Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps
aa r X i v : . [ m a t h . P R ] N ov HEAVY TAIL PHENOMENON AND CONVERGENCE TO STABLE LAWSFOR ITERATED LIPSCHITZ MAPS
MARIUSZ MIREK
In memory of Andrzej Hulanicki
Abstract.
We consider the Markov chain { X xn } ∞ n =0 on R d defined by the stochastic recursion X xn = ψ θ n ( X xn − ), starting at x ∈ R d , where θ , θ , . . . are i.i.d. random variables taking theirvalues in a metric space (Θ , r ) , and ψ θ n : R d R d are Lipschitz maps. Assume that the Markovchain has a unique stationary measure ν . Under appropriate assumptions on ψ θ n , we will showthat the measure ν has a heavy tail with the exponent α > ν ( { x ∈ R d : | x | > t } ) ≍ t − α .Using this result we show that properly normalized Birkhoff sums S xn = P nk =1 X xk , converge inlaw to an α –stable law for α ∈ (0 , Introduction and Statement of Results
We consider the Euclidean space R d endowed with the scalar product h x, y i = P di =1 x i y i , thenorm | x | = p h x, x i , and its Borel σ –field B or ( R d ). An iterated random function is a sequence ofthe form X xn = ψ ( X xn − , θ n ) , (1.1)where n ∈ N , X x = x and θ , θ , . . . ∈ Θ are independent and identically distributed accordingto the measure µ on a metric space Θ = (Θ , r ). We assume that ψ : R d × Θ R d is jointlymeasurable and we write ψ θ ( x ) = ψ ( x, θ ). Then the sequence ( X xn ) n ≥ is a Markov chain withthe state space R d , the initial Dirac distribution δ x , and the transition probability P defined by P ( x, B ) = R Θ B ( ψ θ ( x )) µ ( dθ ) for all x ∈ R d and B ∈ B or ( R d ). Unless otherwise stated we assumethroughout this paper that for every θ ∈ Θ, ψ θ : R d R d is a Lipschitz map with the Lipschitzconstant L θ = sup x = y | ψ θ ( x ) − ψ θ ( y ) || x − y | < ∞ . Matrix recursions(1.2) X xn = ψ θ n ( X xn − ) = M n X xn − + Q n ∈ R d , where θ n = ( M n , Q n ) ∈ Gl ( R d ) × R d = Θ and X x = x ∈ R d ( Gl ( R d ) is the group of d × d invertiblematrices) are probably the best known examples of the situation we have in mind [5, 6, 14, 24, 25].If the Lipschitz constant L θ is contracting in average i.e. R Θ log( L θ ) µ ( dθ ) < R Θ | log( L θ ) | +log + ( | ψ θ ( x ) | ) µ ( dθ ) < ∞ for some x ∈ R d , then (1.1) has a unique (in law) stationary solution S with law ν . In fact, S = lim n →∞ ψ θ ◦ ψ θ ◦ . . . ◦ ψ θ n ( x ) a.s. and does not depend on the startingpoint x ∈ R d (see [6], [27] for more details). Throughout this paper we assume that our Lipschitzmaps ψ θ ’s satisfy above conditions and recursion (1.1) has the stationary solution S with law ν . This research project has been partially supported by Marie Curie Transfer of Knowledge Fellowship HarmonicAnalysis, Nonlinear Analysis and Probability (contract number MTKD-CT-2004-013389) and by KBN grant N201012 31/1020.
We are going to describe the asymptotic behavior of Birkhoff sums S xn = P nk =1 X xk of (nonindependent) random variables X xk . We prove that the sequence S xn normalized appropriatelyconverges to a stable law (see Theorem 1.15).The problem has been recently studied in [4] for the recursion (1.2) with M ∈ R ∗ + × O ( R d )( O ( R d ) is the orthogonal group) and a central limit theorem has been proved. Depending on thegrowth of M and Q , a stable law or a Gaussian law appear as the limit. In the first case the heavytail behavior of the stationary solution of (1.2) at infinity is vital for the proof. (See [5]).On the other hand being linear is not that crucial for ψ θ and so, it is tempting to generalize theresult of [4] for a larger class of possible ψ θ . Lipschitz transformations fit perfectly into the scheme– see examples in section 2 due to Goldie [7] and Borkovec and Kl¨uppelberg [3].To give an idea of our result let us formulate it in the special case of the recursion X xn = max( M n X xn − , Q n ) , (1.3)where ( M n , Q n ) n ∈ N ⊆ R + × R and X x = x ∈ R . For the stationary solution S of (1.3) with law ν ,lim t →∞ t α P ( { S > t } ) exists, and under appropriate assumption, it is positive [7]. Then the limitTheorem 1.15 is: Theorem 1.4.
Assume that ( M n , Q n ) n ∈ N ⊆ R + × R is the sequence of i.i.d. pairs with the law µ such that E ( M α ) = 1 , and E ( M α | log M | ) < ∞ , for some α ∈ (0 , , the conditional law of log | M | ,given M = 0 is non arithmetic, P ( Q > > , and E ( | Q | α ) < ∞ . Let S xn = P nk =1 X xk for n ∈ N .Then given < α < (for simplicity we assume here α = 1 ), there is a sequence d n = d n ( α ) anda constant C α ∈ C such that the random variables n − α ( S xn − d n ) converge in law to the α – stablerandom variable with the characteristic function Υ α ( t ) = exp( C α t α ) , for t > , If α = 2 , there is a sequence d n = d n (2) and a constant C ∈ R such that the random variables ( n log n ) − ( S xn − d n ) converge in law to the random variable with characteristic function Υ ( t ) = exp ( C t ) , for t > . If α ∈ (0 , , then d n = 0 , and if α ∈ (1 , , then d n = nm , where m = R R d xν ( dx ) . Furthermore, ℜ C α < for every α ∈ (0 , . The paper is divided into three parts. In the first one (section 3) we describe the support of thestationary law ν of (1.1) in the terms of the fixed points for maps ψ θ – see Theorem 3.1. Secondly,in section 4 we take care of the tail of ν . (See Theorem 1.8 saying that ν ( { x ∈ R d : | x | > t } ) ≍ t − α ).Finally, sections 5 and 6 are devoted to the proof of the limit Theorem 1.15. The limit law is astable law with exponent α ∈ (0 , ax + b ” model statedin [15] and [4] for one dimensional and multidimensional situation respectively. The case where α > α – stable theorem for α ∈ (0 ,
2) foradditive functionals on metric spaces using martingale approximation method, but our situationdoes not fit into their framework. Convergence to stable laws were also studied by [1] and [9].Now we are ready to formulate assumptions and to state theorems.1.1.
Heavy tail phenomena.
In this section we state conditions that guarantee a heavy tail of ν . Contrary to the affine recursion(1.5) X xn = ψ θ n ( X xn − ) = M n X xn − + Q n ∈ R , where θ n = ( M n , Q n ) ∈ R × R = Θ, we need more than just the behavior of the Lipschitz constant L θ . EAVY TAIL PHENOMENON AND CONVERGENCE TO STABLE LAWS 3
Assumption 1.6 ( Shape of the mappings ψ ) . For every t > , let ψ θ,t : R d R d be definedby ψ θ,t ( x ) = tψ θ ( t − x ) , where x ∈ R d and θ ∈ Θ . ψ θ,t are called dilatations of ψ θ . (H1) For every θ ∈ Θ , there exists a map ψ θ : R d R d such that lim t → ψ θ,t ( x ) = ψ θ ( x ) forevery x ∈ R d , and ψ θ ( x ) = M θ x for every x ∈ supp ν . The random variable M θ takes itsvalues in the group G = R ∗ + × K , where K is a closed subgroup of orthogonal group O ( R d ) . (H2) For every θ ∈ Θ , there is a random variable N θ such that ψ θ satisfies a cancellationcondition i.e. | ψ θ ( x ) − M θ x | ≤ | N θ | , for every x ∈ supp ν . To get the idea what is the meaning of (H1)–(H2) the reader may think of the affine recursion(1.5) with θ = ( M, Q ) ∈ G × R d = Θ or the recursion ψ θ ( x ) = max { M x, Q } , where θ = ( M, Q ) ∈ R ∗ + × R = Θ (see section 2). Then ψ θ ( x ) = M x or ψ θ ( x ) = max { M x, } respectively. It isrecommended to have in mind ψ θ ( x ) = max { M x, Q } to get the first approximation of what thehypotheses mean. Notice that for the max recursion (H2) is not satisfied on R , but only on[0 , ∞ ) ⊇ supp ν .In one dimensional case condition (H2) has a very natural geometrical interpretation, namely itcan be written in an equivalent form M θ x − | N θ | ≤ ψ θ ( x ) ≤ M θ x + | N θ | . It means that the graphof ψ θ ( x )’s lies between the graphs of M θ x − | N θ | and M θ x + | N θ | for every x ∈ supp ν . This allowsus to think that the recursion is, in a sense, close to the affine recursion.For simplicity we write X instead of X θ . Assumption 1.7 ( Moments condition for the heavy tail ) . Let κ ( s ) = E | M | s for s ∈ [0 , s ∞ ) ,where s ∞ = sup { s ∈ R + : κ ( s ) < ∞} . Let ¯ µ be the law of M . (H3) G is the smallest closed semigroup generated by the support of ¯ µ i.e. G = h supp ¯ µ i . (H4) The conditional law of log | M | , given M = 0 is non arithmetic. (H5) M satisfies Cram´er condition with exponent α > , i.e. there exists α ∈ (0 , s ∞ ) such that κ ( α ) = E ( | M | α ) = 1 . (H6) Moreover, E ( | M | α | log | M || ) < ∞ . (H7) For the random variable N defined in (H2) we have E ( | N | α ) < ∞ . Conditions (H4)–(H7) are natural in this context, see [3, 4, 5, 7, 10, 11, 12, 15, 16, 24] and [32].Now we are ready to formulate the main result.
