Limit theorems for some time dependent expanding dynamical systems
aa r X i v : . [ m a t h . D S ] O c t LIMIT THEOREMS FOR SOME TIME DEPENDENTEXPANDING DYNAMICAL SYSTEMS
YEOR HAFOUTADEPARTMENT OF MATHEMATICSTHE HEBREW UNIVERSITY AND THE OHIO STATE UNIVERSITY
Abstract.
In this paper we will prove various probabilistic limit theoremsfor some classes of distance expanding sequential dynamical systems (SDS).Our starting point here is certain sequential complex Ruelle-Perron-Frobenius(RPF) theorems which were proved in [17] and [22] using contraction propertiesof a complex version of the projective Hilbert metric developed in [36]. Wewill start from the growth rate of the variances of the underlying partial sums.This is well understood in the random dynamics setup, when the maps arestationary, but not in the SDS setup, where various growth rates can occur.Some of our results in this direction rely on certain type of stability in theseRPF theorems, which is one of the novelties of this paper. Then we willprovide general conditions for several classical limit theorems to hold true inthe sequential setup. Some of our general results mostly have applications forcomposition of random non-stationary map, while the conditions of the otherresults hold true for general type of SDS. In the latter setup, results such as theBerry-Esse´en theorem and the local central limit theorem were not obtainedso far even for independent but not identically distributed maps, which is aparticular case of the setup considered in this paper. Introduction
Probabilistic limit theorems for dynamical systems and Markov chains is a wellstudied topic. One way to derive such results is relying on some quasi-compactness(or spectral gap) of an appropriate transfer or Markov operator, together with asuitable perturbation theorem (see [33], [34], [16] and [23]). This quasi-compactnesscan often be verified only via an appropriate Ruelle-Perron-Frobenius (RPF) the-orem, which is the main key for thermodynamic formalism type constructions.Probabilistic limit theorems for random dynamical systems and Markov chains inrandom dynamical environments were also studied in literature (see, for instance,[26], [27], [17], [1] ,[12], [13] and references therein). In these circumstances, theprobabilistic behaviour of the appropriate process is determined by compositionsof random operators, and not of a single operator, so no spectral theory can beexploit, and instead, many of these results rely on an appropriate version of the
Date : October 30, 2019.2010
Mathematics Subject Classification.
Key words and phrases. limit theorems; Perron-Frobenius theorem; thermodynamic formalism;sequential dynamical systems; time dependent dynamical systems; random dynamics; randomnon-stationary environments;
RPF theorem for random operators. Relying on certain contraction properties ofrandom complex transfer and Markov operators, with respect to a complex versionof the Hilbert protective metric develpoed in [36] (see also [14] and [15]), we provedin [17] an RPF theorem for random complex operators and presented the appropri-ate random complex thermodynamic formalism type constructions, which was oneof the main keys in the proof of versions of the Berry-Esseen theorem and the localcentral limit theorem for certain processes in random dynamical environment.In recent years (see, for instance, [2], [6], [24], [29] and [35] and referencestherein) there has been a growing interest in proving limit theorems for sequences X n = T n x of random variables generated by an appropriate random variable x and compositions T n = T n − ◦ T n − ◦ · · · ◦ T of different maps T , T , T , ... . Exceptfor the random dynamical system case, in which the maps T j = T ξ j are chosenat random according to a stationary process { ξ j } , the results obtained so far arecertain versions of the central limit theorem (CLT), without close to optimal (dis-tributional) convergence rate and corresponding local CLT’s. In this paper we willprove several limit theorems for some classes of maps T j , as described in the nextparagraphs. We stress that some, but not all, of our general results are counter-parts of the results obtained in Chapter 7 of [17] for sequences of maps instead ofrandom stationary maps (and for non-uniformly distance expanding maps). Wethink that even though there is some overlap with this Chapter 7, it is importantto have all the limit theorems formulated together as “corollaries” of the sequentialRuelle-Perron-Frobenius theorem.Let T j be a sequence of distance expanding maps, and f j , u j sequences of H¨oldercontinuous (or differentiable) functions (uniformly in j ). Let L ( j )0 be the transferoperator generated by T j and the function e f j , and let L ( j ) z ( g ) = L ( j )0 ( g · e zu j ) , z ∈ C be the perturbations of L ( j )0 corresponding to u j . The starting point of this paperis a complex RPF theorem for the sequences of transfer operator L ( j ) z , j ∈ Z .Applying this theorem we obtain several limit theorems (finer than the CLT) forsequences of random variables of the form S ,n u ( x ) = P n − j =0 u j ◦ T j ( x ), where x is distributed according some special (Gibbs) probability measure. We will alsostudy certain stability properties of the RPF triplets, which will yield that thevariance of the above random variables grow linearly fast in n , when the T j , f j and u j lie in some neighborhood of a single distance expanding map T and functions f and u , respectively.Some of the conditions of our general results hold true for non-random sequencesof maps. The other results will be mainly applicable for certain classes random mapswhich are not necessarily stationary, a situation which was not considered so far.Several difficulties arise beyond the random dynamical setup (which was consideredin [17]), when the operators do not have the form L ( j ) z = L θ j ωz for an appropriatemeasure preserving system (Ω , F , P, θ ). For instance, in the stationary case manylimit theorems follow from the existence of the limit Π( z ) = R Π ω ( z ) dP ( ω ) =lim n →∞ n P n − j =0 Π θ j ω ( z ) , where Π ω ( z ) is a certain random pressure function. Inour case we have a sequence of pressure functions Π j ( z ) , j ∈ Z and the limitsΠ( z ) = lim n →∞ n P n − j =0 Π j ( z )do not necessarily exist. When T j = T ξ j , f j = f ξ j and u j = u ξ j are chosen atrandom according to several classes of non-stationary sequences { ξ j } with certain imit theorems 3 mixing properties, we will show that the limits Π( z ) exist and are analytic functionsof z , and then use that in order to apply some of our limit theorems. Acknowledgment.
I would like to thank Prof. Yuri Kifer for suggesting me toapply complex cone methods in the setup of time dependent dynamical systems,and for several references on these systems, as well.2.
Sequential dynamical systems: preliminaries and main results
Our setup consists of a bounded metric space ( X , d ), a family {E j : j ∈ Z } ofcompact subsets of X , and a family of maps T j : E j → E j +1 , j ∈ Z . For any j ∈ Z and n ≥ T nj = T j + n − ◦ · · · ◦ T j +1 ◦ T j . We will assume that one of the following three assumptions holds true2.1.
Assumption.
There exist constants ξ > , γ > , L, n ∈ N , and D > j ∈ Z :(i) For any x ∈ E j ,(2.1) T n j B j ( x, ξ ) = E j + n where B j ( x, ξ ) := { w ∈ E j : d ( w, x ) < ξ } .(ii) For any x, x ′ ∈ E j +1 so that d ( x, x ′ ) < ξ we can write T − j { x } = { y , ..., y k } and T − j { x ′ } = { y ′ , ..., y ′ k } where k ≤ D and for each i = 1 , , , ..., k ,(2.2) d ( y i , y ′ i ) ≤ γ − d ( x, x ′ ) . (iii) There are x ,j , x ,j , ..., x L j ,j , L j ≤ L in E j so that E j = L j [ s =1 B j ( x s,j , ξ ) . Assumption.
The inverse image T − j { x } of any point x ∈ E j +1 under T j is atmost countable, and there exists a constant γ > { x i } and { x ′ i } of any two points x, x ′ ∈ E j +1 under T j can be paired so that (2.2) holdstrue.2.3. Assumption.
There exist two sided sequences ( L j ), ( σ j ), ( q j ) and ( d j ) so that( L j ) is bounded and for each j we have σ j > q j , d j ∈ N , q j < d j and for any x, x ′ ∈ E j +1 we can write T − j { x } = { x , ..., x d j } and T − j { x ′ } = { x ′ , ..., x ′ d j } where for any i = 1 , , ..., q j we have d ( x i , x ′ i ) ≤ L j ρ j +1 ( x, x ′ )while for any i = q j + 1 , ..., d j we have d ( x i , x ′ i ) ≤ σ − j ρ j +1 ( x, x ′ ) . Y. Hafouta
The arguments in [32] show that Assumption 2.1 is satisfied when the maps T j are locally distances expanding, uniformly in j . The second assumption holds true,for instance, when each T j is a map on the unit interval with a countable number ofmonotonicity intervals, and the absolute value of the derivative of T j on each one ofthese intervals is bounded from below by γ . The main example we have in mind isthe case where X j = M are the same compact and connected Riemannian manifoldand each T j : M → M is a diffeomorphism satisfying the conditions in [39] and [4](with constant which are uniform in j ). This means that each T j locally expandsdistance only on some open region. These conditions are satisfied when all themaps T j lie in a C neighborhood of some map T satisfying the above conditions.Assumption 2.2 holds true for intervals maps which only expand distance only onsome pieces of the unit interval.Next, for each integer j , let f j , u j : E j → R be H¨older continuous functions withexponent α which does not depend on j . For each integer j and a complex number z , let L ( j ) z be the linear operator which maps complex valued functions g on E j tocomplex valued functions L ( j ) z g on E j +1 by the formula L ( j ) z g ( x ) = X y ∈ T − j { x } e f j ( y )+ zu j ( y ) g ( y ) . For each j ∈ Z and n ∈ N set L j,nz = L ( j + n − z ◦ · · · ◦ L ( j +1) z ◦ L ( j ) z . Let H j be the (Banach) space of all H¨older continuous functions g : E j → C withexponent α , equipped with the norm k g k α = k g k ∞ + v ( g ), where k g k ∞ = sup | g | and v α ( g ) = v α,ξ ( g ) = sup n | g ( x ) − g ( x ′ ) | d α ( x, x ′ ) : 0 < d ( x, x ′ ) ≤ ξ o . In the case when α = 1 and X j is a Riemannian manifold we will also considerthe case when f j and u j are C functions and in this case we also consider the”variation” ν ( g ) = sup k Dg k , where Dg is the differential of g . We will denotehere by H ∗ j the dual of the Banach space H j . Finally, for any family of functions { g j : E j → C : j ∈ Z } we set for each integer j and n ≥ S n,j g = j + n − X k = j g k ◦ T kj . We will work in this paper under the following2.4.
Assumption.
Under Assumptions 2.1 and 2.2, the norms k f j k and k u j k arebounded in j by some constant B .2.5. Assumption.
Assumption 2.3 holds true and there exist constants
B > s ∈ (0 ,
1) so that we have sup j sup | f j | ≤ B , sup j sup x P y ∈ T − j { x } e f j ( y ) ≤ B ,sup f j − inf f j < ε j and v ( φ j ) < ε ′ j where ε j , ε ′ j are sequence of positive constants satisfying(2.3) sup j e ε j q j L αj − ( d j − q j ) σ − αj d j ≤ s and sup j ε ′ j ≤ B . imit theorems 5 From now on, we will refer to the constants appearing in in Assumptions 2.1,2.2, 2.3, 2.4 and 2.5 as the “initial parameters”.Our starting point in this paper is the following2.6.
Theorem.
