j =1 [2 P ( Z j = 0) − P ( Z j = p ) − P ( Z j = − p )] | < + ∞ , ∀ p ∈ L ( W ) . (48) Proof . Since 1 − Ψ( t ) vanishes on the torus only at t = 0 with an order 2, we have: | N − X j =1 [2 P ( Z j = 0) − P ( Z j = p ) − P ( Z j = − p )] | = 4 | Z T sin π h p, t i ℜ e ( 1 − Ψ N ( t )1 − Ψ( t ) ) dt |≤ Z T sin π h p, t i | − Ψ N ( t )1 − Ψ( t ) | dt ≤ Z T sin π h p, t i| − Ψ( t ) | dt < + ∞ . (cid:3) Proof of Theorem 2.10 (for d = 1: lim n V n,p ( ω ) V n ( ω ) = 1, a.e. for every p ∈ L ( W ).)Recall that V n ( ω ) − V n,p ( ω ) ≥ n − E [ V n − V n,p ] by1 + n − n − X k =1 | n − k − X j =0 ( P ( Z j = p ) − P ( Z j = 0)) + n − k − X j =0 ( P ( Z j = − p ) − P ( Z j = 0)) | LT FOR RANDOM WALKS OF COMMUTING ENDOMORPHISMS 35 which is bounded according to (48).Let δ > 0. The bound E ( V n ( ω ) − V n,p n ) = O ( n − ) implies by the Borel-Cantelli lemmathat ( V n ( ω ) − V n,p n ) tends to 0 a.e. along the sequence k n = [ n δ ] , n ≥ V n ≥ c ( ω ) n (LogLog n ) − by (47), we obtain0 ≤ − V n,p ( ω ) V n ( ω ) ≤ V n ( ω ) − V n,p ( ω ) V n ( ω ) < V n ( ω ) − V n,p ( ω ) c ( ω ) n (LogLog n ) − . Therefore ( V n,p ( ω ) V n ( ω ) ) n ≥ converges to 1 a.e. along the sequence ( k n ).To complete the proof, it suffices to prove that a.s. lim n max k n ≤ j 0, there is an a.e. finite constant c ε ( ω ) such thatsup ℓ ∈ Z R n ( ω, ℓ ) = sup ℓ ∈ Z P n − k =0 Z k ( ω ) = ℓ = c ε ( ω ) n + ε . This implies V k n +1 − V k n = k n +1 X ℓ,j =0 { Z ℓ = Z j } − k n X ℓ,j =0 { Z ℓ = Z j } ≤ k n +1 X ℓ = k n +1 k n +1 X j =1 { Z ℓ = Z j } + k n +1 X j = k n +1 k n +1 X ℓ =1 { Z ℓ = Z j } ≤ k n +1 − k n ) sup p ∈ Z k n +1 X j =1 { Z j = p } ≤ c ε ( ω ) ( k n +1 − k n ) k + εn +1 ≤ K ( ω ) n δ +(2+ δ )( + ε ) . Therefore, V k n +1 − V k n ≤ K ( ω ) n δ +2 ε + εδ and V kn +1 − V kn V kn is a.s. bounded by V k n +1 − V k n k / n (LogLog k n ) − ≤ K ( ω ) n δ +2 ε + εδ ( n δ ) / (LogLog ( n δ )) − ≤ K ( ω ) n δ +2 ε + εδ n − (3+ δ ) ((LogLog ( n δ )) = 2 K ( ω ) n − ε + εδ (LogLog ( n δ )) , which tends to 0, if 2 ε + εδ < (cid:3) d=2: variance and SLLN for V n,p . For d = 2, in the centered case with finite variance, the a.s. convergence V n,p /V n, → p ∈ L ( W ) follows from the strong law of large numbers (SLLN): lim n V n,p / E V n,p = 1,a.s. We adapt the method of [27] to the case p = 0 in the estimation of Var( V n,p ). Seealso [10] for the computation of the variance of V n, . We need two auxiliary results. Lemma 6.3. There is C such that, if p, q are in L and n, k are such that nℓ = p mod D and ( n + k ) ℓ = q mod D , then | P ( Z n + k = q ) − P ( Z n = p ) | ≤ C ( 1( n + k ) + kn ( n + k ) ) , ∀ n, k ≥ . (49) Proof . We have P ( Z n + r = q ) − P ( Z n = p ) = R T G n,r ( t ) dt , with G n,r ( t ) := ℜ e [ e − πi h q,t i Ψ( t ) n + r − e − πi h p,t i Ψ( t ) n ] . The functions e − πi h q,t i Ψ( t ) n + r and e − πi h p,t i Ψ( t ) n are invariant by translation by theelements of Γ and have a modulus < 1, except for t ∈ Γ (cf. Lemma 2.4). To boundthe integral of G n,r , it suffices to bound its integral I n := R U G n,r ( t ) , dt restricted to afundamental domain U of Γ acting on T .Denote by B ( η ) the ball with a small radius η and center 0 in T . If η > U \ B ( η ), we have | Ψ( t ) | ≤ λ ( η ) with