Theorem 1.8.
Assume that ψ θ satisfies assumptions 1.6 and 1.7 for every θ ∈ Θ . Then thereis a unique stationary solution S of (1.1) with law ν , and there is a unique Radon measure Λ on R d \ { } such that lim g ∈ G, | g |→ | g | − α E f ( gS ) = Λ( f ) . (1.9) The convergence is valid for every bounded continuous function f that vanishes in a neighbourhoodof zero. Furthermore the recursion defined in (1.1) has a heavy tail lim t →∞ t α P ( {| S | > t } ) = 1 αm α E ( | ψ ( S ) | α − | M S | α ) , (1.10) where m α = E ( | M | α log | M | ) > . If additionally the support of ν is unbounded, and one of thefollowing condition is satisfied s ∞ < ∞ and lim s → s ∞ E ( | N | s ) κ ( s ) = 0 , (1.11) s ∞ = ∞ and lim s →∞ (cid:18) E ( | N | s ) κ ( s ) (cid:19) s < ∞ , (1.12) then the measure Λ is nonzero. M. MIREK
Remark 1.13.
Contrary to Theorems 1.15 and 3.1 the assumption that ψ θ ’s are Lipschitz is notnecessary for Theorem 1.8. The same conclusion holds if ψ θ : R d R d is continuous for every θ ∈ Θ , and the map Θ ∋ θ ψ θ ( x ) ∈ R d is continuous for every x ∈ R d , 1.6 and 1.7 are satisfiedand S = lim n →∞ ψ θ ◦ ψ θ ◦ . . . ◦ ψ θ n ( x ) exists a.s. and does not depend on x ∈ R d .In view of Letac’s principle [27] the random variable S with law ν is a unique stationary solutionof the recursion (1.1). Theorem 1.8 on one hand generalizes Theorem 1.6 of [5] for multidimensional affine recursionsand on the other, the results of Goldie [7] for a family of one–dimensional recursions modeled on ax + b . (H4)–(H6) were already assumed by Goldie. (H3) was introduced in [5] and the wholeproof is based on it. (H1)–(H2) say that asymptotically (1.1) looks like an affine recursion and itallows us to use the methods of [5].1.2. Limit theorem for Birhhoff sums.
Now we introduce conditions necessary to obtain con-vergence in law of appropriately normalized sums S xn = P nk =1 X xk to an α – stable distribution. Assumption 1.14 ( For the limit theorem ) . (L1) For every θ ∈ Θ , L θ ≤ | M θ | . (L2) For every θ ∈ Θ , there is a random variable Q θ , such that ψ θ satisfies a smoothnesscondition with respect to t > , i.e. | ψ θ,t ( x ) − ψ θ ( x ) | ≤ t | Q θ | , for every x ∈ R d . (L3) For the random variable Q we have E ( | Q | α ) < ∞ . Clearly, if ψ θ ( x ) = M θ x for every x ∈ supp ν , then (L2) implies (H1), and (L2) together with(L3) imply (H2) and (H7). Now we are ready to formulate the limit theorem. Theorem 1.15.
Assume that ψ θ satisfies assumptions 1.6, 1.7 and 1.14 for every θ ∈ Θ . Wedefine S xn = P nk =1 X xk for n ∈ N . Let h v ( x ) = E (cid:16) e i h v, P ∞ k =1 ψ θk ◦ ... ◦ ψ θ ( x ) i (cid:17) for x ∈ R d , where ψ θ k ’swere defined in (H1) of assumption 1.6, and let ν be the stationary measure for the recursion (1.1). • If α ∈ (0 , ∪ (1 , , then there is a sequence d n = d n ( α ) and a function C α : S d − C suchthat the random variables n − α ( S xn − d n ) converge in law to the α –stable random variablewith characteristic function Υ α ( tv ) = exp( t α C α ( v )) , for t > and v ∈ S d − . • If α = 1 , then there are functions ξ, τ : (0 , ∞ ) R and C : S d − C such that the ran-dom variables n − S xn − nξ ( n − ) converge in law to the random variable with characteristicfunction Υ ( tv ) = exp( tC ( v ) + it h v, τ ( t ) i ) , for t > and v ∈ S d − . • If α = 2 , then there is a sequence d n = d n (2) and a function C : S d − R such thatthe random variables ( n log n ) − ( S xn − d n ) converge in law to the random variable withcharacteristic function Υ ( tv ) = exp( t C ( v )) , for t > and v ∈ S d − .If α ∈ (0 , , then d n = 0 , and if α ∈ (1 , , then d n = nm , where m = R R d xν ( dx ) . In all theabove cases the function C α depends on the function h v and the measure Λ defined in Theorem1.8. Moreover, C α ( tv ) = t α C α ( v ) for every t > , v ∈ S d − and α ∈ (0 , ∪ (1 , . Ifsupp Λ spans R d as a linear space, then ℜ C α ( v ) < for every v ∈ S d − . The proof of the above theorem will be based on the spectral method that was initiated byNagaev in [30] and then used and improved by many authors (see [1, 4, 9, 15, 18, 19, 20] andthe references given there). The method is based on quasi–compactness of transition operators
EAVY TAIL PHENOMENON AND CONVERGENCE TO STABLE LAWS 5
P f ( x ) = E ( f ( ψ ( x ))) = R Θ f ( ψ θ ( x )) µ ( dθ ) on appropriate function spaces (see [4, 15, 18, 19, 20]).They are perturbed by adding Fourier characters.The standard use of the perturbation theory requires exponential moments of µ , but there issome development towards µ ’s with polynomial moments [18] and their improvements [20], or evenfractional moments [15], and [4]. They are based on a theorem of Keller and Liverani [24] (we referalso to [29] for an improvement of [24]). It says that the spectral properties of the operator P canbe approximated by those of its Fourier perturbations(1.16) P t,v f ( x ) = E (cid:16) e i h tv,ψ ( x ) i f ( ψ ( x )) (cid:17) = Z Θ e i h tv,ψ θ ( x ) i f ( ψ θ ( x )) µ ( dθ ) , (with convention that P ,v = P ). Indeed,(1.17) P t,v = k v ( t )Π P,t + Q P,t , where lim t → k v ( t ) = 1, Π P,t is a projection on a one dimensional subspace and the spectral radiiof Q P,t are smaller than ̺ <
1, when t ≤ t . To obtain Theorem 1.15 we need to expand thedominant eigenvalue k v ( t ) at 0.When α ∈ (0 , k v ( t ) is neither analytic nor differentiable, hence their asymptotics at zero ismuch harder to obtain. The method used in [4] does not work here and so we propose anotherapproach which is applicable to general Lipschitz models (see section 6).2. Examples
The following examples will help the reader to understand the meaning of the assumptionsformulated in the introduction as well as to feel the breadth of the method.2.1.
An affine recursion.
Let G = R ∗ + × O ( R d ) and take the sequence of i.i.d. random pairs( A n , B n ) n ∈ N ⊆ Θ = G × R d with the same law µ on Θ and define the affine map ψ n ( x ) = A n x + B n ,where x ∈ R d . This example was also widely considered in the context of discrete subgroups of R ∗ + see [5] and [4].2.2. An extremal recursion.
Let G = R ∗ + and Θ = G × R . We consider the sequence of i.i.d.pairs ( A n , B n ) n ∈ N ⊆ Θ with the same law µ on Θ. Let ψ n ( x ) = max { A n x, B n } , where x ∈ R .Assume that P ( B > >
0. Then • lim t → ψ n,t ( x ) = ψ n ( x ), where ψ n ( x ) = max { A n x, } and M n = A n . • The stationary solution S with law ν is given by the explicit formula, S = max ≤ k< ∞ { A A · . . . · A k − B k } , where A = 1 a.s. [7]. • P ( B > > ν ⊆ [0 , ∞ ) and supp ν is unbounded. • In order to check cancellation condition (H2) notice that S ≥ x > | ψ n,t ( x ) − A n x | = | max { A n x, B n } − A n x | { A n x
A model due to Letac.
Let G be as above and take the sequence of i.i.d. random triples( A n , B n , C n ) n ∈ N ⊆ Θ = G × R + × R + with the same law µ on Θ. Consider the map ψ n ( x ) = A n max { x, B n } + C n , where x ∈ R . If C ≥ P ( B >
0) + P ( C > >
0, then the supportof the stationary measure ν is unbounded [7]. The others assumptions are also satisfied.2.4. Another example.