Suppose that one of Assumptions 2.1, 2.2 and 2.3 hold true andthat either Assumption 2.4 or Assumption 2.5 are satisfied (depending on the case)hold true. Then there exists a neighborhood U of , which depends only on theinitial parameters, so that for any z ∈ U there exist families { λ j ( z ) : j ∈ Z } , { h ( z ) j : j ∈ Z } and { ν ( z ) j : j ∈ Z } consisting of a nonzero complex number λ j ( z ) , acomplex function h ( z ) j ∈ H j and a complex continuous linear functional ν ( z ) j ∈ H ∗ j such that:(i) For any j ∈ Z , (2.4) L ( j ) z h ( z ) j = λ j ( z ) h ( z ) j +1 , ( L ( j ) z ) ∗ ν ( z ) j +1 = λ j ( z ) ν ( z ) j and ν ( z ) j ( h ( z ) j ) = ν ( z ) j ( ) = 1 where is the function which takes the constant value . When z = t ∈ R then λ j ( t ) > a and the function h j ( t ) takes values at some interval [ c, d ] , where a > and < c < d < ∞ depend only on the initial parameters. Moreover, ν ( t ) j isa probability measure which assigns positive mass to open subsets of E j and theequality ν j +1 ( t ) (cid:0) L ( j ) t g ) = λ j ( t ) ν ( t ) j ( g ) holds true for any bounded Borel function g : E j → C .(ii) The maps λ j ( · ) : U → C , h ( · ) j : U → H j and ν ( · ) j : U → H ∗ j are analytic and there exists a constant C > , which depends only on the initialparameters such that (2.5) max (cid:0) sup z ∈ U | λ j ( z ) | , sup z ∈ U k h ( z ) j k , sup z ∈ U k ν ( z ) j k (cid:1) ≤ C, where k ν k is the operator norm of a linear functional ν : H j → C . Moreover,there exist a constant c > , which depends only on the initial parameters, so that | λ j ( z ) | ≥ c and min x ∈E j | h ( z ) j ( x ) | ≥ c for any integer j and z ∈ U .(iii) There exist constants A > and δ ∈ (0 , , which depend only on the initialparameters, so that for any j ∈ Z , g ∈ H j and n ≥ , (2.6) (cid:13)(cid:13)(cid:13) L j,nz gλ j,n ( z ) − ν ( z ) j ( g ) h ( z ) j + n (cid:13)(cid:13)(cid:13) ≤ A k g k δ n where λ j,n ( z ) = λ j ( z ) · λ j +1 ( z ) · · · λ j + n − ( z ) . Moreover, the probability measures µ j , j ∈ Z given by dµ j = h (0) j dν (0) j satisfy that ( T j ) ∗ µ j = µ j +1 and that for any n ≥ and f ∈ H j + n , (2.7) (cid:12)(cid:12) µ j ( g · f ◦ T nj ) − µ j ( g ) µ j + n ( f ) (cid:12)(cid:12) ≤ A k g k µ j + n ( | f | ) δ n . Under Assumptions 2.1 Theorem 2.6 was essentially proved in Chapter 5 of [17],and under Assumption 2.3 it was proved in [22]. The proof under Assumption2.2 proceeds similarly to the proof under Assumption 2.1, and for that reason thedetails are omitted.We note that when the { T j } are ”sequentially non-singular” the measures µ j areabsolutely continuous, as stated in the following Y. Hafouta
Proposition.
Let m j , j ∈ Z be a family of probability measures on E j , whichassign positive mass to open sets, so that for each j we have ( T j ) ∗ m j ≪ m j +1 andthat e − f j = d ( T j ) ∗ m j d m j +1 . Then for any j we have λ j (0) = 1 and ν (0) j = m j . When E j = X we can always take m j = m for some fixed m (e.g. a volumemeasure when X is a Riemannian manifold). Theorem 2.7 is proved exactly as in[19] (see Proposition 3.1.2 there).Our first result in this paper is the following stability theorem:2.8. Theorem.
Let r > be so that ¯ B (0 , r ) =: { z ∈ Z : | z | ≤ r } ⊂ U and set K = ¯ B (0 , r ) . Then for any ε > there exists δ > with the following property:if T ,j , j ∈ Z is a family of maps and f ,j , u ,j ∈ H j , where j ∈ Z , are familiesof functions which satisfy one of Assumptions 2.1-2.3 and one of Assumptions 2.4and 2.5 (in accordance with T j and with the same initial parameters), and for any z ∈ K and j ∈ Z , (2.8) kL ( j ) z − L ( j )1 ,z k < δ where L ( j )1 ,z is the operator defined similarly to L ( j ) z but with T ,j , f ,j and u ,j inplace of T j , f j and u j , then for any integer j and z ∈ K we have max (cid:0) | λ j ( z ) − λ ,j ( z ) | , k h ( z ) j − h ( z )1 ,j k , k ν ( z ) j − ν ( z )1 ,j k (cid:1) < ε where { λ ,j ( z ) : j ∈ Z } , { h ( z )1 ,j : j ∈ Z } and { ν ( z )1 ,j : j ∈ Z } , z ∈ U are the RPFfamilies corresponding to the operators L ( j )1 ,z . The relation between this theorem and to “limit theorems” is that it is the keyingredient in the proof of Theorem 2.9 (ii) below.2.1.
Probabilistic limit theorems: general formulations.
In this section wewill formulate general theorems, while some of them will mainly be applicable tothe case of non-stationary random environment, Theorem 2.9, the Berry-Esseentheorem the exponential, the exponential concentration inequalities and the mod-erate deviations theorems stated below are applicable for non-random sequences ofmaps. We especially want to stress that Theorem 2.12 is a modification of the localcentral limit theorem from [17] in the sequential setup, and its main importancein its applications to the non-stationary random environments studied in Section5. The main reason this theorem (and Theorem 2.16) are formulated here is inorder not to overload the exposition with the precise definitions of the differentnon-stationary environment we have in mind.2.1.1.
The variance and the moments.
We begin with the variances:2.9.
Theorem. (i) The variances var µ k ( S k,n u ) , k ∈ Z do not converge to ∞ as n → ∞ if and only if there exists a family of functions Y s : E s → R , s ∈ Z and aconstant C > so that for any s ∈ Z we have Z Y s ( x ) dµ s ( x ) < C and u s − µ s ( u s ) = Y s +1 ◦ T s − Y s , µ s − a.s.When viewed as a random variable, the function Y k is a member of the subspace of L ( E k , µ k ) generated by the functions { u k ◦ T dk − µ k ( u k ) : d ≥ } where T k := Id.Moreover, the function Y k can be chosen to be a member of H k so that k Y k k is imit theorems 7 bounded in k , and in this case the equalities u s ( x ) − µ s ( u s ) = Y s +1 ( T s x ) − Y s ( x ) hold true for any x ∈ E s .(ii) Suppose that E k = X for each k . Let T : X → X be a map so that one ofAssumptions 2.1-2.3 are satisfies with T j = T , and let u, f : X → R be members of H = H j , so that k f k and k u k do not exceed B , and u does not admit a co-boundaryrepresentation with respect to T . In the case when Assumption 2.3 holds true wealso require Assumption 2.5 to hold with T j = T, f j = f and u j = u (with the sameinitial parameters). Then there exists ε > , which depends only on the initialparameters, so that the following holds true: if (2.9) sup k ∈ Z kL ( k ) z − L z k ∞ ≤ ε for any z in some neighborhood of , where L z is the transfer operator generatedby T and f + zu , then inf k var µ k ( S k,n u ) ≥ δ n for some δ > and all sufficiently large n . Suppose that X j = M are all the same Riemannian manifold, and that the maps T j satisfy the conditions from [4]. Assume also that all of the T j ’s lie a sufficientlysmall C -ball of a single map which satisfies these conditions, and that the functions f j and u j all lie in a sufficiently small ball (in the C -norm) around f and u ,respectively. Then (2.9) is satisfied with the norm k g k = sup | g | + sup | Dg | (i.e. wetake α = 1, see Proposition 5.3 in [4]). We note that the size of the neighborhoodsof T, f and u which is needed depends only on the initial parameters and on thelimit σ u = lim n →∞ Var µ ( P n − j =0 u ◦ T j ) >
0, where µ = µ T,f is the appropriate T -invariant Gibbs measure corresponding to the function f . In fact, it is possibleto give a precise formula for the radius of the balls around T, f and u . Anotherexample is intervals maps with finite number of monotonicity intervals which donot depend on j , where on each one of them each T j and T is either expanding orcontracting. If each T j is obtained from T by perturbing each inverse branch of T in some H¨older norm, and f j and u j are small perturbations of f and u in thisnorm, then (2.8) will hold true in the appropriate H¨older norm. Similar examplescan be given in rectangular regions in R d for d > n ≥ σ ,n = q var µ ( S ,n u )and let x be a E -valued random variable which is distributed according to µ .Then, relying on (2.6) and the arguments in [6], it follows that ¯ S ,n u ( x ) − µ ( S ,n u ) σ ,n converges in distribution towards the standard normal law, when σ ,n converges to ∞ as n → ∞ (this essentially means that the quadratic variations of the martingalesconstructed in the proof of Theorem 2.14 converge as n → ∞ , after a propernormalization). When the variances grow faster than n then we are able to provea self normalized Berry-Esseen theorem:2.1.2. Berry-Eseen´en and a local CLT.
Theorem.
Suppose that lim n →∞ σ ,n n − = ∞ Y. Hafouta and set ¯ S ,n u = S ,n u − µ ( S ,n u ) . Then there exists a constant C > so that forany n ≥ and r ∈ R , (2.10) (cid:12)(cid:12)(cid:12) µ { x ∈ E : ¯ S ,n u ( x ) ≤ rσ ,n } − √ π Z r −∞ e − t dt (cid:12)(cid:12)(cid:12) ≤ Cnσ − ,n . In particular, when σ ,n grows linearly fast in n the above left hand side does notexceed C n − for some constant C . Note that we obtain here optimal convergence rate (i.e. rate of order n − ) inthe circumstances of Theorem 2.9 (ii).For the sake of completeness, we will formulate the following theorem whoseproof is carried out exactly as in [20]:2.11. Theorem. (i) By possibly decreasing r , where r comes from Theorem 2.8, wecan define analytic functions Π j : B (0 , r ) → C , j ∈ Z so that Π j (0) = 0 , λ j ( z ) /λ j (0) = e Π j ( z ) and | Π j ( z ) | ≤ c for any z ∈ B (0 , r ) , for some constant c which does not depend on j and z .(ii) There exists a constant R so that for any integer j and n ≥ , (cid:12)(cid:12) µ j ( S j,n u ) − n − X m =0 Π ′ j + m (0) (cid:12)(cid:12) ≤ R where Π ′ j + m (0) is the derivative of Π j + m at z = 0 .(iii) Suppose that µ j ( u j ) = 0 for any integer j . For any k ≥ , s ∈ Z and n ≥ ,set γ j,k,n = n − [ k ] Z E j ( S j,n u ( x )) k dµ j ( x ) and Π j,k,n = n − n − X m =0 Π ( k ) j + m (0) where Π ( k ) j + m (0) is the k -th derivative of the function Π j + m at z = 0 . Then thereexist constants R k > , k ≥ , so that for any even k ≥ , (2.11) max (cid:16)(cid:12)(cid:12) γ j,k,n − C k ( γ j, ,n ) k (cid:12)(cid:12) , (cid:12)(cid:12) γ j,k,n − C k (Π j, ,n ) k (cid:12)(cid:12)(cid:17) ≤ R k n where C k = 2 − k ( k !) − k ! , while with D k = k !3! − ( k − ( k − !) − for any odd k ≥ , max (cid:0)(cid:12)(cid:12) γ j,k,n − D k ( γ j, ,n ) k − γ j, ,n (cid:12)(cid:12) , (cid:12)(cid:12) γ j,k,n − D k (Π j, ,n ) k − Π j, ,n (cid:12)(cid:12)(cid:17) ≤ R k n . An immediate consequence of Theorem 2.11 is that | γ j,k,n | is uniformly boundedin j and n , for each k . Using the Markov inequality, together with (2.11) with k = 4, we obtain that for any ε > j ∈ Z , µ j (cid:8) x : | S j,n u ( x ) − µ j ( S j,n u ) | ≥ εn (cid:9) ≤ Cn − ε − for some C >
0. Therefore, by the Borel-Cantelli Lemma,lim n →∞ S j,n u ( x ) − µ j ( S j,n u ) n = 0 , µ j − a.s.namely, a sequential version of the strong law of large numbers holds true.When T j , f j , u j are chosen at random according to some type of (not necessarilystationary) sequence of random variables, then, in Section 5 we obtain almost sureconverges rate of the form (cid:12)(cid:12)(cid:12) n µ j ( S j,n u ) − p (cid:12)(cid:12)(cid:12) ≤ R ,ω n imit theorems 9 where p is some constant and R ,ω is some random variable. When µ j ( u j ) = 0 wealso derive that for any k ≥ (cid:12)(cid:12) γ j,k,n − γ k (cid:12)(cid:12) ≤ R ω,k n − ln n where R ω,k is some random variable and γ k is some constant. Using (2.12) with k = 2, when σ = γ >
0, and almost optimal convergence rate in the central limittheorem of the form(2.13) sup s ∈ R (cid:12)(cid:12) µ { x ∈ E : ¯ S ,n u ( x ) ≤ s √ n } − √ πσ Z s −∞ e − t σ dt (cid:12)(cid:12) ≤ c ω n − ln n follows (see Section 5.3.4).Next, as usual, in order to present the local central limit theorem we will distin-guish between two cases. We will call the case a lattice one if the functions u j takevalues at some lattice of the form h Z := { hk : k ∈ Z } for some h >
0. We will callthe case a non-lattice if there exist no h which satisfy the latter lattice condition.In the non-lattice case set h = 0 and I h = R \ { } , while in the lattice case set I h = ( − πh , πh ) \ { } .2.12. Theorem.