λ ( η ) < 1, which implies: | I n | ≤ Cλ n ( η ) + Z B ( η ) | G n,r ( t ) | dt. and we have | G n,r ( t ) | := |ℜ e [ e − πi h q,t i Ψ( t ) n + r − e − πi h p,t i Ψ( t ) n ] | ≤ | Ψ( t ) | n | Ψ( t ) r − e πi h q − p,t i | . Since the distribution ν of the centered r.w. W is assumed to have a moment of order2, by Lemma 2.5, for η sufficiently small, there are constants a, b > | Ψ( t ) | < − a k t k , | − Ψ( t ) | < b k t k , ∀ t ∈ B ( η ) . We distinguish two cases:if p = q , then | G n,r ( t ) | ≤ C ( r )(1 − a k t k ) n k t k ;if p = q , | G n,r ( t ) | ≤ C ( r ) | Ψ( t ) | n k t k ≤ C ( r )(1 − a k t k ) n k t k .Now we bound the integral R B ( η ) (1 − a k t k ) n k t k dt dt . By the change of variable( t , t ) → ( s , √ n , s , √ n ), then ( s , s ) → ( ρ cos θ, ρ sin θ ), it becomes successively:1 n Z B ( η ) (1 − a k s k n ) n k s k ds ds ≤ Cn Z R e − aρ ρ dρ = O ( 1 n ) . So for p = q , we get the bound O ( n ). Likewise, if p = q , we get the bound O ( n / ).Recall the L/D is a cyclic group (Lemma 2.2). If n, k satisfy the condition of the lemma,we write: k = r + uv , where r and v (= | L/D | ) are the smallest positive integers suchthat respectively: rℓ = q − p mod D , vℓ = 0 mod D . Then, using the previous boundsand writing the difference P ( Z n + k = q ) − P ( Z n = p ) as P ( Z n + k = q ) − P ( Z n + r +( u − v = q ) + P ( Z n + r +( u − v = q ) − P ( Z n + r +( u − v = q )+ ... + P ( Z n + r + v = q ) − P ( Z n + r = q ) + P ( Z n + r = q ) − P ( Z n = p ) , the telescoping argument gives (49). (cid:3) LT FOR RANDOM WALKS OF COMMUTING ENDOMORPHISMS 37 Lemma 6.4. For d ≥ , let M ( d ) n := { ( m , ..., m d ) ∈ N d : P i m i = n } . Let ˜ M (5) n := { ( m , ..., m ) ∈ M (5) n , m , m > } . If | λ | , | α | , | β | , | γ | < , then ∞ X n =0 λ n X ( m ,...,m ) ∈ ˜ M (5) n α m β m γ m = 1(1 − λ ) λ βγ (1 − λα )(1 − λβ )(1 − λγ ) . (50) Proof . We have for λ, α , ..., α d such that | λα | , ..., | λα d | ∈ [0 , ∞ X n =0 λ n X ( m ,...,m d ) ∈ M ( d ) n α m α m ... α m d d = d Y i =1 − λα i ) . (51)Indeed, the left hand of (51) is the sum over Z of the discrete convolution product of thefunctions G i defined on Z by G i ( k ) = 1 [0 , ∞ [ ( k )( λα i ) k , hence is equal to X k ∈ Z ( G ∗ ... ∗ G d )( k ) = d Y i =1 (cid:0) X k ∈ Z G i ( k ) (cid:1) = d Y i =1 − λα i ) . For d = 5 we find ∞ X n =0 λ n X ( m ,...,m ) ∈ M (5) n α m β m γ m = 1(1 − λ ) − λα )(1 − λβ )(1 − λγ ) . The left hand of (50) reads ∞ X n =0 λ n X ( m ,...,m ) ∈ M (5) n α m β m γ m − ∞ X n =0 λ n X ( m ,m ,m ,m ) ∈ M (4) n α m γ m − ∞ X n =0 λ n X ( m ,m ,m ,m ) ∈ M (4) n α m β m + ∞ X n =0 λ n X ( m ,m ,m ) ∈ M (3) n α m = 1(1 − λ ) [ 1(1 − λα )(1 − λβ )(1 − λγ ) − − λα )(1 − λγ ) − − λα )(1 − λβ ) + 1(1 − λα ) ] = 1(1 − λ ) λ βγ (1 − λα )(1 − λβ )(1 − λγ ) . (cid:3) Proof of Theorem 2.10 (Case d = 2)A) Below we consider: X ≤ i 0; hence: P ( Z j − Z i = r, Z j − Z i = s ) = P ( Z m + m = r, τ m Z m + m = s )= X ℓ [ P ( Z m = ℓ ) P ( Z m = r − ℓ ) P ( Z m = s − r + ℓ )]= Z e − πi ( h r,u i + h s − r,v i ) X ℓ e − πi h ℓ, t − u + v i Ψ( t ) m Ψ( u ) m Ψ( v ) m dt du dv. (53)The last equation can be shown by approximating the probability vector ( p j ) j ∈ J byprobability vectors with finite support and