Take the sequence of i.i.d. random triples ( A n , B n , C n ) n ∈ N ⊆ Θ = R ∗ + × R + × R + with the same law µ on Θ, such that B n − A n C n <
0. Consider the map ψ n ( x ) = √ A n x + B n x + C n , where x ∈ R . If P ( B >
0) + P ( C > >
0, then the support of thestationary measure ν is unbounded [7]. Notice that ψ n ( x ) can be written in the equivalent form ψ n ( x ) = p A n ( x + U n ) + V n , where U n = B n A n and V n = C n − B n A n >
0. Now we can easily verifythat ψ n ( x ) is Lipschitz and (L2) is satisfied. Indeed, | ψ n ( x ) − ψ n ( y ) || x − y | = A n | ( x + U n ) − ( y + U n ) || x − y | (cid:16)p A n ( x + U n ) + V n + p A n ( y + U n ) + V n (cid:17) ≤ p A n . Next observe that ψ n ( x ) = √ A n | x | , and | ψ n,t ( x ) − ψ n ( x ) | = (cid:12)(cid:12) A n ( x + tU n ) + t V n − A n x (cid:12)(cid:12)p A n ( x + tU n ) + t V n + √ A n x ≤ tA n U n | x |√ A n | x | + t A n U n + t V n t √ V n ≤ t (cid:18) B n √ A n + C n √ V n (cid:19) , this shows that (L2) is fulfilled.For the above examples statements 1.8, 1.15 and 3.1 apply straightforwardly.2.5. An autoregressive process with ARCH(1) errors.
Now we consider an example de-scribed by Borkovec and Kl¨uppelberg in [3]. For x ∈ R , let ψ ( x ) = (cid:12)(cid:12)(cid:12) γ | x | + p β + λx A (cid:12)(cid:12)(cid:12) , where γ ≥ , β > , λ > A is a symmetric random variable with continuous Lebesguedensity p , finite second moment and with the support equal the whole of R . (see section 2. in [3]for more details). Now consider the sequence ( ψ n ( x )) n ∈ N of i.i.d. copies of ψ ( x ) and observe that • lim t → ψ n,t ( x ) = ψ n ( x ), where ψ n ( x ) = M n | x | and M n = (cid:12)(cid:12)(cid:12) γ + √ λA n (cid:12)(cid:12)(cid:12) . • | ψ n,t ( x ) − M n | x || = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ | x | + p βt + λx A n (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) γ + √ λA n (cid:12)(cid:12)(cid:12) | x | (cid:12)(cid:12)(cid:12) ≤ | t |√ β | A n | ,so (L2) holds with | Q n | = √ β | A n | . Notice that (H2) holds for every x ∈ [0 , ∞ ) with | N n | = √ β | A n | .In [3] the authors showed that it is possible to choose parameters γ ≥ , β > , λ > E (log M n ) < E ( M αn ) = 1 for some 0 < α ≤
2. Observe that P (cid:0)(cid:8) M n ∈ R ∗ + (cid:9)(cid:1) = 1. Weare not able to verify conditions (1.11) and (1.12) to conclud that Λ is not zero, but this propertyfollows from [3] and so Theorem 1.15 applies.3. Stationary measure
Support of the stationary measure.
Let C ( R d ) be the set of continuous functions on R d and C b ( R d ) be the set of bounded and continuous functions on R d . Recall that unless otherwisestated we assume (as in Introduction) that for every θ ∈ Θ, ψ θ : R d R d is a Lipschitz map withthe Lipschitz constant L θ < ∞ .Let L µ Θ = { ψ θ ◦ . . . ◦ ψ θ n ( · ) : ∀ n ∈ N ∀ ≤ i ≤ n θ i ∈ supp µ } i.e. L µ Θ is the closed semigroup gener-ated by the maps ψ θ , where θ ∈ supp µ . Given ψ θ with L θ <
1, let ψ • θ be the unique fixed point of ψ θ . Then we can formulate the main Theorem of this section: EAVY TAIL PHENOMENON AND CONVERGENCE TO STABLE LAWS 7
Theorem 3.1.
Assume that R Θ log( L θ ) µ ( dθ ) < , R Θ | log( L θ ) | + log + ( | ψ θ ( x ) | ) µ ( dθ ) < ∞ forsome x ∈ R d and Θ ∋ θ ψ θ ( x ) ∈ R d is continuous for every x ∈ R d . If S = { ψ • θ ∈ R d : ψ θ ( ψ • θ ) = ψ • θ , where ψ θ ∈ L µ Θ and L θ < } ⊆ R d , then supp ν = S , where ν is the law of thestationary solution S for the recursion (1.1). Theorem 3.1 generalizes similar theorems for affine random walks, see [5] and [14] for moredetails. Notice that conditions R Θ log( L θ ) µ ( dθ ) < R Θ | log( L θ ) | + log + ( | ψ θ ( x ) | ) µ ( dθ ) < ∞ for some x ∈ R d , in view of [6] (see also [27]), give the existence of the stationary measure ν forthe recursion (1.1).Before proving Theorem 3.1, we need two lemmas. Given, ψ θ with L θ <
1, the Banach fixedpoint theorem implies the existence of a unique fixed point ψ • θ ∈ R d of the map ψ θ . Moreover, forevery x ∈ R d lim n →∞ ψ nθ ( x ) = ψ • θ . (3.2) Lemma 3.3.
Assume that for the map ψ θ we have L θ < and ψ ∈ L µ Θ . Then there exists (3.4) lim n →∞ ( ψ ◦ ψ nθ ) • = ψ ( ψ • θ ) , where ( ψ ◦ ψ nθ ) • ∈ R d is the fixed point of the map ψ ◦ ψ nθ , for n ∈ N .Proof. Notice that for n sufficiently large ψψ nθ = ψ ◦ ψ nθ is contracting. Fix ε >
0, then there exist N ε ∈ N such that L ψ L nθ − L ψ L nθ < ε for all n ≥ N ε , where L ψ is the Lipschitz constant associated to ψ .For every m ∈ N we have | ( ψψ nθ ) m ( ψ • θ ) − ψ ( ψ • θ ) | ≤ ∞ X k =1 ( L ψ L nθ ) k ! · | ψ ( ψ • θ ) − ψ • θ | = L ψ L nθ · | ψ ( ψ • θ ) − ψ • θ | − L ψ L nθ . By (3.2) we can find m ∈ N such that | ( ψψ nθ ) • − ( ψψ nθ ) m ( ψ • θ ) | < ε . Then | ( ψψ nθ ) • − ψ ( ψ • θ ) | ≤ | ( ψψ nθ ) • − ( ψψ nθ ) m ( ψ • θ ) | + | ( ψψ nθ ) m ( ψ • θ ) − ψ ( ψ • θ ) |≤ ε + | ψ ( ψ • θ ) − ψ • θ | · L ψ L nθ − L ψ L nθ ≤ ε (1 + | ψ ( ψ • θ ) − ψ • θ | ) , for all n ≥ N ε . Since ε is arbitrary, (3.4) is established. (cid:3) Lemma 3.5. If ψ θ : R d R d is continuous for every θ ∈ Θ (not necessarily Lipschitz) and Θ ∋ θ ψ θ ( x ) ∈ R d is continuous for every x ∈ R d , then for every θ ∈ supp µψ θ [ supp ν ] ⊆ supp ν, where the measure ν is µ stationary i.e. R R d R Θ f ( ψ θ ( x )) µ ( dθ ) ν ( dx ) = R R d f ( x ) ν ( dx ) for any f ∈ C b ( R d ) .Proof. Suppose for contradiction that ψ θ (supp ν ) supp ν. Then for some θ ∈ supp µ and x ∈ supp ν , there exists an open neighborhood U of ψ θ ( x ) such that U ∩ supp ν = ∅ . Notice, that µ ( { θ ∈ Θ : R R d U ( ψ θ ( x )) ν ( dx ) > } ) = 0, since the measure ν is µ stationary. By the assumptions { θ ∈ Θ : R R d U ( ψ θ ( x )) ν ( dx ) > } is an open subset of Θ. ψ − θ [ U ] is an open neighborhood of x ∈ supp ν , so θ ∈ { θ ∈ Θ : R R d U ( ψ θ ( x )) ν ( dx ) > } , but this contradicts θ ∈ supp µ. (cid:3) Proof of Theorem 3.1.
It is a consequence of Lemma 3.3 and 3.5. Compare also with Lemma 2.7in [5]. (cid:3)
M. MIREK
Simple properties of recursions and their stationary measures.Lemma 3.6.
Assume that Y xn,t = ψ θ ,t ◦ ψ θ ,t ◦ . . . ◦ ψ θ n ,t ( x ) for any n ∈ N and t > . Then, | Y xn,t − Y yn,t | ≤ n Y i =1 L θ i | x − y | , (3.7) | Y xn,t − Y xn + m,t | ≤ n Y i =1 L θ i | x − ψ θ n +1 ,t ◦ . . . ◦ ψ θ n + m ,t ( x ) | , (3.8) | x − ψ θ n +1 ,t ◦ . . . ◦ ψ θ n + m ,t ( x ) | ≤ m X k =1 n + k − Y i = n +1 L θ i ! | x − ψ θ n + k ,t ( x ) | , (3.9) for any x, y ∈ R d and m, n ∈ N .Proof. It is easy to see that | ψ θ,t ( x ) − ψ θ,t ( y ) | = | tψ θ ( t − x ) − tψ θ ( t − y ) | ≤ L θ | x − y | for any x, y ∈ R d , so (3.7), (3.8) and (3.9) follow by induction. (cid:3) The next Lemma is obvious in view of what has just been established.