Suppose that for any compact interval J ⊂ I h we have (2.14) lim n →∞ √ n sup t ∈ J kL ,nit k /λ ,n (0) = 0 and that there exists c > so that σ ,n = var µ ( S ,n u ) ≥ c n for any sufficientlylarge n . Then for any continuous function g : R → R with compact support we have lim n →∞ sup r ∈ R h (cid:12)(cid:12)(cid:12) √ πσ ,n Z g ( S ,n u ( x ) − µ ( S ,n u ) − r ) dµ ( x ) − (cid:16) Z ∞−∞ g ( t ) dm h ( t ) (cid:17) e − r σ ,n (cid:12)(cid:12)(cid:12) = 0 where in the non-lattice case R h = R and m h is the Lebesgue measure, while in thelattice case R h = h Z = { hk : k ∈ Z } and m h is the measure assigning unit massto each one of the members of R h . When the limit σ = lim n →∞ n σ ,n exists thenwe can replace σ ,n with σ √ n . Next, let Y , ..., Y m be compact metric spaces, set Y m +1 := Y and let S j : Y j → Y j +1 , j = 1 , , ..., m be maps so that Assumption 2.1 holds true with E ′ j = Y k j and T ′ j = S k j in place of E j and T j , respectively, where k = k j = j mod m .Let r i , v i : Y i → R be H¨older continuous functions whose k · k norms are boundedby B . Set S = S m ◦ · · · ◦ S ◦ S and for each real t let the transfer operator L it be defined by L it g ( x ) = X y ∈ S − { x } e P m j =1 r j ( y )+ it P m j =1 v j ( y ) g ( y ) . In Section 4.3 we will show that (2.14) holds true under the following2.13.
Assumption.
The the spectral radius of L it is strictly less than 1 for any t ∈ I h . Moreover, for any compact interval J ⊂ I h there exists δ ∈ (0 ,
1) so thatfor any constant B J > s we have(2.15) lim n →∞ |{ ≤ m < n : B J kL m,sm it − L sit k < − δ ∀ t ∈ J }| ln n = ∞ where | Γ | is the cardinality of a finite set Γ. Note that in Assumption 2.13 there is an underlying assumption that E m = Y for any m so that B J kL m,sm it − L sit k < − δ . In the non-lattice case, the conditionabout the spectral radius of L it means that the function P m j =1 v j ◦ S j − ◦ · · · · S ◦ S is non-arithmetic (or aperiodic) with respect to the map S defined above, while inthe lattice case it means that h satisfies a certain maximality condition with respectto this function (see [16] and [23]).Assumption 2.13 holds true when T k , f k , u k are chosen at random according tosome (not necessarily stationary) sequence of random variables, see Section 5 anda discussion at the end of this Section. Non random examples can be constructedas follows: assume that kL a k + im ,m it − L it k ≤ δ k α ( t )for any sufficiently large k and 0 ≤ i ≤ m − ( a k +1 − a k − m ), for some sequence δ k which converges to 0 as k → ∞ and a continuous function α ( t ), where ( a k ) ∞ k =1 is asequence of natural numbers so that lim k →∞ ( a k +1 − a k ) = ∞ and a k ≤ c e c k r forsome r ∈ (0 ,
1) and c , c > k ’s.2.1.3. Exponential concentration inequalities and moderate and large deviations.
Now we will discuss several large deviations type results. We begin with the fol-lowing2.14.
Theorem.
There exist constants
C, C > , which depend only on the initialparameters, so that for any natural n there is a martingale { M ( n ) j = W ( n )1 + ... + W ( n ) j : j ≥ } whose differences W ( n ) j are bounded by C , and k S ,n u ( x ) − E [ S ,n u ( x )] − M ( n ) n k L ∞ ≤ C where x is a random member of E which is distributed according to µ . Moreover,for any t ≥ we have µ { x ∈ X : | S ,n u ( x ) − µ ( S ,n u ) | ≥ t + C } ≤ e − t nC . By taking t = εn we obtain estimates of the form µ { x ∈ X : | S ,n u ( x ) − µ ( S ,n u ) | ≥ εn } ≤ e − ε n C for any ε > n ≥ εn ≥ C .The following moderate deviations principle also holds true:2.15. Theorem.
Set σ n := var µ ( S ,n u ) and suppose that lim n →∞ σ n n + ε = ∞ for some < ε < . Let ( a n ) ∞ n =1 be a strictly increasing sequence of real numbersso that lim n →∞ a n = ∞ and lim n →∞ n − ε a n = 0 , and set W n = S ,n u − µ ( S ,n u ) σ n a n .Then for any Borel set Γ ⊂ R , − inf x ∈ Γ I ( x ) ≤ lim inf n →∞ a n µ { x : W n ( x ) ∈ Γ } and (2.16) lim sup n →∞ a n µ { x : W n ( x ) ∈ Γ } ≤ − inf x ∈ ¯Γ I ( x ) where I ( x ) = − x , Γ o is the interior of Γ and ¯Γ is its closer. imit theorems 11 The scaling sequence ( a n ) ∞ n =1 in theorem 2.15 is not optimal even when σ n growslinearly fast in n . In the following circumstances we can also derive more accuratemoderate deviations principle together with a local large deviations principle:2.16. Theorem.
Suppose that for some δ > for any z ∈ B (0 , δ ) the followinglimit Π( z ) = lim n →∞ n n − X j =0 Π j ( z ) exists. Then Π( z ) is analytic on z ∈ B (0 , δ ) and we have (2.17) Π( z ) = lim n →∞ n ln µ ( e zS ,n u ) = lim n →∞ n ln l n ( L ,nz ) where under Assumption 2.1 l n ( g ) = P L n i =1 g ( x i,n ) for any g : E n → C , and the x i,n ’s come from Assumption 2.1 (iii), while under either Assumption 2.2 or As-sumption 2.3 l n ( g ) = g ( x n ) for an arbitrary point x n . Moreover:(i) The limit p = lim n →∞ n Z S j,n ( x ) dµ j ( x ) exists, and it does not depend on j , and when µ j ( u j ) = 0 for each j the centralizedasymptotic moments (defined in Theorem 2.11) γ k = lim n →∞ γ j,k,n exist and theydo not depend on j . Furthermore, with σ = γ we have γ k = C k σ k for even k ’s,while for odd k ’s we have γ k = D k σ k − γ , where the C k ’s the the D k ’s are definedin Theorem 2.11. In addition, p = Π ′ (0) , γ = σ = Π ′′ (0) and γ = Π ′′′ (0) . In particular, in the circumstances of Theorem 2.9 (ii) we have σ > .(ii) Suppose that σ > . Then for any strictly increasing sequence ( b n ) ∞ n =1 ofreal numbers so that lim n →∞ b n n = 0 and lim n →∞ b n √ n = ∞ and a Borel set Γ ⊂ R (2.16) holds true with W n = S ,n u − µ ( S ,n u ) b n and I ( x ) = x σ .(iii) Let L ( t ) be the Legendre transform of Π( t ) . Then, (2.16) holds true for anyBorel set Γ ⊂ [Π ′ ( − δ ) , Π ′ ( δ )] with W n = S ,n u − µ ( S ,n u ) n and I ( t ) = L ( t ) . Note that Π ′ ( − δ ) < Π ′ ( δ ) when σ > t → Π( t ) isstrictly convex in some real neighbourhood of the origin. Theorem 2.16 (ii) is amoderate deviations principle (i.e. with quadratic rate function I ( x )) which allowsscaling sequences ( b n ) ∞ n =1 of optimal order, as can be viewed from Theorem 2.16(iii), which is a local large deviations principle. Note also that the equalityΠ( z ) = lim n →∞ n ln l n ( L ,nz )can be interpreted for real z ’s as a sequential analogy of the usual pressure functionof the potential f + zu = f j + u j , in the case of a single map T and functions f and u (see [8] in the subshift case). Remark also that, in fact, our proof shows that ifone of the limits in (2.17) exists, then all of them exists and they are equal. The limits Π( z ) does not seem to exist in general, not even for a single z . When T k , f k , u k are chosen at random according to some (not necessarily stationary) se-quence of random variables, we provide in Section 5 quite general conditions guaran-teeing that Assumption 2.13 holds true and that the limits Π( z ) exist. In particular,in the circumstances considered in Section 5, the limit σ = lim n →∞ n var µ j ( S j,n u )exist and it does not depend on j . In the circumstances of Theorem 2.9 (ii), wewill derive that σ >