using the continuity of Ψ. ( i < i < j ≤ j ) Setting m = i , m = i − i , m = j − i , m = j − j , m = n − j , we have: Z j − Z i = τ m Z m + m + m , Z j − Z i = τ m + m Z m , with m , m ≥ , m , m > P ( Z m + τ m + m Z m = q ) = P ℓ P ( Z m = ℓ, τ m + m Z m = q − ℓ ) = P ℓ P ( Z m = ℓ ) P ( τ m + m Z m = q − ℓ ) = P ℓ P ( Z m = ℓ ) P ( τ m Z m = q − ℓ ) = P ( Z m + m = q ), wehave: P ( Z j − Z i = r, Z j − Z i = s ) = P ( Z m + m + m = r, τ m Z m = s )= P ( Z m + τ m + m Z m = r − s, τ m Z m = s )= P ( Z m + τ m + m Z m = r − s ) P ( Z m = s ) = P ( Z m + m = r − s ) P ( Z m = s ) . Hence, in case 2a), we get: P ( Z j − Z i = r, Z j − Z i = s ) = P ( Z m + m = r − s ) P ( Z m = s ) . (54) ( i < j ≤ i < j ) The events Z j − Z i = r and Z j − Z i = s are independent. LT FOR RANDOM WALKS OF COMMUTING ENDOMORPHISMS 39 Following the method of [27], now we estimate Var( V n,p ) = E ( V n,p ) − ( E V n,p ) . Recallthat V n,p = P ≤ i,j 0, for the bound of (1a), we can neglectthe corresponding centering terms, since they are subtracted and non negative. (1a) For r, s ∈ Z , let a r,s ( n ) := P ≤ i ≤ i 1. Indeed we have: λ n u n = (1 − λ )( + ∞ X k = n λ k ) u n ≤ (1 − λ ) + ∞ X k = n λ k u k ≤ (1 − λ ) + ∞ X k =0 λ k u k ≤ C (1 − λ ) . For λ = 1 − n in the previous inequality, we get: u n ≤ Cn (1 − n ) − n ≤ C ′ n . (2a) Let b r,s ( n ) be the sum X ≤ i
Likewise, we have: n X k =1 X m + m + m + m = k m ( m + m + m ) ≤ n n X k =1 X m + m + m = k m k ≤ n n X k =1 k / k X m =1 km ≤ n n X k =1 Log k √ k ≤ C n / Log n. Therefore, we obtain b ( n, p ) = O ( n ).The previous estimations imply Var( V n,p ) = O ( n ). By (26), for p ∈ L ( W ), lim n V n,p / E V n,p =1 a.e. follows as in [27] and therefore lim n V n,p /V n, = 1 a.e. (cid:3) Appendix II: mixing, moments and cumulants For the sake of completeness, we recall in this appendix some general results on mixing ofall orders, moments and cumulants (see [25] and the references given therein). Implicitlywe assume existence of moments of all orders when they are used.For a real random variable Y (or for a probability distribution on R ), the cumulants(or semi-invariants) can be formally defined as the coefficients c ( r ) ( Y ) of the cumulantgenerating function t → ln E ( e tY ) = P ∞ r =0 c ( r ) ( Y ) t r r ! , i.e., c ( r ) ( Y ) = ∂ r ∂ r t ln E ( e tY ) | t =0 . Similarly the joint cumulant of a random vector ( X , ..., X r ) is defined by c ( X , ..., X r ) = ∂ r ∂t ...∂t r ln E ( e P rj =1 t j X j ) | t = ... = t r =0 . (55)This definition can be given as well for a finite measure ν on R r . Its cumulant is noted c ν ( x , ..., x r ). The joint cumulant of ( Y, ..., Y ) ( r copies of Y ) is c ( r ) ( Y ).For any subset I = { i , ..., i p } ⊂ J r := { , ..., r } , we put m ( I ) = m ( i , ..., i p ) := E ( X i · · · X i p ) , s ( I ) = s ( i , ..., i p ) := c ( X i , ..., X i p ) . The cumulants of a process ( X j ) j ∈J , where J is a set of indexes, is the family { c ( X i , ..., X i r ) , ( i , ..., i r ) ∈ J r , r ≥ } . The following formulas link moments and cumulants and vice-versa: c ( X , ..., X r ) = s ( J r ) = X P ( − p − ( p − m ( I ) · · · m ( I p ) , (56) E ( X · · · X r ) = m ( J r ) = X P s ( I ) · · · s ( I p ) , (57)where in both