Lemma 3.10.
Under the assumptions of the previous Lemma, if (H2), (H5), (H7) and (L1) aresatisfied, then for every β ∈ (0 , α ) , x ∈ R d , sup n ∈ N (cid:0) E | X xn,t | β (cid:1) β = sup n ∈ N (cid:0) E | Y xn,t | β (cid:1) β < ∞ , where X xn,t = ψ θ n ,t ◦ ψ θ n − ,t ◦ . . . ◦ ψ θ ,t ( x ) for any n ∈ N and t > . In particular, we obtain (cid:0) E | S | β (cid:1) β < ∞ for every β ∈ (0 , α ) , where S is the stationary solution of (1.1). The tail measure
This section deals with a heavy tail phenomenon for Lipschitz recursions satisfying assumptions1.7 modeled on analogous hypotheses for matrix recursions (1.5). (H1) and (H2) say that recursion(1.1) is in a sense close to the affine recursion with the linear part M ∈ R ∗ + × K . This allows us touse techniques of [5], in particular a generalized renewal theorem.Conditions 1.7 are typical for considerations of this type and they decide of asymptotic behaviorof stationary measure; especially condition (H5) is crucial. Goldie and Gr¨ubel [8] showed that P ( { S > t } ) can decay exponentially fast to zero if (H5) is not satisfied.A closed subgroup of R ∗ + × O ( R d ) containing R ∗ + is necessarily G = R ∗ + × K , where K is a closedsubgroup of the orthogonal group O ( R d ), see e.g. Appendix C in [5] and Appendix A in [4]. Let drr be the Haar measure of R ∗ + and let ρ be the Haar measure of K such that ρ ( K ) = 1. Anyelement g ∈ R ∗ + × K can be uniquely written as g = rk , where r ∈ R ∗ + and k ∈ K , and so the Haarmeasure λ on R ∗ + × K is R G f ( g ) λ ( dg ) = R R ∗ + R K f ( rk ) ρ ( dk ) drr . Clearly, G is unimodular.Define convolution of a function f with a measure µ on the group G as f ∗ µ ( g ) = Z G f ( gh ) µ ( dh ) . Given f ∈ C b ( R d ), let¯ f ( g ) = E ( f ( gS )) , and χ f ( g ) = ¯ f ( g ) − ¯ f ∗ ¯ µ ( g ) . EAVY TAIL PHENOMENON AND CONVERGENCE TO STABLE LAWS 9
The functions ¯ µ and χ f are bounded and continuous. We are going to express the function ¯ f inthe terms of the potential U = P ∞ k =0 ¯ µ ∗ k . Notice that for any n ∈ N ∪ { } E ( f ( gM θ M θ · . . . · M θ n S )) = Z G E ( f ( ghS ))¯ µ ∗ n ( dh ) = ¯ f ∗ ¯ µ ∗ n ( g ) . Remark 4.1.
Conditions (H5), (H6) imply that the function κ ( s ) = E ( | M | s ) is well defined on [0 , α ] and κ (0) = κ ( α ) = 1 . Since κ is convex, we have E (log( | M | )) < , and m α = E ( | M | α log | M | ) > . For more details we refer to [7] . Let ¯ µ α ( dg ) = | g | α ¯ µ ( dg ). In view of Remark 4.1 ¯ µ α is a probability measure with positive meanand ¯ µ ∗ nα ( dg ) = | g | α ¯ µ ∗ n ( dg ) for all n ∈ N . Let U α = P ∞ k =0 ¯ µ ∗ kα be the potential kernel built out ofthe measure ¯ µ α .The aim of this section is to prove Theorem 4.3 which implies Theorem 1.8. Given any Radonmeasure Λ on R d \ { } , let define F Λ = { f : R d R : f is measurable function such that Λ(Dis( f )) = 0, andsup x ∈ R d | x | − α | log | x || ε | f ( x ) | < ∞ for some ε > } , (4.2)where Dis( f ) is the set of all discontinuities of function f . Theorem 4.3.
Suppose that 1.6 and 1.7 are satisfied. Then there is a unique stationary solution S of (1.1) with the law ν , and there is a unique Radon measure Λ on R d \ { } such that lim | g |→ | g | − α E f ( gS ) = lim | g |→ | g | − α Z R d f ( gx ) ν ( dx ) = Z R d \{ } f ( x )Λ( dx ) , (4.4) for every function f ∈ F Λ . The measure Λ is homogeneous with degree α i.e. R R d f ( gx )Λ( dx ) = | g | α Λ( f ) for every g ∈ G . There exists a measure σ Λ on S d − such that Λ has the polar decompo-sition Z R d \{ } f ( x )Λ( dx ) = Z ∞ Z S d − f ( rx ) σ Λ ( dx ) drr α +1 , (4.5) where σ Λ ( S d − ) = m α E ( | ψ ( S ) | α − | M S | α ) and m α = E ( | M | α log | M | ) ∈ (0 , ∞ ) . Furthermore,recursion defined in (1.1) has a heavy tail lim t →∞ t α P ( {| S | > t } ) = 1 αm α E ( | ψ ( S ) | α − | M S | α ) . (4.6) If additionally the support of ν is unbounded, and one of the following condition is satisfied s ∞ < ∞ and lim s → s ∞ E ( | N | s ) κ ( s ) = 0 , (4.7) s ∞ = ∞ and lim s →∞ (cid:18) E ( | N | s ) κ ( s ) (cid:19) s < ∞ , (4.8) then the measures Λ and σ Λ are nonzero. We divide the proof into three steps. Step 1. (Existence of the tail measure Λ) and Step 3.(Nontriviality of the tail measure Λ) go along the same lines as the Main Theorem 1.6. in [5] so wegive only outlines of proofs. The proof of the existence of a polar decomposition for the measureΛ is shorter here and it is given in the Step 2..
Proof. Step 1. Existence of the tail measure Λ . Assumptions (1.7) imply the existence of thestationary solution S for the recursion (1.1) with the law ν (see [6], [27] for more details). Now,for an ε ∈ (0 , H ε = { f ∈ C b ( R d ) : ∀ x,y ∈ R d | f ( x ) − f ( y ) | ≤ C f | x − y | ε , and f vanishes in a neighbourhood of 0 } . Given f ∈ H ε for some ε ∈ (0 ,
1] and ε < α , we write χ f,α ( g ) = | g | − α χ f ( g ). Using cancellationcondition (H2) and arguing in a similar way as in [5] (see Lemma 2.19) we obtain that the function χ f,α ( g ) = | g | − α χ f ( g ) is d R i ( direct Riemann integrable on G , definition of d R i functions can befound in [5]). Now we can use a renewal theorem for closed subgroups of R ∗ + × K , where K is ametrizable group not necessarily Abelian, (see Appendix A of [5], also [13] and [31]). It is appliedto the function χ f,α ( g ), to obtainlim | g |→ | g | − α ¯ f ( g ) = lim | g |→ U α ( χ f,α )( g ) = 1 m α Z G χ f,α ( g ) λ ( dg ) . (4.9)The formula Λ( f ) = 1 m α Z G χ f,α ( g ) λ ( dg ) = 1 m α Z G | g | − α ( E ( f ( gS ) − f ( gM S ))) λ ( dg ) , defines a nonnegative Radon measure on R d \ { } , which is α homogeneous. Convergence in (4.9)holds also for f ∈ F Λ (compare with the proof of Theorem 2.8 in [5]). Step 2. Polar decomposition for the measure Λ . Being homogeneous Λ can be nicely expressed inpolar coordinates. Let Φ : R d \ { } 7→ (0 , ∞ ) × S d − be defined as follows Φ( x ) = (cid:16) | x | , x | x | (cid:17) andits inverse Φ − : (0 , ∞ ) × S d − R d \ { } by Φ − ( r, z ) = rz . Next we define the measures σ s on S d − σ s ( F ) = s Λ s (cid:0) Φ − [[1 , ∞ ) × F ] (cid:1) , where Λ s ( f ) = m α R G | g | − s ( E ( f ( gS ) − f ( gM S ))) λ ( dg ) for s < α and F ∈ B or ( S d − ). B or ( X )means the Borel σ –field of X . Fix 0 < β < γ and notice that for any [ β, γ ) × F ∈ B or ((0 , ∞ )) ⊗B or ( S d − ), (cid:0) Λ s ◦ Φ − (cid:1) ([ β, γ ) × F ) = σ s ( F ) Z γβ drr s +1 . The above proves (4.5) with the measure Λ s instead of Λ. Now notice that σ s ( S d − ) = s Λ s (cid:0) Φ − (cid:2) [1 , ∞ ) × S d − (cid:3)(cid:1) = 1 m α E ( | ψ ( S ) | s − | M S | s ) . Hence (4.5) holds. Furthermore,lim t →∞ t α P ( {| S | > t } ) = lim t →∞ t α Z ∞ t drr α +1 σ Λ ( S d − ) = 1 αm α E ( | ψ ( S ) | α − | M S | α ) , and (4.6) also holds. Step 3. Nontriviality of the tail measure Λ . In order to prove that measure Λ is nontrivial in viewof (4.5) we have to show that σ Λ = 0. Suppose for a contradiction that σ Λ ( S d − ) = 0. Applying themethod from section 3 from [5] together with condition (H2) we obtain that the stationary solution S is bounded, but it contradicts with the fact that the support of the measure ν is unbounded andthis finishes the proof of Theorem 4.3. (cid:3) EAVY TAIL PHENOMENON AND CONVERGENCE TO STABLE LAWS 11