0. Under certain circumstances, we will also show that (2.12)holds true, and then derive (2.13).3.
Sequential stability and non-singular maps
We will prove here Theorem 2.8. We will first recall some of the parts of theproof of Theorem 2.6. The crucial part of the proof is to show that the exists asequence of complex cones C j , containing the constant function , and a constant n so that for any n ≥ n and j ∈ Z L j,nz C j ⊂ C j + n . Under Assumption 2.1, the linear functional θ j ( g ) = L − j P L j i =1 g ( x i,j ) belongs tothe dual of the complex cone C j , while under Assumptions 2.2 and 2.3 the linearfunctions θ j ( g ) = g ( x j ) belongs to this dual, where x j is an arbitrary point in E j .Next, let K , T ,j , f ,j and u ,j be as in the statement of Theorem 2.8. Let ε > ν j,m and ν ,j,m the m -th derivative at z = 0 of the maps z → ν ( z ) j and z → ν ( z )1 ,j , respectively. Since k ν j ( z ) k and k ν ,j ( z ) k are both bounded by someconstant C , which does not depend on z and j , it follows from Lemma 2.8.2 in [17]that for any z ∈ ¯ B (0 , r ) and k ≥ (cid:13)(cid:13) ν j ( z ) − k X m =0 ν j,m m ! z m (cid:13)(cid:13) < ( k + 2) C − k − and the same inequality holds true with ν ,j in place of ν j . Let k = k ε be thesmallest positive integer so that ( k + 2) C − k − < ε . Then it is sufficient to showthat there exists δ > kL ( j ) z − L ( j )1 ,z k < δ for any j and z ∈ K then forany m = 0 , , ..., k ε we have(3.1) k ν j,m − ν ,j,m k < r − m k − ε ε. First, applying (2.6), we get that for any sufficiently large n , z ∈ U and j ∈ Z ,max (cid:16) k ν ( z ) j − F ( j, n, z ) k , k ν ( z )1 ,j − F ( j, n, z ) k (cid:17) ≤ C δ n where the linear functionals F ( j, n, z ) and F ( j, n, z ) are given by F ( j, n, z ) = θ n + j ( L j,nz ( · )) θ n + j ( L j,nz ) and F ( j, n, z ) = θ n + j ( L j,n ,z ( · )) θ n + j ( L j,n ,z ) . Here C and δ ∈ (0 ,
1) depend only on the initial parameters
B, L, α, γ, n , ξ and D and the linear functional θ n + j was defined at the beginning of this section. Notethat the denominators in the definition of F ( j, n, z ) and F ( j, n, z ) indeed do notvanish since L j,nz ( · )) and L j,n ,z ( · )) belong to C j + n . Let n = n ( ε ) be the smallest imit theorems 13 positive integer so that C δ n r − m m ! < ε for any 0 ≤ m ≤ k ε . Then by the Cauchyintegral formula, for any 0 ≤ m ≤ k ε we havemax (cid:16) k ν j,m − F ( m ) ( j, n , k , k ν ,j,m − F ( m )1 ( j, n , k (cid:17) ≤ r − m m ! C δ n < ε where F ( m ) ( j, n,
0) = d m F ( j,n,z ) z m (cid:12)(cid:12) z =0 and F ( m )1 ( j, n, z ) is defined similarly with F ( j, n, z ) in place of F ( j, n, z ). Observe next that L − j + n l j + n ( L j,n ) ≥ e −k S j,n f k ∞ ≥ e − Bn and the same inequality holds true with L j,n , in place of L j,n . Therefore, the de-nominators in F ( m ) ( j, n ,
0) and F ( m )1 ( j, n ,
0) are bounded from below by e − Bn m ,which depends only on ε and m . Using that for any families of operators A , ..., A n and B , ..., B n , we have A ◦ A ◦ · · · ◦ A n − B ◦ B ◦ · · · ◦ B n = n − X i =1 A ◦ · · · ◦ A i − ( A i − B i ) B i +1 ◦ · · · ◦ B n and that L j,n z are analytic in z and uniformly bounded in j and z ∈ K in theoperator norm k · k (by some constant which depends only on n = n ( ε ) and theinitial parameters), we find thatsup j ∈ Z sup z ∈ K kL ( j ) z − L ( j )1 ,z k < δ for some δ >
0, then for any j and m ≥ k F ( m ) ( j, n , − F ( m )1 ( j, n , k < C ( m, r, ε ) δ where C ( m, r, ε ) depends only on m, ε, r and the initial parameters. Taking asufficiently small δ completes the proof of the claim about the stability of { ν ( z ) j : j ∈ Z } (which was stated in Theorem 2.8). Since λ j ( z ) = ν ( z ) j +1 ( L ( j ) z ) and λ ,j ( z ) = ν ( z )1 ,j +1 ( L ( j )1 ,j ), we drive that for any ε > δ > j ∈ Z sup z ∈ K | λ j ( z ) − λ ,j ( z ) | < ε if sup j ∈ Z sup z ∈ K kL ( j ) z − L ( j )1 ,z k < δ . Finally, by (2.6) for any j ∈ Z , n ≥ z ∈ U we have (cid:13)(cid:13)(cid:13) h ( z ) j − L j − n,nz λ j − n,n ( z ) (cid:13)(cid:13)(cid:13) ≤ Aδ n and similar inequality holds true with h ( z )1 ,j , λ ,j − n,n ( z ) := n − Y i =0 λ j − n + i ( z )and L j − n,n ,z . Let n be so that Aδ n < ε . By taking fixing a sufficiently large n , we can also assume that | λ j − n ,n (0) | ≥ C kL j − n ,n k ≥ C e − Bn for someconstant C which does not depend on j . Using now the stability of λ j ( z ), for any q > m ≥ δ = δ ( q, m, n ) so that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d m L j ( z ) dz m (cid:12)(cid:12)(cid:12) z =0 − d m L ,j ( z ) dz m (cid:12)(cid:12)(cid:12) z =0 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < q if kL ( j ) z − L ( j )1 ,z k < δ for any integer j and z ∈ K , where L j ( z ) = L j − n ,n z λ j − n ,n ( z )and L ,j ( z ) is defined similarly but with L j − n ,n ,z and λ ,j − n ,n ( z ). Using this wecan approximate each one of the derivatives of h ( z ) j at z = 0 by the correspondingderivative of h ( z )1 ,j , which, as in the proof of the stability of { ν j ( z ) : j ∈ Z } , is enoughto drive the stability of { h ( z ) j : j ∈ Z } (for z ∈ K ).4. Limit theorems: proofs
First, since λ j ( · ) are analytic function and | λ j ( z ) | is bounded uniformly in j and z , and a ≤ λ j (0) ≤ b for some positive constants a and b which do not depend on j , it is indeed possible to construct functions Π j ( · ) which satisfies the conditionsstated in Theorem 2.11 (i).Next, set ˜ L ( j ) z ( g ) = L z ( gh (0) j ) λ j (0) h (0) j +1 . Then ˜ L ( j )0 = for any j and all the results stated in Theorem 2.6 and in the restof Section 2 can be applied with the ˜ L ( j ) z . Indeed we can just consider the triplets(4.1) ˜ λ j ( z ) = a j ( z ) λ j ( z ) a j +1 ( z ) λ j (0) , ˜ h ( z ) j = a j ( z ) h ( z ) j h (0) j and ˜ ν ( z ) j = ( a j ( z )) − h (0) j ν ( z ) j where a j ( z ) = ν ( z ) j ( h (0) j ) (which is nonzero since h (0) j ∈ C j ). Notice that ˜ ν (0) j = µ j ,that ˜ h (0) j ≡ λ j (0) = 1. Moreover, there exists a constant C so that for any j and z we have | a j ( z ) | ≤ c . Since a j (0) = 0 it follows that that exist constant a, b > z ∈ B (0 , a ) we have b ≤ | a j ( z ) | ≤ c . Therefore, there existpositive constants a c so that for any z ∈ B (0 , a ) and n ≥ (cid:12)(cid:12) n − X m =0 Π j + m ( z ) − n − X m =0 ˜Π j + m ( z ) (cid:12)(cid:12) = | ln a j ( z ) − ln a j + n ( z ) | ≤ c where ˜Π j ( z ) , j ∈ Z are analytic functions which are defined (simultaneously) insome neighborhood V of the origin, which depends only on the initial parameters,so that e ˜Π j ( z ) = ˜ λ j ( z ) for each z in V . By the Cauchy integral formula we obtainthat for any k there exists a constant E k so that (cid:12)(cid:12) n − X m =0 Π ( k ) j + m (0) − n − X m =0 ˜Π ( k ) j + m (0) (cid:12)(cid:12) ≤ E k . We conclude that in the proofs of Theorems 2.9, 2.10 and 2.11 we can assume that L ( j ) z = , λ j (0) = 1, h (0) j ≡ ν (0) j = µ j . imit theorems 15 The growth of the variance of S ,n u . We begin with the proof of Theorem2.9 (i). It is sufficient to prove this part in the case when µ s ( u s ) = 0 for any s , forotherwise we will just replace u s with u s − µ s ( u s ). First, if there exists a familyof functions { Y k : k ∈ Z } as in the statement of Theorem 2.9 (i), then clearlyvar µ k ( S k,n u ) is bounded in n , for each integer k . On the other hand, suppose thatlim inf n →∞ var µ ( S ,n u ) < ∞ . Since ( T j ) ∗ µ j = µ j +1 and S j +1 ,m u ◦ T j = S j,m +1 u − u j for any j and m ≥
1, itfollows from (2.7) that lim inf n →∞ var µ k ( S k,n u ) < ∞ for any integer k . Let L ( k ) ∞ be the closed linear subspace of L ( E k , µ k ) generatedby the functions u k , u k +1 ◦ T k , u k +2 ◦ T k , u k +3 ◦ T k , ... . We conclude that for any k there exists a subsequence of var µ k ( S k,n u ) which converges weakly in L ( E k , µ k ) toa member − Y k of L ( E k , µ k ). Using a diagonal argument we can assume that theindexes ( n m ) ∞ m =1 of the above subsequences do not depend on k .Next, for any ζ ∈ L ( E k , µ k ) we havelim n →∞ µ k ( S k,n m · ζ ) = − µ k ( Y k ζ ) . Therefore, since ˜ L ( k )0 is the dual of the map g → g ◦ T k with respect to the spaces L ( E k , µ k ) and L ( E k +1 , µ k +1 ), we have µ k ( Y k +1 ◦ T k · ζ ) − µ k ( Y k · ζ ) = µ k +1 ( Y k +1 · ˜ L ( k )0 ζ ) − µ k ( Y k · ζ )= − lim m →∞ (cid:0) µ k +1 ( S k +1 ,n m u · ˜ L ( k )0 ζ ) − µ k ( S k,n m u · ζ ) (cid:1) = − lim m →∞ (cid:0) µ k ( S k +1 ,n m u ◦ T k · ζ ) − µ k ( S k,n m u · ζ ) (cid:1) = µ k ( u k · ζ ) − lim m →∞ µ k ( u k + n m ◦ T n m k · ζ )where we used that S k +1 ,n m u ◦ T k = S k,n m +1 u − u k . Since ζ ∈ L ( k ) ∞ and µ s ( u s ) = 0for any s , it follows from (2.7) thatlim m →∞ µ k ( u k + n m ◦ T n m k · ζ ) = 0and hence Y k +1 ◦ T k − Y k = u k as members of L ( E k , µ k ). Now we will show that Y k can be chosen to be H¨oldercontinuous so that the norms k Y k k are bounded in k . Let the function W k be givenby W k = ∞ X j =1 ˜ L k − j,j u k − j . This function is we defined, and is a member of H k because of (2.6), and ourassumption that µ j ( u j ) = 0 for any j . In fact, the exponential convergence (2.6)implies that k W k k is bounded in k . Moreover, for each k we have W k − ˜ L ( k − W k − = ˜ L ( k − u k − . Next, substituting both sides of the equality Y k ◦ T k − − Y k − = u k into ˜ L ( k − , itfollows that Y k − ˜ L ( k − Y k − = ˜ L ( k − u k − . Since the functions { W k : k ∈ Z } also satisfy the above relations, the family offunctions { d k : k ∈ Z } given by d k = Y k − W k satisfies that for any k ∈ Z , d k +1 = ˜ L ( k )0 d k , µ k − a.s. . Since ˜ L ( k )0 is the dual of g → g ◦ T k with respect to the spaces L ( E k , µ k ) and L ( E k +1 , µ k +1 ), ˜ L ( k )0 ( v ◦ T k ) = v for any v : E k +1 → C and ( T k ) ∗ µ k = µ k +1 , itfollows that for any integer k , d k +1 ◦ T k = d k or, equivalently, since Y k +1 ◦ T k = Y k + u k , Y k − W k = d k = Y k +1 ◦ T k − W k +1 ◦ T k = Y k + u k − W k +1 ◦ T k . Therefore, u k = W k +1 ◦ T k − W k . This equality holds true µ k -almost surely, butsince both sides are continuous and µ k assigns positive mass to open sets we derivethat it holds true for any point in E k . The proof of Theorem 2.9 (i) is complete. (cid:3) Now we will prove Theorem 2.9 (ii). By Theorem 2.11 there exists a constant R so that (cid:12)(cid:12) n var µ ( S ,n u ) − n n − X j =0 λ ′′ j + n (0) (cid:12)(cid:12) ≤ R n where we used that λ ′′ j (0) = Π ′′ j (0) (recall our assumption that µ j ( u j ) = 0). Let T, f, u satisfy the conditions stated in Theorem 2.9 (i), and let ( λ ( z ) , h ( z ) , ν ( z ) ) bethe RPF triplet corresponding to the operators L z given by L z g ( x ) = X y ∈ T − { x } e f ( y )+ zu ( y ) g ( y ) . Then (see [25]), since u does not admit a co-boundary representation, σ = λ ′′ (0) = lim n →∞ n var µ ( S n u ) > . Note that σ = λ ′′ (0) ≥ c > c which depends only on the initialparameters (as Theorem 2.6 holds true also when T j , f j and u j do not depend on j ). Let ε > δ as in Theorem 2.8. Then | λ j ( z ) − λ ( z ) | ≤ ε for any j , andso by the Cauchy integral formula, | λ ′′ j (0) − λ ′′ (0) | = | λ ′′ j (0) − σ | ≤ cε for some constant c which depends only on the initial parameters. We concludethat (cid:12)(cid:12)(cid:12) n n − X j =0 λ ′′ j + n (0) − σ (cid:12)(cid:12)(cid:12) ≤ cε and therefore when cε < σ we obtain that1 n var µ ( S ,n u ) ≥ σ − cε − R n which completes the proof of Theorem 2.9 (ii). (cid:3) imit theorems 17 Self normalized Berry-Essen theorem.