formulas, P = { I , I , ..., I p } runs through the set of partitions of J r = { , ..., r } into p ≤ r nonempty intervals, with p varying from 1 to r . Now let be given a random field of real random variables ( X k ) k ∈ Z d and a summableweight R from Z d to R + . For Y := P ℓ ∈ Z d R ( ℓ ) X ℓ we obtain from the definition (55): c ( r ) ( Y ) = c ( Y, ..., Y ) = X ( ℓ ,...,ℓ r ) ∈ ( Z d ) r c ( X ℓ , ..., X ℓ r ) R ( ℓ ) · · · R ( ℓ r ) . (58) Limiting distribution and cumulants For our purpose, we state in terms of cumulants a particular case of a theorem of M.Fr´echet and J. Shohat, generalizing classical results of A. Markov. Using the formulaslinking moments and cumulants, a special case of their “generalized statement of thesecond limit-theorem” can be expressed as follows: Theorem 7.1. [13] Let ( Z n , n ≥ be a sequence of centered real r.v. such that lim n c (2) ( Z n ) = σ , lim n c ( r ) ( Z n ) = 0 , ∀ r ≥ , (59) then ( Z n ) tends in distribution to N (0 , σ ) . (If σ = 0 , then the limit is δ ). Theorem 7.2. (cf. Theorem 7 in [24] ) Let ( X k ) k ∈ Z d be a random process and ( R n ) n ≥ a summation sequence on Z d . Let ( Y n ) be defined by Y n = P ℓ R n ( ℓ ) X ℓ , n ≥ . If X ( ℓ ,...,ℓ r ) ∈ ( Z d ) r c ( X ℓ , ..., X ℓ r ) R n ( ℓ ) ...R n ( ℓ r ) = o ( k Y n k r ) , ∀ r ≥ , (60) then Y n k Y n k tends in distribution to N (0 , when n tends to ∞ .Proof . Let β n := k Y n k = k P ℓ R n ( ℓ ) X ℓ k and Z n = β − n Y n .Using (58), we have c ( r ) ( Z n ) = β − rn P ( ℓ ,...,ℓ r ) ∈ ( Z d ) r c ( X ℓ , ..., X ℓ r ) R ( ℓ ) ...R ( ℓ r ). The the-orem follows then from the assumption (60) by Theorem 7.1 applied to ( Z n ). (cid:3) Definition 7.3. A measure preserving N d (or Z d )-action T : n → T n on a probabilityspace ( X, A , µ ) is r -mixing, r > , if for all sets B , ..., B r ∈ A lim min ≤ ℓ<ℓ ′≤ r k n ℓ − n ℓ ′ k→∞ µ ( r \ ℓ =1 T − n ℓ B ℓ ) = r Y ℓ =1 µ ( B ℓ ) . Notation 7.4. For f in the space L ∞ ( X ) of measurable essentially bounded func-tions on ( X, µ ) with R f dµ = 0, we apply the definition of moments and cumulantsto ( T n f, ..., T n r f ) and put m f ( n , ..., n r ) = Z X T n f · · · T n r f dµ, s f ( n , ..., n r ) := c ( T n f, ..., T n r f ) . (61)To use the property of mixing of all orders, we need the following lemma. Lemma 7.5. For every sequence ( n k , ..., n kr ) in ( Z d ) r , there are a subsequence with pos-sibly a permutation of indices (still written ( n k , ..., n kr ) ), an integer κ ( r ) ∈ [1 , r ] , a subdi-vision r < r < ... < r κ ( r ) − < r κ ( r ) ≤ r of { , ..., r } and a constant integral vector LT FOR RANDOM WALKS OF COMMUTING ENDOMORPHISMS 43 a j such that lim k min ≤ s = s ′ ≤ κ ( r ) k n kr s − n kr s ′ k = ∞ , (62) n kj = n kr s + a j , for r s < j < r s +1 , s = 1 , ..., κ ( r ) − , and for r κ ( r ) < j ≤ r. (63) If ( n k , ..., n kr ) satisfies lim k max i = j k n ki − n kj k = ∞ , then κ ( r ) > .Proof . Remark that if sup k max i = j k n ki − n kj k < ∞ , then κ ( r ) = 1 so that (62) is voidand (63) is void for the indexes such that r s +1 = r s + 1. The proof of the lemma is byinduction. The result is clear for r = 2. Suppose that the subsequence for the sequenceof ( r − n k , ..., n kr − ) is built.Let 1 ≤ r < r < ... < r κ ( r − ≤ r − { , ..., r − } , asstated above for the sequence ( n k , ..., n kr − ). If ( n k , ..., n kr − ) satisfies lim k max n k , ..., n kr ). If lim k k n kr − n ki k = + ∞ , for i = 1 , ..., r − 1, then we take1 ≤ r < r < ... < r κ ( r − < r κ ( r ) = r as new subdivision of { , ..., r } . If lim inf k k n kr − n ki s k < + ∞ for some s ≤ κ ( r − n kr ) wehave n kr = n ki s + a r , where a r is a constant integral vector. After changing the labels, weinsert n r in the subdivision of { , ..., r − } and obtain the new subdivision of { , ..., r } .For the last condition on κ , suppose that lim k max 1. If, on the contrary, thesequence ( n k , ..., n kr − ) satisfies lim k max 1, sothat κ ( r ) ≥ κ ( r − > (cid:3) Lemma 7.6. If a Z d -dynamical system on ( X, A , µ ) is mixing of order r ≥ , then, forany f ∈ L ∞ ( X ) , lim max i = j k n i − n j k→∞ s f ( n , ..., n r ) = 0 .Proof . The notation s f was introduced in (61). Suppose that the above convergencedoes not hold. Then there is ε > r -tuples ( n k = 0 , ..., n kr ) such that | s f ( n k , ..., n kr ) | ≥ ε and max i = j k n ki − n kj k → ∞ (we use stationarity).By taking a subsequence (but keeping the same notation), we can assume that, fortwo fixed indexes i, j , lim k k n ki − n kj k = ∞ . By Lemma 7.5, there is a subdivision1 = r < r < ... < r κ ( r ) − < r κ ( r ) ≤ r and constant integer vectors a j such thatlim k min ≤ s = s ′ ≤ κ ( r ) k n kr s − n kr s ′ k = ∞ , (64) n kj = n kr s + a j , for r s < j < r s +1 , s = 1 , ..., κ ( r ) − , and for r κ ( r ) < j ≤ r. (65)Let dµ k ( x , ..., x r ) denote the probability measure on R r defined by the distribution ofthe random vector ( T n k f ( . ) , ..., T n kr f ( . )). We can extract a converging subsequence fromthe sequence ( µ k ), as well as for the moments of order ≤ r . Let ν ( x , ..., x r ) (resp. ν ( x i , ..., x i p )) be the limit of µ k ( x , ..., x r ) (resp. of its marginal measures µ k ( x i , ..., x i p )for { i , ..., i p } ⊂ { , ..., r } ). Let ϕ i , i = 1 , ..., r , be continuous functions with compact support on R . Mixing of order r and condition (64) imply ν ( ϕ ⊗ ϕ ⊗ ... ⊗ ϕ r ) = lim k Z R d ϕ ⊗ ϕ ⊗ ... ⊗ ϕ r dµ k = lim k Z r Y i =1 ϕ i ( f ( T n ki x )) dµ ( x )= lim k Z (cid:2) κ ( r ) − Y s =1 Y r s ≤ j This research was carried out during visits of the first authorto the IRMAR at the University of Rennes 1 and of the second author to the Centerfor Advanced Studies in Mathematics at Ben Gurion University. The first author waspartially supported by the ISF grant 1/12. The authors are grateful to their hosts fortheir support. They thank Y. Guivarc’h, S. Le Borgne and M. Lin for helpful discussionsand B. 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Math.Comp. 30, no. 135, 598-609 (1976). doi: 10.2307/2005329[35] Voln´y, D., Wang, Y.: An invariance principle for stationary random fields under Hannan’s condi-tion. Stochastic Process. Appl. 124, no. 12, 4012-4029 (2014). doi: 10.1016/j.spa.2014.07.015 Guy Cohen,Dept. of Electrical Engineering,Ben-Gurion University, Israel E-mail address : [email protected] Jean-Pierre Conze,IRMAR, CNRS UMR 6625,University of Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France E-mail address ::