The example below shows that for (4.7) and (4.8) the hypothesis that the support of the measure ν is unbounded is crucial. Consider ψ n ( x ) = A n max { x, B n } + C n and assume that P (cid:0)(cid:8) A n = (cid:9)(cid:1) = , P ( { A n = 2 } ) = and P (cid:0)(cid:8) B n = (cid:9)(cid:1) = P ( { C n = − } ) = 1. Then E (log A n ) < E ( A αn ) =1, where α ≈ , ν is supported by the set (cid:8) − , (cid:9) though the function ψ n ( x ) is unbounded.5. Fourier operators and their properties
As it is mentioned in subsection 1.2, for the limit Theorem 1.15, we study the Markov operator P associated to the recursion (1.1) as well as the perturbations P t,v of P defined in (1.16). To expandthe dominant eigenvalue k v ( t ) defined in (1.17) we need some information about the eigenfunctionsΠ P,t t ≥
0. While t varies, the normalization of Π P,t
P,t T t,v . They are used in [4] to obtain an explicit expression forΠ P,t
1, but they are written there by the formula that does not work beyond the affine recursion.However, a careful analysis of operators T t,v suggests to write them abstractly as(5.1) T t,v = ∆ − t ◦ P t,v ◦ ∆ t , where ∆ t is the dilatation ∆ t f ( x ) = f ( tx ). The “abstract” T t,v ’s do the same job making themethod applicable to a much more general context.We start by introducing two Banach spaces C ρ ( R d ) and B ρ,ǫ,λ ( R d ) of continuous functions [26](see also [4, 15, 18, 19] and [20]). C ρ = C ρ ( R d ) = (cid:26) f ∈ C ( R d ) : | f | ρ = sup x ∈ R d | f ( x ) | (1 + | x | ) ρ < ∞ (cid:27) , B ρ,ǫ,λ = B ρ,ǫ,λ ( R d ) = { f ∈ C ( R d ) : k f k ρ,ǫ,λ = | f | ρ + [ f ] ǫ,λ < ∞} , where [ f ] ǫ,λ = sup x = y | f ( x ) − f ( y ) || x − y | ǫ (1 + | x | ) λ (1 + | y | ) λ . Remark 5.2. If ǫ + λ < ρ , then [ f ] ǫ,λ < ∞ implies | f | ρ < ∞ . As a simple application of Arzel`a– Ascoli theorem we obtain that the injection operator B ρ,ǫ,λ ֒ → C ρ is compact. From now on we assume that ψ θ satisfies 1.6, 1.7 and 1.14 for every θ ∈ Θ. For the sakeof simplicity we write ψ instead of ψ θ . On C ρ and B ρ,ǫ,λ we consider the transition operator P f ( x ) = E ( f ( ψ ( x ))) and its perturbations P t,v f ( x ) = E (cid:16) e i h tv,ψ ( x ) i f ( ψ ( x )) (cid:17) , defined in (1.16), where x ∈ R d , v ∈ S d − and t ∈ [0 , P ,v = P . For conveniencewe write ψ t = ψ for t = 0, ( ψ was defined in (H1) of assumption (1.6)). We will also use the familyof Fourier operators T t,v defined in (5.1). Notice that T t,v f ( x ) = (cid:0) ∆ − t ◦ P t,v ◦ ∆ t (cid:1) f ( x ) = E (cid:16) e i h v,ψ t ( x ) i f ( ψ t ( x )) (cid:17) , for x ∈ R d , where t ∈ [0 ,
1] and v ∈ S d − . For simplicity we will write T v = T ,v . T t,v areperturbations of T v .Clearly, for t > T t,v and P t,v , as being dilations of each other, have the same periphericaleigenvalues k v ( t ), but for t = 0, the relation between T v and P is not that close. Therefore, by considering T v and T t,v we obtain some extra information when t →
0. In particular, the eigen-function h v of T v with the eigenvalue 1 plays a vital role in approximating peripherical eigenvectorsof P t,v .To treat both P t,v and T t,v in a unified way we write F s,t,v f ( x ) = E (cid:16) e i h sv,ψ t ( x ) i f ( ψ t ( x )) (cid:17) = Z Θ e i h sv,ψ θ,t ( x ) i f ( ψ θ,t ( x )) µ ( dθ ) . Notice that F s, ,v f ( x ) = E (cid:16) e i h sv,ψ ( x ) i f (cid:0) ψ ( x ) (cid:1)(cid:17) = R Θ e i h sv,ψ θ ( x ) i f (cid:0) ψ θ ( x ) (cid:1) µ ( dθ ) , and F ,t,v f ( x ) = E ( f ( ψ t ( x ))) = R Θ f ( ψ θ,t ( x )) µ ( dθ ) , for x ∈ R d , where s, t ∈ [0 ,
1] and v ∈ S d − . Observe that, F s, ,v = P s,v and F ,t,v = T t,v .Now by the definition (5.1) it is easy to see, that for every n ∈ N and t ∈ [0 , P nt,v ◦ ∆ t = ∆ t ◦ T nt,v . (5.3)Moreover, if f ∈ C ρ is eigenfunction of operator T t,v with eigenvalue k v ( t ), then ∆ t f is an eigen-function of the operator P t,v with the same eigenvalue. The main result of this section is thefollowing Proposition 5.4.
Assume that < ǫ < , λ > , λ + 2 ǫ < ρ = 2 λ and λ + ǫ < α , then thereexist < ̺ < , δ > and t > such that ̺ < − δ , and for every t ∈ [0 , t ] and every v ∈ S d − • σ ( P t,v ) and σ ( T t,v ) are contained in D = { z ∈ C : | z | ≤ ̺ } ∪ { z ∈ C : | z − | ≤ δ } . • The sets σ ( P t,v ) ∩ { z ∈ C : | z − | ≤ δ } and σ ( T t,v ) ∩ { z ∈ C : | z − | ≤ δ } consist ofexactly one eigenvalue k v ( t ) , where lim t → k v ( t ) = 1 , and the corresponding eigenspace isone dimensional. • For any z ∈ D c and every f ∈ B ρ,ǫ,λ (cid:13)(cid:13) ( z − P t,v ) − f (cid:13)(cid:13) ρ,ǫ,λ ≤ D k f k ρ,ǫ,λ , and (cid:13)(cid:13) ( z − T t,v ) − f (cid:13)(cid:13) ρ,ǫ,λ ≤ D k f k ρ,ǫ,λ , where D > is universal constant which does not depend on t ∈ [0 , t ] . • Moreover, we can express operators P t,v and T t,v in the following form P nt,v = k v ( t ) n Π P,t + Q nP,t , and T nt,v = k v ( t ) n Π T,t + Q nT,t , for every n ∈ N , where Π P,t and Π T,t are projections onto the one dimensional eigenspacesmentioned above. Q P,t and Q T,t are the complementary operators to projections Π P,t and Π T,t respectively, such that Π P,t Q P,t = Q P,t Π P,t = 0 and Π T,t Q T,t = Q T,t Π T,t = 0 . Fur-thermore, k Q nP,t k B ρ,ǫ,λ = O ( ̺ n ) and k Q nT,t k B ρ,ǫ,λ = O ( ̺ n ) for every n ∈ N . The operators Π P,t , Π T,t , Q P,t and Q T,t depend on v ∈ S d − , but this is omitted for simplicity. • The above operators can be expressed in the terms of the resolvents of P t,v and T t,v . Indeed,for appropriately chosen ξ > and ξ > we have k v ( t )Π F,t = 12 πi Z | z − | = ξ z ( z − F t,v ) − dz, Π F,t = 12 πi Z | z − | = ξ ( z − F t,v ) − dz,Q F,t = 12 πi Z | z | = ξ z ( z − F t,v ) − dz, where F t,v = P t,v or F t,v = T t,v . Proposition 5.4 is a consequence of the perturbation theorem of Keller and Liverani [23] see also[28]. Before we apply their theorem we will check in a number of Lemmas that its assumptions aresatisfied.
EAVY TAIL PHENOMENON AND CONVERGENCE TO STABLE LAWS 13
Lemma 5.5.
For every n ∈ N F ns,t,v f ( x ) = E (cid:16) e i h sv,S xn,t i f (cid:0) X xn,t (cid:1)(cid:17) , (5.6) where X xn,t is defined as in Lemma 3.10 and S xn,t = P nk =1 X xk,t .Proof. Formula (5.6) is obvious, since { X xn,t } n ≥ is a Markov chain. (cid:3) Remark 5.7.
The formula (5.6) implies that for every < ρ < α , there exists a constant C > independent of s, t ∈ [0 , and v ∈ S d − such that for every n ∈ N (cid:12)(cid:12) F ns,t,v f (cid:12)(cid:12) ρ ≤ C | f | ρ . (5.8)Let denote Π n = L θ · . . . · L θ n for n ∈ N and Π = 1. The inequality | e ix − | ≤ | x | ε for0 < ε ≤ x ∈ R will be used repeatedly. Lemma 5.9.