Set σ ,n = p var µ ( S ,n u ), andsuppose that lim n →∞ σ ,n n − = ∞ . By replacing u j with u j − µ j ( u j ), we can assume without a loss of generality that µ j ( u j ) = 0, which implies that µ j ( S ,n ) = 0 for any n ≥
1, since ( T j ) ∗ µ j = µ j +1 for any j . By Theorem 2.11,(4.3) (cid:12)(cid:12) var µ ( S ,n u ) − n − X j =0 Π ′′ j (0) (cid:12)(cid:12) ≤ R for some constant R , and therefore,(4.4) lim n →∞ n − n − X j =0 Π ′′ j (0) = ∞ . Next, since λ j (0) = 1 and µ j = ν (0) j for any n ≥ j we have ( L ,n ) ∗ µ n = µ and therefore for any z ∈ C ,(4.5) µ ( e zS ,n u ) = µ n ( L ,n e zS ,n ) = µ n ( L ,nz ) . Consider now the analytic function ϕ ,n : U → C given by(4.6) ϕ ,n ( z ) = µ n ( L ,nz ) λ ,n ( z ) = Z L ,nz ( x ) λ ,n ( z ) dµ n ( x )where U is the neighborhood of 0 specified in Theorem 2.6. Then by (4.5) for any z ∈ U , j ∈ Z and n ≥ µ j ( e zS ,n u ) = λ ,n ( z ) ϕ ,n ( z ) = e P n − j =0 Π j ( z ) ϕ ,n ( z ) . Next, by Theorem 2.11 (ii) we have λ ′ ,n (0) = P n − j =0 Π ′ j (0) = E µ j S j,n u , and there-fore by (4.6),(4.8) ϕ ′ ,n (0) = 0 . Now, by taking a ball which is contained in the neighborhood U specified inTheorem 2.6, we can always assume that U = B (0 , r ) is a ball around 0 withradius r >
0. We claim that there exists a constant
A > n ∈ N and z ∈ U ,(4.9) | ϕ ,n ( z ) | ≤ A. Indeed, by (2.6), there exist constants A , k > c ∈ (0 ,
1) such that for any z ∈ U and n ≥ k ,(4.10) (cid:13)(cid:13) L ,nz λ ,n ( z ) − h n ( z ) (cid:13)(cid:13) ≤ A δ n By Theorem 2.6, for any n ≥ j ∈ Z and z ∈ U we have k h n ( z ) k ≤ C , and (4.9)follows.Next, applying Lemma 2.8.2 in [17] with k = 1 we deduce from (4.8) and (4.9)that there exists a constant B > | ϕ ,n ( z ) − ϕ ,n (0) | = | ϕ ,n ( z ) − | ≤ B | z |
28 Y. Hafouta for any z ∈ U = B (0 , r ) = U . Moreover, using (4.3) and the above Lemma2.8.2 there exist constants t , c > s ∈ [ − t , t ] and a sufficientlylarge n ,(4.12) (cid:12)(cid:12)(cid:12) n − X j =0 Π j ( is ) − s σ ,n (cid:12)(cid:12)(cid:12) ≤ c | s | n + 12 R s where we also used that P n − j =0 Π ′ j (0) = E µ S ,n u = 0 and that | Π j ( z ) | ≤ C forsome C which does not depend on j and z . Set s n = σ ,n n c (which is bounded in n ). Then, by (4.12), there exist constants q, q > q s n ≤ min( t , r ) andthat for any sufficiently large n and s ∈ [ − q s n , q s n ],(4.13) ℜ (cid:16) n − X j =0 Π j ( is ) (cid:17) ≤ − qs σ ,n . Next, by the Esseen-inequality (see [37]) for any two distribution functions F : R → [0 ,
1] and F : R → [0 ,
1] with characteristic functions ψ , ψ , respectively, and T > x ∈ R | F ( x ) − F ( x ) | ≤ π Z T (cid:12)(cid:12) ψ ( t ) − ψ ( t ) t (cid:12)(cid:12) dt + 24 πT sup x ∈ R | F ′ ( x ) | assuming that F is a function with a bounded first derivative. Set T n = q σ ,n c n ,which converges to ∞ as n → ∞ . For any real t set t n = σ − ,n t . Let t ∈ [ − T n , T n ].Then by(4.7), | µ ( e itσ − ,n S ,n u ) − e − t | ≤ e P n − j =0 ℜ (Π j ( it n )) | ϕ ,n ( it n ) − | (4.15) + | e P n − j =0 ℜ (Π j ( it n )) − e − t | := I ( n, t ) + I ( n, t ) . By (4.13) we have e P n − j =0 ℜ (Π j ( it n )) ≤ e − qt and therefore by (4.11), I ( n, t ) ≤ B e − qt t σ − ,n . Using the mean value theorem, together with (4.12) applied with s = t n , takinginto account that e P n − j =0 ℜ Π j ( it n ) ≤ e − qt , we derive that I ( n, t ) ≤ c nσ − ,n ( | t | + t ) e − c t for some constants c , c >
0. Let F be the distribution function of S ,n u ( x ),where x is distributed according to µ , and let F be the standard normal distri-bution. Since the functions | t | e − qt and ( | t | + t ) e − c t are integrable, applying(4.14) with these functions and the above T = T n we complete the proof of Theorem2.10, taking into account that σ − ,n converges faster to 0 than T n . (cid:3) Local limit theorem and Edgeworth expansion of order three.
Theo-rem 2.12 follows from the arguments in the proof of Theorem 2.2.3 in [17], whichprovides conditions for a central local limit theorem to hold true in the case when theasymptotic variance lim n →∞ n − var( S ,n u ) exists, and it is positive (using Fouriertransforms). Still, when there exist constants c , c > n , var( S ,n u ) ≥ c n imit theorems 19 then, by replacing any appearance of σ √ n with var( S ,n u ), we derive that in orderto prove Theorem 2.12 it is sufficient to show that there exists constants a, b > δ ∈ (0 ,
1) so that for any n ≥ t ∈ [ − δ, δ ],(4.16) | µ ( e itS ,n u ) | ≤ ae − bnt . When σ ,n grows linearly fast in n then by (4.12), there exists a constant t > t ∈ [ − t , t ] and a sufficiently large n we have(4.17) n − X j =0 ℜ (Π j ( it )) ≤ − qns which together with (4.7) yields (4.16).Now we will show that Assumption 2.13 implies that (2.14) holds true. Indeed,using the spectral methods in [23], it follows that for any J ⊂ I h and s ≥ t ∈ J k L sit k ≤ cr s for some c = c ( J ) and r = r ( J ) ∈ (0 ,
1) which may depend on J but not on s .Under our assumptions we have kL j,nit k ≤ B − γ − ) ≤ B J , where B J is somenumber which depends only on J . Under Assumptions 2.1 and 2.2 such a constantexists due to Lemma 5.6.1 in [17], while under Assumption 2.3 it exists in view of(5.3) in [22]. Write { ≤ m : B J kL m,sm it − L sit k α < − δ ∀ t ∈ J } = { m < m < .... } . for some strictly increasing infinite sequence of nonnegative integers ( m i ) ∞ i =1 . Fixsome s so that B J cr s < δ , and for each k set i k = sm k . Then m i k + sm ≤ m i k +1 and by (2.15) we have lim n →∞ k n ln n = ∞ where k n = max { k : m i k ≤ n − m s } . Set l k = m i k and write: L ,nit = L l kn + sm ,n − l kn − sm it ◦L l kn ,sm it ◦· · ·◦L l ,sm it ◦L l + sm ,l − l − sm it ◦L l ,sm it ◦L ,l it . The blocks of the form L l i ,sm it satisfy kL l i ,sm it k α < B − J (1 − δ ) + cr s < B − J (1 − δ )and the norm of the other block does not exceed B J . Therefore, for any n ≥ t ∈ J kL ,nit k ≤ (cid:0) − δ (cid:1) k n and since µ ( e itS ,n u ) = µ n ( L ,nit ) and k n grows faster than logarithmically in n ,we conclude that (2.14) holds true.Next, when (2.14) holds true, then the following theorem is proved exactly as in[20]:4.1. Theorem.
There exists a sequence of polynomials P j,n ( s ) = m X k =0 a j,n,k s j , n ≥ with random coefficients, whose degree m does not depend on j and n , so that forany n ≥ with Π j,n, = n P n − m =0 Π ′′ j + m (0) > we have sup s ∈ R (cid:12)(cid:12)(cid:12) √ πµ j { x ∈ E j : S j,n ( x ) ≤ √ ns } − p Π j,n, Z s −∞ e − t j,n, dt − n − P j,n ( s ) e − s (cid:12)(cid:12)(cid:12) = o ( n − ) . In the case when T j , f j and u j do not depend on j , such (and high-order) ex-pansions were obtained in [5] and [30]. In [20] we obtained high order Edgeworthexpansions for random dynamical systems, and, in concrete examples, we havemanaged to show that the additional required condition (Assumption 2.4 there)holds true, using some sub-additivity arguments (relying on Kingman’s theorem),together with the arguments in [3], which is impossible to generalize for generalsequential dynamical systems (as there is no subadditivity of any kind). Thereforewe will not formulate here an appropriate theorem, even though it is possible toobtain high-order Edgeworth expansions under appropriate version of the latterAssumption 2.4 (plus additional assumptions in the spirit of Assumption 2.13).4.4. Martingale approximation, exponential concentration inequalitiesand moderate deviations.
As we have explained at the beginning of Section4, we can assume in course of the proof of Theorem 2.14 that L ( j )0 = for each j ,which means that λ j (0) = 1 and h (0) j ≡ . Moreover, we can clearly assume that µ j ( u j ) = 0 for each j . Fix n ≥
1, and let x be a random member of E which isdistributed according to µ . Consider the random variables X , X , ..., X n given by X m = X ( n ) m = u n − m ( T n − m x ) . For each m , consider the E m -valued random variable Z m = T m x . Then Z m +1 is afunction of Z m and so the family of σ -algebras F k = F ( n ) k , k = 1 , , ..., n , which isgiven by F k = σ { Z n , Z n − , ..., Z n − k } = σ { Z n − k } is increasing in k . For the sake of convenience, set F m = F n when m > n . Next,we claim that. E [ X k |F k − l ] = L n − k,l ( u n − k ) ◦ T n − k + l . Indeed, since for any j we have ( L ( j )0 ) ∗ µ j +1 = µ j and ( T j ) ∗ µ j = µ j +1 , for anymeasurable and bounded function g : E n − k + l → R we have E [ g ( Z n − k + l ) L n − k,l ( u n − k ) ◦ T n − k + l ] = Z g ( T n − k + l x ) (cid:0) L n − k,l ( u n − k ) ◦ T n − k + l (cid:1) ( x ) dµ ( x )= Z g ( x ) (cid:0) L n − k,l ( u n − k ) (cid:1) ( x ) dµ n − k + l ( x ) = µ n − k + l (cid:0) L n − k,l ( g ◦ T ln − k · u n − k ) (cid:1) = µ n − k ( g ◦ T ln − k · u n − k ) = µ ( g ◦ T n − k + l · u n − k ◦ T n − k ) = E [ g ( Z n − k + l ) X k ] . By (2.6) and since µ j ( u j ) = 0 we derive that for each 0 ≤ l ≤ k ≤ n we have k E [ X k |F k − l ] k L ∞ ≤ C δ l imit theorems 21 for some constants C > δ ∈ (0 , X m = 0 for any m > n and then for any j ≥ W j = X j + X s ≥ j +1 E [ X s |F j +1 ] − X s ≥ j E [ X s |F j ] . Then { W j : j ≥ } is a martingale difference with respect to the filtration {F j : j ≥ } , whose differences are bounded by some constant C >
0. Observe that k S ,n u ( x ) − n X j =1 W j k L ∞ = k n X j =1 X j − n X j =1 W j k L ∞ ≤ C for some other constant C . Set M n = P nj =1 W j . Let (Ω , F , P ) be a probabilitymeasure so large so that all the random variables defined above are defined on(Ω , F , P ), and denote by E P with expectation with respect to P . Then by theHoeffding-Azuma inequality (see [31]), for any λ > (cid:0) E P [ e λM n ] , E P [ e − λM n ] (cid:1) ≤ e λ nC and so, by the Markov inequality, for any t ≥ µ { x : | S ,n u ( x ) | ≥ C + t ) = P ( | n X j =1 X j | ≥ C + t ) ≤ P ( | M n | ≥ t ) ≤ P ( M n ≥ t ) + P ( − M n ≥ t ) ≤ P ( e λ t M n ≥ e tλ t ) + P ( e − λ t M n ≥ e tλ t ) ≤ e − tλ t + λ t nC = 2 e − t Cn where λ t = t Cn , which together with the previous estimates completes the proofof Theorem 2.14.4.4.1. Moderate deviations theorems via the method of cumulants . Relyingon (2.7) and using that ( T j ) ∗ µ j = µ j +1 , we derive the following multiple correlationestimate holds: for any s and functions f i : E j + m i , where i = 0 , , ..., s and 0 ≤ m < m < m < ... < m s ,(4.19) (cid:12)(cid:12)(cid:12) µ j ( s Y i =0 f i ◦ T m i j ) − s Y j =0 µ j + m i ( f i ) (cid:12)(cid:12)(cid:12) ≤ dM s s X i =1 δ m i − m i − where d is some constant and M = max {k f i k α : 0 ≤ i ≤ s } .Recall next that the k -th cumulant of a (bounded) random variable W is givenby Γ k ( W ) = 1 i k d k dt k (cid:0) ln E e itW (cid:1)(cid:12)(cid:12) t =0 . Relying on 4.19 we can apply Lemma 14 in [11], we derive that(4.20) | Γ k ( S j,n u ( x j ) − µ j ( S j,n u )) | ≤ n ( k !) c k where x j is distributed according to µ j and c is some constant. Suppose thatlim n →∞ σ ,n n + ε = ∞ for some 0 < ε < , where σ ,n := var µ ( S ,n u ). Then, Theorem 2.15 follows fromTheorem 1.1 in [10], applied with Z n = S j,n u x j − µ j ( S j,n u ) σ ,n , ∆ n = n − ε and γ = 2,taking into account that for any 0 < ε < and k ≥ n − kε ≤ n − ε ( k − .Note that several other types of moderate deviations type results follow from the above estimates of the cumulants, see [10]. Remark also that by Corollary 2.1 in[38] for any n ≥ (cid:12)(cid:12)(cid:12) µ { x ∈ E : S ,n u ( x ) − µ ( S ,n u ) ≤ rσ ,n } − √ π Z r −∞ e − t dt (cid:12)(cid:12)(cid:12) ≤ cn − ε for some constant c which depends only on the initial parameters. This providesanother proof of the CLT when the variances grow sufficiently fast (note: the rate n − ε is not optimal).4.5. Logarithmic moment generating functions.
In this section we will proveTheorem 2.16. Suppose that the limitsΠ( z ) = lim n →∞ n n − X j =0 Π j ( z )exist in some open disk B (0 , δ ) around 0 in the complex plane. Then Π( z ) isanalytic in z since it is a pointwise limit of a sequence of analytic functions whichis uniformly bounded in n (such limits are indeed analytic, as a consequence of theCauchy integral formula). Next, we claim that we can construct a branch of thelogarithm of µ ( e zS ,n u ) on B (0 , δ ) so that for any z ∈ B (0 , δ ),(4.22) Π( z ) = lim n →∞ n ln µ ( e zS ,n u ) . In view of (4.2), we can prove the above in the case when L ( j ) z = . In this case,we have µ ( e zS ,n u ) = µ n ( L ,n e zS ,n ) = µ n ( L ,nz )and so by (2.6) we have lim n →∞ (cid:12)(cid:12)(cid:12) µ ( e zS ,n u ) λ ,n ( z ) − µ n ( h ( z ) n ) (cid:12)(cid:12)(cid:12) = 0 . Since h (0) n = 1 and the norms k h n ( z ) k α are uniformly bounded in n and z , thereexist positive constants δ , c and c so that for any z ∈ B (0 , δ ) and n ∈ N wehave c ≤ | µ n ( h ( z ) n ) | ≤ c , which implies that for any sufficiently large n , C ≤ (cid:12)(cid:12)(cid:12) µ ( e zS ,n u ) λ ,n ( z ) (cid:12)(cid:12)(cid:12) ≤ C where C and C are some positive constants. Therefore, a branch of the logarithmof µ ( e zS ,n u ) can be defined so that (4.22) holds true. Note that we also used that P n − j =0 Π j ( z ) is a branch of λ ,j ( z ).In order to prove that Π( z ) = lim n →∞ n ln l n ( L ,nz )we first observe that since λ j = ν ( z ) j +1 ( L ( j ) z ), then by (4.3.26) in [17] we have | λ j ( z ) − g j,n ( z ) | ≤ Cδ n where g j,n ( z, j ) = l n + j +1 ( L j,n +1 z ) l n + j +1 ( L j +1 ,nz ) imit theorems 23 and l j + n was defined in the statement of Theorem 2.16. We also used that thenorms kL ( j ) z k are bounded in j and z ∈ U . Replacing n with n − j − (cid:12)(cid:12)(cid:12) n − X j =0 Π j ( z ) − n − X j =0 ln g j,n − j − ( z, j ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) n − X j =0 Π j ( z ) − (ln l n ( L ,nz ) − ln l ( L (0) z ) (cid:12)(cid:12)(cid:12) is bounded in n .All the statements from Theorem 2.16 (i) follow from Theorem 2.11. The state-ments in parts (ii) and (iii) of Theorem 2.16 follow from the Gartner-Ellis theorem(see [9]), as noted in [43].5. Applicatio to non-stationary random environemnts
The case of independent and identically distributed maps T j has been widelystudied, and extension to stationary maps T j were then considered. From thispoint of view it was natural to ask about limit theorems for sequences of mapwhich are not random, but naturally the results that can be obtained are limited,for instance, one could not expect to have a local central limit theorem (withoutsome kind of normalization) in such a generality. This leads us to consider randommaps which are not stationary. In this setup, even the case when the maps areindependent but not identically distributed was not studied. Even in this simplecase one does not have an underlying skew product map, so the system can not behandled using a single operators as was done in [1] for iid maps.5.1. Random non-stationary mixing environmens: a local CLT.
Let { ξ n : n ∈ Z } be a family of random variables taking values at some measurable space Y ,which are defined on the same probability space (Ω , F , P ). Let ( X , d ) be a compactmetric space and let E ⊂ Y × X a set measurable with respect to the product σ -algebra, so that the fibers E y = { x ∈ X : ( y, x ) ∈ E} , ω ∈ Ω are compact.The latter yields (see [7] Chapter III) that the mapping y → E y is measurable withrespect to the Borel σ -algebra induced by the Hausdorff topology on the space K ( X )of compact subspaces of X and the distance function dist( x, E y ) is measurable in y for each x ∈ X . Furthermore, the projection map π Y ( y, x ) = y is measurable and itmaps any F ×B -measurable set to a F -measurable set (see “measurable projection”Theorem III.23 in [7]). For each y ∈ Y , let f y , u y : E y → R be functions so that thenorms k f y k α and k u y k α are bounded in y . Let T y be a family of maps from E y to X so that P -a.s. we have(5.1) T ξ j : E ξ j → E ξ j +1 and the family { T j = T ξ j : j ∈ Z } satisfies all the conditions specified in Section2 (with non-random constants) with the spaces E j = E ξ j . Note that for (5.1) tohold true, we can just assume that E y = X for each y . We also assume here thatthe maps ( y, x ) → f y ( x ) , u y ( x ) and ( y, x ) → T y ( x ) are measurable with respect tothe σ -algebra induced on { ( y, x ) : y ∈ Y , x ∈ E y } from the product σ -algebra on Y × X . Then by Lemma 5.1.3 in [17], the norms k f y k and k u y k are measurablefunctions of y . Let the transfer operators L ( y ) z which maps functions g on E y tofunctions on T y ( E y ) by the formula L ( y ) z g ( x ) = X a ∈ T − y { x } e f y ( a )+ zu y ( a ) g ( a ) Under Assumption 2.1, by Lemma 4.11 in [32] there exist L ( y ) and x i,y = x i,y which are measurable y , so that E y = L ( y ) [ j =1 B y ( x i,y , ξ )where B y ( x, ξ ) is an open open around x ∈ E y with radius ξ . In the above circum-stance we also assume that L ( y ) is bounded in y (e.g. when E y = X ). Consider therandom operators L ( j ) z given by L ( j ) z = L ( ξ j ) z . We remark that the the RPF triplets λ j ( z ), h ( z ) j and ν ( z ) j from Theorem 2.6 are measurable in ω (in view of the limitingexpressions of λ j ( z ), h ( z ) j and ν ( z ) j from Chapter 4 of [17]).In the rest of this section, we will impose restrictions on the process { ξ j } , whichwill guarantee that the local central limit theorem described in 2.12 holds true,where it is sufficient to to derive that Assumption 2.13 holds true. In order toachieve that, we will rely on the following5.1. Assumption.
The sequence { ξ j : j ∈ Z } satisfies the following φ -mixing typecondition: there exists a sequence φ ( n ) , n ≥ P ∞ n =1 φ ( n ) < ∞ and for any j ∈ Z , n ≥ A ∈ σ { ξ m : m ≤ j } and B ∈ σ { ξ m : m ≥ j + n } ,(5.2) | P ( A ∩ B ) − P ( A ) P ( B ) | ≤ P ( A ) φ ( n )where σ { X i : i ∈ I} is the σ -algebra generated by a family of random variables { X i : i ∈ I} .(ii) There exist points y , y , ..., y m ∈ Y so that for any sufficiently large s ∈ N and all sufficiently small open neighborhoods U i of y i , i = 1 , , ..., m and s ≥ n →∞ P nm =1 P (cid:8) ξ msm + i ∈ U i ; ∀ ≤ i ≤ m s }√ n ln n = ∞ where we set U i = U k i for any i > m , if i = m k i + r some 0 ≤ r < m .Under Assumption 5.1, set L z = L ( y m ) z ◦ · · · ◦ L ( y ) z ◦ L ( y ) z . Theorem.
Suppose that Assumption 5.1 holds true and that the following twoconditions hold true:(A1) For any compact J ⊂ R , the family of maps { y → L ( y ) it } , where t over J ,is equicontinuous at the points y , y , ..., y m , and E y does not depend on y , when y lies in some open neighborhood of one of the y i ’s.(A2) The spectral radius of L it is strictly less than for any t ∈ I h , where in thenon-lattice case we set I h = R \ { } , while in the lattice case we set I j = ( − πh , πh ) .Then Assumption 2.13 holds true. After showing that Assumption 2.13 holds true we can apply Theorem 2.12 with T j = T ξ j , f j = f ξ j and u j = u ξ j when, P -a.s. the variance var µ ( S ,n u ) growslinearly fast in n , which holds true when E y = X and kL ( y ) z − L z k α < ε for any y ∈ Y and complex z in some neighborhood of 0, where L z and ε are specified inTheorem 2.9 (ii). Proof of Theorem 5.2.
Let J ⊂ be a compact set and let B J > kL j,nit k /λ j,n (0) ≤ B J . Let s ∈ N and δ ∈ (0 , imit theorems 25 of our Assumption (A1), there exist open neighborhoods U j of y j , j = 1 , , ..., m so that kL ( y ) it − L ( y j ) it k < ε for any t ∈ J , where ε satisfies that sm B J ε < − δ .Using that for any families of operators A , ..., A m and B , ..., B m we have A ◦ A ◦ · · · ◦ A m − B ◦ B ◦ · · · ◦ B m = m X j =1 A ◦ A ◦ · · · ◦ A j − ( A j − B j ) B j +1 ◦ B j +2 ◦ · · · ◦ B m we obtain that for any t ∈ J and j ∈ Z so that ξ j + k ∈ U k for any 1 ≤ k ≤ m s , wehave(5.4) B J kL j +1 ,m sit − L sit k α ≤ sm B J ε < − δ where L j,m sit = L ( ξ j + m s ) it ◦ · · · ◦ L ( ξ j +2 ) it ◦ L ( ξ j +1 ) it . Next, set Γ m = { ω : ξ msm + i ( ω ) ∈ U i ; ∀ ≤ i ≤ m s } and S n = [ n − sm sm ] X m =1 I Γ m where I Γ is the indicator of a set Γ. Then S n does not exceed the number of j ’sbetween 1 and n so that (5.4) holds true. In the above circumstances, we can useTheorem 2.4 in [18] with the random vectors { I Γ m : 1 ≤ j ≤ sm } , m = 1 , , ... and derive from (2.11) there that for any t ≥ P {| S n − E S n | ≥ t + c } ≤ e − c t n where c and c are positive constants which may depend only on m and s . Taking t of the form t = t n = θ √ n ln n for an appropriate θ , we derive from the Borel-Cantelli lemma that P -a.s. for any sufficiently large n we have | S n − E S n | ≤ t n + c, P − a.s.which together with (5.3) yields thatlim n →∞ S n √ n ln n = ∞ and so (2.15) holds true P -a.s. with the operators L ( j ) it , where, in fact, in ourcircumstances the numerator inside the limit expression in (2.15) grows faster than √ n ln n . (cid:3) Examples.