Assume that < ǫ < , λ > , λ + ǫ < α , and ρ = 2 λ . Then there exist constants C > , C > and < ̺ < independent of s, t ∈ [0 , and v ∈ S d − such that for every f ∈ B ρ,ǫ,λ and n ∈ N (cid:2) F ns,t,v f (cid:3) ǫ,λ ≤ C ̺ n [ f ] ǫ,λ + C | f | ρ . (5.10) Proof.
By the definition of the seminorm [ · ] ǫ,λ we have F ns,t,v f ( x ) − F ns,t,v f ( y ) = E (cid:16) e i h sv,S xn,t i (cid:0) f (cid:0) X xn,t (cid:1) − f (cid:0) X yn,t (cid:1)(cid:1)(cid:17) (5.11) + E (cid:16)(cid:16) e i h sv,S xn,t i − e i h sv,S yn,t i (cid:17) f (cid:0) X yn,t (cid:1)(cid:17) . (5.12)To obtain (5.10) we have to estimate (5.11) and (5.12) separately. Indeed, (cid:12)(cid:12)(cid:12) E (cid:16) e i h sv,S xn,t i (cid:0) f (cid:0) X xn,t (cid:1) − f (cid:0) X yn,t (cid:1)(cid:1)(cid:17)(cid:12)(cid:12)(cid:12) | x − y | ǫ (1 + | x | ) λ (1 + | y | ) λ (5.13) ≤ [ f ] ǫ,λ · E (cid:12)(cid:12) X xn,t − X yn,t (cid:12)(cid:12) ǫ (cid:0) (cid:12)(cid:12) X xn,t (cid:12)(cid:12)(cid:1) λ (cid:0) (cid:12)(cid:12) X yn,t (cid:12)(cid:12)(cid:1) λ | x − y | ǫ (1 + | x | ) λ (1 + | y | ) λ ! ≤ [ f ] ǫ,λ · E Π ǫn (cid:0) (cid:12)(cid:12) X n,t (cid:12)(cid:12) + Π n | x | (cid:1) λ (cid:0) (cid:12)(cid:12) X n,t (cid:12)(cid:12) + Π n | y | (cid:1) λ (1 + | x | ) λ (1 + | y | ) λ ! ≤ [ f ] ǫ,λ · E (cid:16) Π ǫn (cid:0) Π n + (cid:12)(cid:12) X n,t (cid:12)(cid:12) + 1 (cid:1) λ (cid:17) ≤ λ [ f ] ǫ,λ · (cid:16) E (cid:0) Π λ + ǫn (cid:1) + E (cid:16) Π ǫn (cid:12)(cid:12) X n,t (cid:12)(cid:12) λ (cid:17) + E (Π ǫn ) (cid:17) . Now let ̺ = max (cid:8) κ ( ǫ ) , κ (2 λ + ǫ ) , κ ǫ λ + ǫ (2 λ + ǫ ) (cid:9) <
1. Applying the H¨older inequality to the lastexpression, we obtain 3 λ [ f ] ǫ,λ · (cid:16) E (cid:0) Π λ + ǫn (cid:1) + E (cid:16) Π ǫn (cid:12)(cid:12) X n,t (cid:12)(cid:12) λ (cid:17) + E (Π ǫn ) (cid:17) (5.14) ≤ λ [ f ] ǫ,λ · (cid:18) κ (2 λ + ǫ ) n + t λ (cid:0) κ ǫ λ + ǫ (2 λ + ǫ ) (cid:1) n E (cid:16)(cid:12)(cid:12) X n (cid:12)(cid:12) λ + ǫ (cid:17) λ λ + ǫ + κ ( ǫ ) n (cid:19) ≤ λ ̺ n [ f ] ǫ,λ · (cid:18) t λ E (cid:16)(cid:12)(cid:12) X n (cid:12)(cid:12) λ + ǫ (cid:17) λ λ + ǫ (cid:19) ≤ C ̺ n [ f ] ǫ,λ , where by Lemma 3.10 the constant C = 3 λ sup n ∈ N (cid:18) E (cid:16)(cid:12)(cid:12) X n (cid:12)(cid:12) λ + ǫ (cid:17) λ λ + ǫ (cid:19) is finite. In order to estimate (5.12) notice that we have (cid:12)(cid:12) S xn,t − S yn,t (cid:12)(cid:12) ≤ n X k =1 (cid:12)(cid:12)(cid:12) X xk,t − X yk,t (cid:12)(cid:12)(cid:12) ≤ n X k =1 Π k | x − y | ≤ B n | x − y | , where B n = P nk =0 Π k . Assume that | y | ≤ | x | , then (cid:12)(cid:12)(cid:12) E (cid:16)(cid:16) e i h sv,S xn,t i − e i h sv,S yn,t i (cid:17) f (cid:0) X yn,t (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) | x − y | ǫ (1 + | x | ) λ (1 + | y | ) λ (5.15) ≤ | f | ρ · E (cid:12)(cid:12)(cid:12) e i h sv,S xn,t − S yn,t i − (cid:12)(cid:12)(cid:12) (cid:0) (cid:12)(cid:12) X yn,t (cid:12)(cid:12)(cid:1) ρ | x − y | ǫ (1 + | x | ) λ (1 + | y | ) λ ≤ s ǫ | f | ρ · E B ǫn (cid:0) (cid:12)(cid:12) X n,t (cid:12)(cid:12) + Π n (cid:1) ρ (1 + | y | ) ρ (1 + | y | ) λ ! ≤ · ρ s ǫ | f | ρ · E (cid:16) B ǫn + t ρ B ǫn (cid:12)(cid:12) X n (cid:12)(cid:12) ρ + B ǫn Π ρn (cid:17) ≤ C | f | ρ , where the constant C = sup n ∈ N · ρ · E (cid:0) B ǫn + B ǫn (cid:12)(cid:12) X n (cid:12)(cid:12) ρ + B ǫn Π ρn (cid:1) is finite by the similar argumentas in the previous case. This completes the proof of the Lemma. (cid:3) Lemma 5.16.
Assume that < ǫ < , λ > , λ + ǫ < α , ρ = 2 λ and λ + 2 ǫ < ρ . Then thereexist finite constants C > and C > independent of s, t ∈ [0 , and of v ∈ S d − such that forevery f ∈ B ρ,ǫ,λ | ( F s,t,v − F s, ,v ) f | ρ ≤ C t ǫ k f k ρ,ǫ,λ , (5.17) | ( F s,t,v − F ,t,v ) f | ρ ≤ C s ǫ k f k ρ,ǫ,λ . (5.18)Notice that this Lemma also applies to the special case when F , ,v = T v . Proof.
In order to prove (5.17) we write( F s,t,v − F s, ,v ) f ( x ) = E (cid:16) e i h sv,ψ t ( x ) i (cid:0) f ( ψ t ( x )) − f (cid:0) ψ ( x ) (cid:1)(cid:1)(cid:17) (5.19) + E (cid:16)(cid:16) e i h sv,ψ t ( x ) i − e i h sv,ψ ( x ) i (cid:17) f (cid:0) ψ ( x ) (cid:1)(cid:17) . (5.20)Now we estimate (5.19) and (5.20) separately. By the definition of map ψ we know that ψ (0) = 0,so (cid:12)(cid:12) ψ ( x ) (cid:12)(cid:12) ≤ | M || x | . Then condition (L2) implies that | ψ t ( x ) | ≤ t | Q | + | M || x | and so (cid:12)(cid:12) E (cid:0) e i h sv,ψ t ( x ) i (cid:0) f ( ψ t ( x )) − f (cid:0) ψ ( x ) (cid:1)(cid:1)(cid:1)(cid:12)(cid:12) (1 + | x | ) ρ ≤ E (cid:12)(cid:12) f ( ψ t ( x )) − f (cid:0) ψ ( x ) (cid:1)(cid:12)(cid:12) (1 + | x | ) ρ ! (5.21) ≤ [ f ] ǫ,λ · E (cid:12)(cid:12) ψ t ( x ) − ψ ( x ) (cid:12)(cid:12) ǫ (1 + | ψ t ( x ) | ) λ (cid:0) (cid:12)(cid:12) ψ ( x ) (cid:12)(cid:12)(cid:1) λ (1 + | x | ) ρ ! ≤ t ǫ [ f ] ǫ,λ · E (cid:18) | Q | ǫ (1 + t | Q | + | M || x | ) λ (1 + | x | ) ρ (cid:19) ≤ λ t ǫ [ f ] ǫ,λ · E (cid:0) | Q | ǫ + t λ | Q | λ + ǫ + | Q | ǫ | M | λ (cid:1) ≤ D t ǫ [ f ] ǫ,λ . The quantity D = 3 λ · E (cid:0) | Q | ǫ + | Q | λ + ǫ + | Q | ǫ | M | λ (cid:1) is finite according to H¨older’s inequality,(H5) and (L3). EAVY TAIL PHENOMENON AND CONVERGENCE TO STABLE LAWS 15
For (5.20), we have (cid:12)(cid:12)(cid:12) E (cid:16)(cid:16) e i h sv,ψ t ( x ) i − e i h sv,ψ ( x ) i (cid:17) f (cid:0) ψ ( x ) (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) (1 + | x | ) ρ (5.22) ≤ | f | ρ · E (cid:12)(cid:12)(cid:12) e i h sv,ψ t ( x ) − ψ ( x ) i − (cid:12)(cid:12)(cid:12) (1 + | M || x | ) ρ (1 + | x | ) ρ ≤ s ǫ t ǫ | f | ρ · E ( | Q | ǫ (1 + | M | ) ρ ) ≤ ρ +1 s ǫ t ǫ | f | ρ · E ( | Q | ǫ + | Q | ǫ | M | ρ ) ≤ D t ǫ | f | ρ , where the constant D = 2 ρ +1 · E ( | Q | ǫ + | Q | ǫ | M | ρ ) is also finite by the H¨older inequality, (H5) and(L3). Combining (5.21) with (5.22) we obtain (5.17) with C = max { D , D } .In order to prove (5.18) notice that | ( F s,t,v − F ,t,v ) f ( x ) | (1 + | x | ) ρ ≤ E (cid:12)(cid:12) e i h sv,ψ t ( x ) i f ( ψ t ( x )) − f ( ψ t ( x )) (cid:12)(cid:12) (1 + | x | ) ρ ! (5.23) ≤ E (cid:18) | e i h sv,ψ t ( x ) i − || f ( ψ t ( x )) − f (0) | (1 + | x | ) ρ (cid:19) + E (cid:18) | e i h sv,ψ t ( x ) i − || f (0) | (1 + | x | ) ρ (cid:19) ≤ s ǫ (cid:18) [ f ] ǫ,λ · E (cid:18) | ψ t ( x ) | ǫ (1 + | ψ t ( x ) | ) λ (1 + | x | ) ρ (cid:19) + | f | ρ · E (cid:18) | ψ t ( x ) | ǫ (1 + | x | ) ρ (cid:19)(cid:19) ≤ λ +1 s ǫ k f k ρ,ǫ,λ · E (cid:18) | ψ t ( x ) | ǫ + | ψ t ( x ) | λ +2 ǫ + | ψ t ( x ) | ǫ (1 + | x | ) ρ (cid:19) ≤ C s ǫ k f k ρ,ǫ,λ , where C = 2 λ +1 · E (cid:16) (1 + | M | + | Q | ) ǫ + (1 + | M | + | Q | ) λ +2 ǫ + (1 + | M | + | Q | ) ǫ (cid:17) is finite by (H5)and (L3). Hence (5.23) proves (5.18) and finally it completes the proof of the Lemma. (cid:3) Lemma 5.24.