We will provide here several examples in which Assumption 5.1is satisfied. First, when considering the simple case when Y = Z , then for any U ⊂ ( − ,
1) which contains 0 we have n X m =1 P (cid:8) ξ mm s + i ∈ U i ; ∀ ≤ i ≤ m s } = n X m =1 P { ξ mm s + i = y i ; ∀ ≤ i ≤ m s } where we set y i = y r if i = m k + r for some integers k ≥ ≤ r < m . When ξ i ’s form an inhomogenious Markov chain (e.g. when they are independent) with n -th step transition probabilities p ( i,n ) j,k = P ( ξ i + n = k | ξ i = j ) (so that (5.2) holdstrue), then we only require that n X m =1 m s Y i =1 P ( ξ mm s = y ) p ( mm s + i, y i ,y i +1 grows faster than √ n ln n in n , for any s . For instance, we could require that p ( j, y i ,y i +1 ≥ δ j and P ( ξ j = y ) ≥ r j for some δ j , r j > i ’s, which will give uslinear growth if δ i ’s and r j ’s are bounded from below by some positive constant,while in general we can impose certain restrictions on the δ i and r j ’s to obtain thedesired growth rate (e.g when δ j is at least of logarithmic order in j and r j decayssufficiently slow to 0 as j → ∞ ). Note that in these circumstances condition (i)from Theorem 5.2 trivially holds true since L ( y ) z are locally constant in y around y , ..., y m .A close but more general situation is the case when the maps y → T y , f y , u y arelocally constant around y , y , ..., y m and { ξ j } is an inhomogeneous Markov chainso that, P ( ξ i +1 ∈ U | ξ i = x ) = Z U p i ( x, y ) dη ( y ) , ∀ x ∈ V for any sufficiently small neighborhoods U and V some y i and y j , respectively. Here η is some probability measure on Y which assigns positive mass to open sets, and p i ( x, y ) are functions which are bounded from below by some positive constants δ i .In this case we have n X m =1 P (cid:8) ξ mm s + i ∈ U i ; ∀ ≤ i ≤ m s } ≥ (cid:0) m Y j =1 η ( U j ) (cid:1) s n X m =1 m s Y i =1 δ mm s + i . Imposing some restrictions on the δ i ’s we will get that the above right handside grows faster than √ n ln n . For instance, when δ j ≤ c ln j then, for any s , Q m si =1 δ mm s + i is of order θ m in m , where θ = m s , and so n X m =1 m s Y i =1 δ mm s + i is at least of order n ln θ n in n . When Y is compact and the densities p i ( x, y ) arebounded from below and above by some positive constant then condition (5.2) holdstrue with φ ( n ) of the form φ ( n ) = ae − nb for some a, b > δ j ’s are bounded from below. Still, relying on (5.2) our arguments allow thatlim inf i →∞ δ i = 0’s, as i → ∞ , as described above.Finally, let ( Q , T, m ) be a mixing Young tower (see [41] and [42]) whose tails ν { R ≥ n } are of order n − a for some a > ν is the original measure on thetower, R is the return time function and m is the invariant mixing probability mea-sure). Consider the case when ξ j ( q ) = H j ( T j q ), where q is distributed accordingto m and H j is a function that is constant on elements of the partition definingthe tower. Then by (7.6) in [21] the inequality (5.2) holds true with a summuablesequence φ ( n ) if the tails of the tower decay polynomially and sufficiently fast,and the function H j is measurable with respect to the partition which defines thetower. Next, suppose that T has a periodic point q with period m and that H j ( T m q ) := y m does not depend on j for each m = 0 , , ..., m −
1. We will show imit theorems 27 now that (5.3) holds true with the above y i ’s. Let U i be a neighborhood of y i , where i = 0 , , ...m −
1, and let s ≥
1. Observe that ( ξ msm + i ( q )) sm i =1 takes the value( y , y , ..., y ) ⊗ s when q lies in a set of the form T − msm A s , for some open neigh-borhood A s of the periodic point q . Here the power s stands for concatenation: a ⊗ s = aaa...a . Since m is T -invariant we derive that n X m =1 P (cid:8) ξ msm + i ∈ U i ; ∀ ≤ i ≤ m s } ≥ n m ( A s )and so (5.3) holds true (as m ( A s ) > H j ’s are H¨older continuous uniformly in j can be considered, since then we canapproximate (in the proof of Theorem 5.2) the H j ’s by functions which are constanton the above partitions, and use again (7.6) in [21] .We can also consider the case when each ξ j depends only the first coordinate inthe following model: Non-stationary subshifts of finite type . Let d j , j ∈ Z be a family of positiveintegers so that d j ≤ d for some d ∈ N and all j ’s. Let A j = A j ( a, b ) be a family ofmatrices of sizes d j × d j +1 whose entries are either 0 or 1, so that all the entries of A j + n · · · A j + n − · A j +1 are positive, for some n ≥ j ’s. Let the compactspace X be given by X = { , , ..., d } N ∪{ } and for each j , and let d ( x, y ) = 2 − min { n ≥ x n = y n } . For each integer j set E j = { ( x j + m ) ∞ m =0 ∈ X : x j + m ≤ d j + m and A j + m ( x j + m , x j + m +1 ) = 1 , ∀ m ≥ } and define T j : E j → E j +1 by T j ( x j , x j +1 , x j +2 , .. ) = ( x j +1 , x j +2 , ... ) . We also set γ = 2 and ξ = 1, so the inequality d ( x, y ) < ξ means that x = y . Inthis nonstationary subshift case we have the following result:5.3. Theorem. (i) There exist constants C , C > so that fir any symbols a j , ..., a j + r , where j ∈ Z and r ≥ we have C ≤ µ j { ( x m ) ∞ m = j : x i = a i ∀ ∈ [ j, j + r ] } /e S nj φ ( a ( j ) ) − ln λ j,r (0) ≤ C where a ( j ) ∈ X j is any point so that a ( j ) i = a i for any j ≤ i ≤ j + r .(ii) There exist constants C > and δ ∈ (0 , so that for any integer j , r, s, n ≥ , symbols a j , ..., a j + r and b j + r + n , ..., b j + r + n + s and cylinder sets A = { ( x m ) ∞ m = j : x i = a i ∀ ∈ [ j, j + r ] } ⊂ E j and B = { ( x m ) ∞ m = j : x i = b i ∀ ≤ i ∈ [ j + r + n, j + r + n + s ] } ⊂ E j we have (5.5) | µ j ( A ∩ B ) − µ j ( A ) µ j ( B ) | ≤ Cµ j ( A ) µ j ( B ) δ n . Namely, the σ -algebras generated by the cylinder sets are exponentially fast ψ -mixing (uniformly in j ). This result is proved similarly to [8].5.3.
Random sequential dynamical environments.
Random sequential distance expanding environments . Let Y , E y , T y , f y and u y be as in the beginning of Section 5.1. We assume here that Y is a metric space and let B be its Borel σ -algebra. Let Y j ⊂ Y , j ≥ θ j : Y j → Y j +1 , j ∈ Z be family of measurable maps and P j , j ∈ Z befamily of probability measures on Y which are supported on Y j , respectively, so that( θ j ) ∗ P j = P j +1 for each j . For each j and m ≥ θ mj = θ j + m − ◦ · · · ◦ θ j +1 ◦ θ j and consider the case when ζ j = θ j y , where y is distributed according to P . Inthis section we will start with a certain type of one sided sequences and consideriterates of the form L ( ζ,n )0 := L ( ζ j − )0 ◦ · · · ◦ L ( ζ )0 ◦ L ( ζ )0 . Namely, we view here the ζ j ’s as a sequential dynamical random environment andconsider (random) one sided sequences of maps T j and functions f j and u j of theform T j = T ζ j , f j = f ζ j and u j = u ζ j . Note that we can not apply directly Theorem 2.6 and all the other results fromSection 2 since we only have one sided sequences. In order to overcome this difficultywe will need the following. Let ˆ Y be the space of all sequences ˆ y = ( y k ) ∞ k = −∞ ∈ Y Z so that y k +1 = S k y k for each k , let σ : ˆ Y → Y be the shift map given by σx =( x k +1 ) ∞ k = −∞ and set ξ k = σ k ξ where ξ is distributed according to the measure ˆ P induced on ˆ Y by the sequence of finite dimensional distributions given byˆ P k { y : y i ∈ A i ; ∀ − k ≤ i ≤ k } = P − k (cid:8) k \ i = − k ( θ i − k ) − A i (cid:9) . Note that the Kolmogorov extension theorem indeed can be applied (i.e. the family { ˆ P k } is consistent) since ( θ j ) ∗ P j = P j +1 for each j . Henceforth, we will refer tothe process { ξ j : j ∈ Z } as the “invertible extension” of the process { ζ j : j ≥ } .Now, we can view T j , f j and u j as functions of ξ j : they depend only on the 0-thcoordinate of ξ j (so now T j , f j and u j are defined also for negative j ’s). Henceforth, { λ j ( z ) : j ∈ Z } , { h ( z ) j : j ∈ Z } and { ν ( z ) j : j ∈ Z } will denote the RPF tripletscorresponding to the (random) family of operators L ( j ) z = L ( ξ j ) z := L ( π ξ j ) z , where π y = y .In the following section we will provide general conditions under which the resultsstated in Section 2 hold true. Note that formally, we will show that the limittheorems stated in Section 2 hold true with the extension { ξ j : j ∈ Z } as the randomenvironment, but when the random Gibbs measure µ j given by dµ j = h (0) j dν (0) j depend only on ξ = { ζ j : j ≥ } then we can formulate all the limit theoremswithout passing to the invertible extension. First, taking a careful look at thearguments in Chapter 4 of [17], we see that the functional ν ( z ) j depend only on ζ j .Therefore, µ j depends only on ξ j if the function h (0) j is deterministic. We refer thereaders to Theorem 5.7 (and its proof) for conditions which guarantee that h (0) j = h for any j , for some deterministic function h (take there T j in place of S j ). imit theorems 29 The LCLT . We assume here that for any Lipschitz function g on Y , aninteger s ≥ t ≥ P (cid:8)(cid:12)(cid:12) n − X j =0 g ◦ θ js − n − X j =0 E P g ◦ θ js (cid:12)(cid:12) ≥ t + c (cid:9) ≤ c e − c t n where c , c and c are some positive constants which may depend on g . Theinequality (5.6) holds true when θ j ’s are maps satisfying the assumptions fromSection 2 and P j ’s are the appropriate (sequential) Gibbs measures correspondingto these maps.In this section we will prove the following:5.4. Theorem.
Suppose that (5.6) holds true, that there exists a point y ∈ T ∞ j =0 Y j so that θ j y = y for each j and that the maps θ j are H¨older continuous withexponent β ∈ (0 , and H¨older constant less or equal to K , for some constants β and K which do not depend on j . Assume, in addition, that for any open neighbourhood V of y we have (5.7) lim n →∞ P nj =1 P j ( V ) √ n ln n = ∞ and that conditions (A1) and (A2) from Theorem 5.2 hold true with L it = L ( y ) it .Then Assumption 2.15 holds true P -a.s. with m = 1 and the above L it . Note that when verifying Assumption 2.15, we can use the one sided sequence { ζ j } without passing to its invertible extension. When θ j , j ≥ P j to be the appropriate j -thGibbs measure. In this case, by Lemma 5.10.3 in [17] we have P j (cid:0) B j ( y , r ) (cid:1) ≥ Cr q for some q >
0, which implies that (5.7) holds true, since in this case the numeratorgrows linearly fast in n . Note that a common fixed point exists, for instance when all Y j are, the same torus and all S j ’s vanish at the origin, and when S j ’s form a non-stationary subshift of finite type so that the matrices A j , j ≥ A j ( a, a ) = 1 for some a ∈ N (and then we can take y = ( a, a, a, ... )). Proof of Theorem 5.4.