The unique eigenvalue of modulus for operator P acting on C ρ is and theeigenspace is one dimensional. The corresponding projection on C · is given by the map f ν ( f ) .Proof. The proof can be found in section 3 of [4]. (cid:3)
Recall, that for every n ∈ N , T nv f ( x ) = E (cid:16) e i h v, P nk =1 ψ k ◦ ... ◦ ψ ( x ) i f (cid:0) ψ n ◦ . . . ◦ ψ ( x ) (cid:1)(cid:17) , where ψ k ( x ) = ψ θ k ( x ), and ψ θ k ’s were defined in (H1) of assumption 1.6. Let h v ( x ) = E (cid:16) e i h v, P ∞ k =1 ψ k ◦ ... ◦ ψ ( x ) i (cid:17) . (5.25) Lemma 5.26.
The function h v defined in (5.25) belongs to B ρ,ǫ,λ , and h v ( tx ) = h tv ( x ) for every x ∈ R d and t > . Moreover, | h v ( x ) | ≤ , (5.27) | h v ( x ) − h v ( y ) | ≤ − κ ( δ ) | x − y | δ , (5.28) for every x, y ∈ R d and every < δ ≤ such that < δ < α . Proof.
Inequality, (5.27) is obvious, (5.28) follows from the definition of function h v and inequality (cid:12)(cid:12) ψ k ( x ) − ψ k ( y ) (cid:12)(cid:12) ≤ | M || x − y | for k ∈ N , where ψ k was defined in (H1) of assumption 1.6. Inorder to prove that h v ( tx ) = h tv ( x ) it is enough to show that for a fixed s > x ∈ R d , ψ ( sx ) = sψ ( x ). Indeed, for every ε > η > (cid:12)(cid:12) tψ ( t − sx ) − ψ ( sx ) (cid:12)(cid:12) < ε forevery 0 < t < sη . Hence if t = rs and 0 < r < η then (cid:12)(cid:12) srψ ( r − x ) − ψ ( sx ) (cid:12)(cid:12) < ε . Letting r tend to0 we obtain sψ ( x ) = ψ ( sx ). (cid:3) Lemma 5.29.
The unique eigenvalue of modulus for operator T v acting on C ρ is with theeigenspace C · h v ( x ) , where function h v was defined in (5.25).Proof. Notice that lim n →∞ ψ n ◦ . . . ◦ ψ ( x ) = 0 a.e., ( ψ k ’s were defined in (H1) of assumption 1.6).Take f ∈ C ρ , then by the Lebesgue dominated convergence theorem we have T nv f ( x ) = E (cid:16) e i h v, P nk =1 ψ k ◦ ... ◦ ψ ( x ) i (cid:0) f (cid:0) ψ n ◦ . . . ◦ ψ ( x ) (cid:1) − f (0) (cid:1)(cid:17) + E (cid:16) e i h v, P nk =1 ψ k ◦ ... ◦ ψ ( x ) i f (0) (cid:17) −−−→ n →∞ f (0) h v ( x ) , for every x ∈ R d . Since h v (0) = 1 the above convergence shows that 1 is a simple eigenvalue forthe action of T v on B ρ,ǫ,λ with h v as the unique associated eigenfunction (up to multiplicativeconstant). It also proves that 1 is the unique peripheral eigenvalue. (cid:3) Proof of Proposition 5.4.
Remark 5.7 implies that ker( T v − I ) = ker( T v − I ) . By Lemmas 5.9,5.24 and 5.29, we have thanks to [17] and [21], that for every n ∈ N ∀ f ∈ B ρ,ǫ,λ P n f = P n ,v f = ν ( f ) · Q nP, f, and T nv f = T n ,v f = f (0) · h v + Q nT, f, where Q P, and Q T, are the complementary operators to the projections Π P, (Π P, f = ν ( f ) · T, (Π T, f = f (0) · h v ) respectively.Besides, in view of Lemmas 5.9 and 5.16, (in particular (5.17) and (5.18) imply ∀ f ∈ B ρ,ǫ,λ | ( T t,v − T v ) f | ρ ≤ C t ǫ k f k ρ,ǫ,λ , and | ( P t,v − P ) f | ρ ≤ C t ǫ k f k ρ,ǫ,λ , respectively for every t ∈ [0 , P t,v and T t,v to get Proposition5.4. (cid:3) Remark 5.30. If z ∈ σ ( P t,v ) or z ∈ σ ( T t,v ) and | z | > ̺ , where < ̺ < is defined as in Lemma5.9, then z does not belong to the residual spectrum of the operator P t,v or T t,v (see [17] and [21] ).But thanks to the improvement of [23] given in [28] (see also [20] ) condition on the essential spectralradius of P t,v (and T t,v ) is not required for t = 0 . Rate of convergence and fractional expansions
Rate of convergence of projections.
As it has been already mentioned, to write downan expansion of k v ( t ) sufficiently good for the limit Theorem 1.15, we study the periphericaleigenfunctions of P t,v and, when t varies, the normalization is important. We have two naturalcandidates Π P,t t Π T,t h v one being a multiple of the other∆ t Π T,t h v ( x ) = c t Π P,t ( x ) . EAVY TAIL PHENOMENON AND CONVERGENCE TO STABLE LAWS 17
Notice that, if P t,v f t = k v ( t ) f t , then( k v ( t ) − · ν ( f t ) = ν (( P t,v − f t )= Z (cid:16) e it h v,ψ θ ( x ) i − (cid:17) f t ( ψ θ ( x )) µ ( dθ ) ν ( dx )= Z (cid:16) e it h v,x i − (cid:17) f t ( x ) ν ( dx ) . Therefore, for both Π
P,t t Π T,t h v we have( k v ( t ) − · ν (Π P,t
1) = ν (cid:16)(cid:16) e it h v, ·i − (cid:17) · (Π P,t (cid:17) , (6.1)and ( k v ( t ) − · ν (∆ t Π T,t f ) = ν (cid:16)(cid:16) e it h v, ·i − (cid:17) · (∆ t Π T,t f ) (cid:17) . (6.2)We may use either Π P,t t Π T,t h v and approximate it by h v . Wechoose the second possibility and we prove the following Theorem 6.3.
Let h v be the eigenfunction for operator T v defined in (5.25). Then for any < δ ≤ such that ǫ < δ < α , there exist C > and D > such that k ∆ t (Π T,t − Π T, ) h v k ρ,ǫ,λ ≤ Ct δ , (6.4) and ν (∆ t Π T,t h v − ≤ Dt δ , (6.5) for every < t ≤ t . Moreover, for every x ∈ R d and every < t ≤ t | Π T,t ( h v )( tx ) − Π T, ( h v )( tx ) | ≤ Ct δ (1 + | x | ) ρ . (6.6) Remark 6.7.