Let s ≥
1. As in the proof of Theorem 5.2, it is sufficient toshow that for any neighborhood U of y in Y we havelim n →∞ P n − sm =0 I Γ m ( ω ) √ n ln n = ∞ , P − a.s.where Γ m = { ω : ( ζ ms ( ω ) , ζ ms +1 ( ω ) , ..., ζ ms + s − ( ω )) ∈ U × U × · · · × U } , and I A isthe indicator function of a set A . Equivalently, we need to show that for P -almostany y we have lim n →∞ P n − sm =0 I ∆ m ( y ) √ n ln n = ∞ where ∆ m = s − \ j =0 ( θ ms + j ) − U = ( θ ms ) − U m,s and U m,s = U ∩ θ − ms U ∩ ( θ ms ) − U ∩ ... ∩ ( θ s − ms ) − U . Since y is a common fixedpoint, for any r > B m ( y , r β K − β ) ⊂ θ − m B m +1 ( y , r ) where for each m , x ∈ Y m and δ > B m ( x, δ ) denotes an open ball in Y m around x with radius δ . Therefore, U m,s contains an open ball V m,s = B m ( y , r s ) = B ( y , r s ) ∩ Y := V s around y in Y m , whose radius does not depend on m (here B ( y , r s ) is the corresponding ball in Y ). Hence, it is sufficient to show that(5.8) lim n →∞ P n − sm =0 I V s ◦ θ ms √ n ln n = ∞ , P − a.s. . Let f be Lipschitz function so that I B (0 ,r s ) ≤ f ≤ I V s = I B (0 , r s ) . Then n − s X m =0 I V s ◦ θ ms ≥ n − s X m =0 f ◦ θ ms . Note that E P f ◦ θ ms = E P ms f = R f dP ms ≥ P ms ( B (0 , r s )). Taking t of the form t = t n = c √ n ln n in (5.6), for an appropriate c , and using the Borel-Cantelli lemmawe derive that P -a.s. for any sufficiently large n , (cid:12)(cid:12) n − s X m =0 f ◦ θ ms − n − s X m =0 E P f ◦ θ ms (cid:12)(cid:12) ≤ C √ n ln n where C is some constant. The letter inequality together with the former inequali-ties and (5.7) imply (5.8). (cid:3) Remark.
The proof of Theorem 5.4 proceeds similarly if we assume thatthere exists y , y , ..., y m − ∈ Y so that for any j and i = 0 , , ..., m − S j + i y i = y i +1 , where y m := y , that (5.3) holds true, with y i − in place of y i appearing there and with ζ mm s + i in place of ξ mm s + i , and that for any familyof Lipschitz functions g j , j ≥ | g j | ≤ P (cid:8)(cid:12)(cid:12) n − X j =0 g j ◦ θ j − n − X j =0 E P g j ◦ θ js (cid:12)(cid:12) ≥ t + c (cid:9) ≤ c e − c t n for some positive constants c and c (which do not depend on the g j ’s). In thiscase we require that the spectral radius of L it = L ( y m − ) it ◦ · · · ◦ L ( y ) it ◦ L ( y ) it is less than 1 for any t ∈ I h . In particular we can consider non-stationary subshiftsof finite type, with the property that for some a , a , ...a m ∈ N and all j ’s wehave A j ( a i , a i +1 ) = 1, where a m +1 := a , which means that the periodic word( a , a , a , ...a m , a , a , ..., a m , ... ) = ( a , a , ..., a m ) ⊗ N belongs to all of the E j ’s.5.3.3. Existence of limiting logarithmic moment generating functions . Weassume here that Y is a compact metric space. Let S j : Y → Y be a family of mapssatisfying all the conditions specified in Section 2 with E j = Y , and consider thecase when θ j = S j for each j . Let r j : Y → R be a family of maps so that theH¨older norms k r j k α are bounded in j , and let ( λ j (0) , h (0) j , ν (0) j ) be the RPF tripletcorresponding to the operators L ( j )0 given by L ( j )0 g ( x ) = X y ∈ S − j { x } e r j ( y ) g ( y ) . imit theorems 31 In these circumstances, we take P j = µ j , where µ j is given by d µ j = h (0) j d ν (0) j .Namely, we consider here a random sequential environment generated by a twosequence of distance expanding maps S j .We will first need the following5.6. Theorem. (i) The random pressure function Π j ( z ) depends only on ζ j .(ii) When the maps S j are H¨older continuous with the same exponent and H¨olderconstants which are bounded in j then P -a.s. we have lim n →∞ (cid:12)(cid:12)(cid:12) n n − X j =0 Π j ( z ) − n n − X j =0 E P Π j ( z ) (cid:12)(cid:12)(cid:12) = 0 . In particular, when the distribution of ζ j does not depend on j (i.e, when P j = µ j does not depend on j ) then the limit Π( z ) from Theorem 2.16 exists and Π( z ) = E P Π ( z ) . The distribution of ζ j does not depend on j when, for instance, each one ofthe S j ’s has the form S j ( x ) = ( m ( j )1 x , ..., m ( j ) d x d )mod 1 for some positive integers m ( j ) i , where x = ( x , ..., x d ) ∈ E j = T d belongs to some d -dimensional Torus (here ν (0) j = Lebesgue and h (0) j ≡ n on Y which assigns positive mass to open sets, so that ( S j ) ∗ n ≪ n forany integer j and r j be defined by e − r j = d ( S j ) n d n , then by Theorem 2.7 we have λ j (0) = 1 and ν (0) j = n . In this case, the claim that the distribution of ζ j does notdepend on j means that h (0) j , j ≥ j , and in Theorem 5.7 wewill show that h (0) j does not depend on j and ω when the S j ’s are drawn at randomaccording to some, not necessarily stationary, classes of processes (so we will have”random random” non-stationary neighborhoods). Proof of Theorem 5.6.
Set g j,n ( z, ζ j ) = l n + j +1 ( L j,n +1 z ) l n + j +1 ( L j +1 ,nz ) . Since λ j = ν ( z ) j +1 L ( j ) z and the norms kL ( j ) z k α are bounded in j and z ∈ U , applying(2.6) we obtain that for any n ≥ j ∈ Z ,(5.10) | λ j ( z ) − g j,n ( z, ζ j ) | ≤ Cδ n where C > δ ∈ (0 ,
1) are constants, and therefore λ j ( z ) and Π j ( z ) dependonly on ζ j . We also derive from the above that there exists a constant r > j and complex z so that | z | < r we have(5.11) | Π j ( z ) − ln g j,n ( z, ζ j ) | ≤ C δ n where C > λ j (0) ≥ a for some constant a > g j,n ( z, ζ j ) = ln g j,n ( z, ζ j ) . Let ε > k be so that Cδ k < ε . In the circumstances of Theorem 5.6 (i),the maps g j,k ( z, ζ j ) are H¨older continuous functions of ζ j with the same exponentand with H¨older constants which are bounded in j . Therefore, by Theorem 2.14, there exist constants c , c and c , which may depend on k , so that for any ε > P (cid:8) | n − X j =0 ˆ g j,k ( z, ζ j ) − n − X j =0 E ˆ g j,k ( z, ζ j ) | ≥ c + t (cid:9) ≤ c e − c t n . By taking t = t n which grows faster than √ n but slower linearly in n , we derivefrom Borel-Cantelli lemma thatlim n →∞ (cid:12)(cid:12) n n − X j =0 ˆ g j,k ( z, ζ j ) − n n − X j =0 E ˆ g j,k ( z, ζ j ) (cid:12)(cid:12) = 0 , P − a.s.and so for any ε ,lim sup n →∞ (cid:12)(cid:12) n n − X j =0 Π j ( z ) − n n − X j =0 E Π j ( z ) (cid:12)(cid:12) < ε, P − a.s.which completes the proof of the theorem. (cid:3) Next, let Q be a compact metric space and let ¯ S j : X → X and ¯ r j : X → R satisfy the conditions specified in Section 2 with E j = Q . Let (¯ λ (0) j , ¯ h (0) j , ¯ ν (0) j ) be theRPF triplets corresponding to the transfer operator generated by ¯ S j and ¯ r j and let¯ µ j = ¯ h j ν (0) j be the appropriate Gibbs measure. Let η j = ¯ S j η = ¯ S j − ◦· · ·◦ ¯ S ◦ ¯ S η be a sequence of random variables, where η is distributed according to ¯ µ .5.7. Theorem.
Suppose that there exists a probability measure n on Y which assignspositive mass to open sets, so that ( S j ) ∗ n ≪ n for any integer j and r j is definedby e − r j = d ( S j ) n d n . Assume also that S j , j ≥ are random, and that they have form S j = S η j . Thenthere exists a function h : X → R so that, ˆ P -almost surely, the distribution P j = µ j of all the ζ j ’s is κ := hd n (i.e. in the extension we have µ j = µ ξ j = κ for each j ).Proof. Note first that the assumptions in the statement of the theorem mean that S j and r j are chosen at random by the invertible extension ˆ η j , j ∈ Z of η j , j ≥ S j , j < j -the coordinate in this extension. In thiscase, we only need to show that there exists a function h : X → R so that for any j ≥ h (0) j = h ( ˆ P -a.s.).Let β j j ≥ η , and consider the setup of i.i.d. maps S β j , j ≥
0. Suppose that the above i.i.d. process is defined on a probability space(Ω , F , P ) so that β j = θ j β for some P preserving map θ . Consider the skewproduct map S ( ω, x ) = ( θω, S β ( ω ) x ). Then, as in [1], there exists an S -invariantmeasure of the form P × ( hd n ) for some strictly positive continuous function h : X → R so that n ( h ) = 1. Exactly as in Section 4.1 in [19], it follows that for¯ µ -almost any q ∈ Q , L ( q )0 ¯ h = ¯ h. Fix some y ∈ Q and set Γ y = { q ∈ Q : L ( q )0 h ( y ) = h ( y ) } . imit theorems 33 Then ¯ µ ( Q \ Γ y ) = 0 for each y . Since ¯ ν (0) j = n we have¯ µ j ( Q \ Γ y ) = Z Q\ Γ y ¯ h j ( t )¯ h ( t ) d ¯ µ ( t ) = 0and therefore ¯ µ (cid:8) ∞ \ j =0 ( ¯ S j ) − Γ y (cid:9) = 1and therefore for any y and j ≥ L (ˆ η j )0 h ( y ) = h ( y ) , a.s. . Since both sides are continuous in y and Y is compact we conclude that L (ˆ η j )0 h = h, ˆ P − a.s.for any j ≥
0. Replacing ˆ η with ˆ η k for any integer k , and using that ˆ η k + j = σ j ˆ η k for any j ≥ L (ˆ η k )0 h = h for any k . Since ¯ λ j (0) = 1 for any j , using (2.6) we derive thatlim n →∞ L (ˆ η k − )0 ◦ L (ˆ η k − )0 ◦ · · · ◦ L (ˆ η k − n )0 h = n ( h ) h k = h k and so h k = h for any k . Note that (2.6) holds true for H¨older continuous func-tions, but for real z ’s the converges itself, without rates, holds true for continuousfunctions by monotonicity arguments due to Walters (see the proof of Proposition3.19 in [32] and [40]). (cid:3) Converges rate towards the moments . Suppose that S j , j ∈ Z is anonstationary subshift of finite type so that h (0) j does not depend on j , i.e. that( S j ) ∗ µ = µ for some probability measure µ (e.g. when S j is random). Moreover,assume that T j , f j and u j depend only on the j -the coordinate X j . Then by (5.11)the random variables Π j ( z ) can be approximated exponentially fast in the L ∞ normby functions of the coordinates at places j, j + 1 , ..., j + n . Taking into accountTheorem 5.3 (ii), we conclude that all the conditions of Theorem 2.4 in [18] holdtrue with ℓ = 1, with Π j ( z ) in place of ξ j (from there) and with any boundedfunction F which identifies with the function G ( x ) = x on a compact set whichcontains all the possible values of all of the Π j ( z )’s. In particular, (2.11) from [18]holds true, and so for any t ≥ r ≥ n ≥ P (cid:8) | n − X j =0 Π j ( z ) − n E Π ( z ) | ≥ t + Cnδ r + r (cid:9) ≤ e − C t nr where C is some positive constant. Taking r of the form r = a ln n for an appropriate a we derive that for some constant C >
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