The use of the family { T t,v } t> above is much more efficient than that of { P t,v } t> .Indeed, the difference | P t,v f ( x ) − P f ( x ) | involves the term (cid:12)(cid:12) e i h tv,ψ ( x ) i − (cid:12)(cid:12) which depends on x , while | T t,v f ( x ) − T v f ( x ) | , leads to (cid:12)(cid:12)(cid:12) e i h v,ψ t ( x ) i − e i h v,ψ ( x ) i (cid:12)(cid:12)(cid:12) ≤ | t || Q | independently of x by assumption (L2).This is the main new idea in comparison to [4] and it allows to prove (6.6). In (6.6) < δ ≤ satisfies ǫ < δ < α while ρ can be chosen small. This cannot be proved for Π P,t directly. Moreover,in Subsection 6.2, we shall deduce from (6.6) the following important property (6.8) Z R d (cid:16) e it h v,x i − (cid:17) (Π T,t ( h v )( tx ) − h v ( tx )) ν ( dx ) = o ( t α ) as t → , or o ( t log t ) when α = 2 . Therefore, for < α < k v ( t ) − t α ≈ t α Z R d (cid:16) e it h v,x i − (cid:17) h v ( tx ) ν ( dx ) , when t → (if α = 2 we have t | log t | instead of t α in the above denominators) and the right handside of (6.9) is further studied in the next subsection. Before we prove Theorem 6.3 we need two lemmas.
Lemma 6.10.
Assume that the function f satisfies | f ( x ) | ≤ C for any x ∈ R d , and | f ( x ) − f ( y ) | ≤ C | x − y | δ for any < δ ≤ and x, y ∈ R d , where constant C > depends on δ . Then for every δ ∈ ( ǫ, α ) [( T t,v − T v ) f ] ǫ,λ ≤ C t δ − ǫ , (6.11) | ( T t,v − T v ) f | ρ ≤ C t δ , (6.12) where C > and C > do not depend on < t ≤ t . Proof.
In order to show (6.11) we have to estimate the seminorm [( T t,v − T v ) f ] ǫ,λ . Notice, that[( T t,v − T v ) f ] ǫ,λ ≤ sup x = y, | x − y |≤ t | ( T t,v − T v ) f ( x ) − ( T t,v − T v ) f ( y ) || x − y | ǫ (1 + | x | ) λ (1 + | y | ) λ (6.13) + sup x = y, | x − y | >t | ( T t,v − T v ) f ( x ) − ( T t,v − T v ) f ( y ) || x − y | ǫ (1 + | x | ) λ (1 + | y | ) λ . For the first term in (6.13) ( | x − y | ≤ t ) we observe that( T t,v − T v ) f ( x ) − ( T t,v − T v ) f ( y ) == E (cid:16)(cid:16) e i h v,ψ t ( x ) i − e i h v,ψ t ( y ) i (cid:17) f ( ψ t ( x )) (cid:17) (6.14) + E (cid:16) e i h v,ψ t ( y ) i ( f ( ψ t ( x )) − f ( ψ t ( y ))) (cid:17) (6.15) − E (cid:16)(cid:16) e i h v,ψ ( x ) i − e i h v,ψ ( y ) i (cid:17) f (cid:0) ψ ( x ) (cid:1)(cid:17) (6.16) − E (cid:16) e i h v,ψ ( y ) i (cid:0) f (cid:0) ψ ( x ) (cid:1) − f (cid:0) ψ ( y ) (cid:1)(cid:1)(cid:17) . (6.17)We will estimate (6.14), (6.15), (6.16) and (6.17) separately. By assumptions on the function f observe, that for every 0 < δ ≤ ǫ < δ < α we have E (cid:12)(cid:12) e i h v,ψ t ( x ) i − e i h v,ψ t ( y ) i (cid:12)(cid:12) | f ( ψ t ( x )) || x − y | ǫ (1 + | x | ) λ (1 + | y | ) λ ! ≤ C E (cid:18) | ψ t ( x ) − ψ t ( y ) | δ | x − y | ǫ (cid:19) (6.18) ≤ C E (cid:0) | M | δ (cid:1) | x − y | δ − ǫ ≤ C E (cid:0) | M | δ (cid:1) t δ − ǫ . Similarly, we obtain the estimate of the second term. Indeed, E (cid:12)(cid:12) e i h v,ψ t ( y ) i ( f ( ψ t ( x )) − f ( ψ t ( y ))) (cid:12)(cid:12) | x − y | ǫ (1 + | x | ) λ (1 + | y | ) λ ! ≤ E (cid:18) | f ( ψ t ( x )) − f ( ψ t ( y )) || x − y | ǫ (cid:19) (6.19) ≤ C E (cid:0) | M | δ (cid:1) | x − y | δ − ǫ ≤ C E (cid:0) | M | δ (cid:1) t δ − ǫ . Remaining (6.16) and (6.17) are estimated in the similar way, since (cid:12)(cid:12) ψ ( x ) − ψ ( y ) (cid:12)(cid:12) ≤ | M || x − y | bydefinition of ψ . Now consider the second term of (6.13) ( | x − y | > t ) and notice, that( T t,v − T v ) f ( x ) − ( T t,v − T v ) f ( y )= E (cid:16)(cid:16) e i h v,ψ t ( x ) i − e i h v,ψ ( x ) i (cid:17) f ( ψ t ( x )) (cid:17) (6.20) + E (cid:16) e i h v,ψ ( x ) i (cid:0) f ( ψ t ( x )) − f (cid:0) ψ ( x ) (cid:1)(cid:1)(cid:17) (6.21) − E (cid:16)(cid:16) e i h v,ψ t ( y ) i − e i h v,ψ ( y ) i (cid:17) f ( ψ t ( y )) (cid:17) (6.22) − E (cid:16) e i h v,ψ ( y ) i (cid:0) f ( ψ t ( y )) − f (cid:0) ψ ( y ) (cid:1)(cid:1)(cid:17) . (6.23)As before, we will estimate (6.20), (6.21), (6.22) and (6.23) separately using (L2) and (L3). Indeed,for every 0 < δ ≤ ǫ < δ < α we have E (cid:12)(cid:12)(cid:12) e i h v,ψ t ( x ) i − e i h v,ψ ( x ) i (cid:12)(cid:12)(cid:12) | f ( ψ t ( x )) || x − y | ǫ (1 + | x | ) λ (1 + | y | ) λ ≤ C E (cid:12)(cid:12) ψ t ( x ) − ψ ( x ) (cid:12)(cid:12) δ | x − y | ǫ ! (6.24) ≤ C E (cid:18) t δ | Q | δ | x − y | ǫ (cid:19) ≤ C E (cid:0) | Q | δ (cid:1) t δ − ǫ . EAVY TAIL PHENOMENON AND CONVERGENCE TO STABLE LAWS 19
Similarly, we obtain the estimate for the second term. Indeed, E (cid:12)(cid:12)(cid:12) e i h v,ψ ( x ) i (cid:0) f ( ψ t ( x )) − f (cid:0) ψ ( x ) (cid:1)(cid:1)(cid:12)(cid:12)(cid:12) | x − y | ǫ (1 + | x | ) λ (1 + | y | ) λ ≤ E (cid:12)(cid:12) f ( ψ t ( x )) − f (cid:0) ψ ( x ) (cid:1)(cid:12)(cid:12) | x − y | ǫ ! (6.25) ≤ C E (cid:12)(cid:12) ψ t ( x ) − ψ ( x ) (cid:12)(cid:12) δ | x − y | ǫ ! ≤ C E (cid:18) t δ | Q | δ | x − y | ǫ (cid:19) ≤ C E (cid:0) | Q | δ (cid:1) t δ − ǫ . Also remaining (6.22) and (6.23) can be estimated analogously. Hence, in view of (6.18), (6.19),(6.24) and (6.25), we obtain (6.11). For (6.12) notice that( T t,v − T v ) f ( x ) = E (cid:16)(cid:16) e i h v,ψ t ( x ) i − e i h v,ψ ( x ) i (cid:17) f ( ψ t ( x )) (cid:17) (6.26) + E (cid:16) e i h v,ψ ( x ) i (cid:0) f ( ψ t ( x )) − f (cid:0) ψ ( x ) (cid:1)(cid:1)(cid:17) . We have, E (cid:12)(cid:12)(cid:12) e i h v,ψ t ( x ) i − e i h v,ψ ( x ) i (cid:12)(cid:12)(cid:12) | f ( ψ t ( x )) | (1 + | x | ) ρ ≤ C E (cid:16)(cid:12)(cid:12) ψ t ( x ) − ψ ( x ) (cid:12)(cid:12) δ (cid:17) (6.27) ≤ C E (cid:0) t δ | Q | δ (cid:1) ≤ C E (cid:0) | Q | δ (cid:1) t δ , and E (cid:12)(cid:12)(cid:12) e i h v,ψ ( x ) i (cid:0) f ( ψ t ( x )) − f (cid:0) ψ ( x ) (cid:1)(cid:1)(cid:12)(cid:12)(cid:12) (1 + | x | ) ρ ≤ E (cid:0)(cid:12)(cid:12) f ( ψ t ( x )) − f (cid:0) ψ ( x ) (cid:1)(cid:12)(cid:12)(cid:1) (6.28) ≤ C E (cid:16)(cid:12)(cid:12) ψ t ( x ) − ψ ( x ) (cid:12)(cid:12) δ (cid:17) ≤ C E (cid:0) t δ | Q | δ (cid:1) ≤ C E (cid:0) | Q | δ (cid:1) t δ . Combining (6.27) with (6.28) we obtain (6.12) which completes the proof of the Lemma. (cid:3)
Lemma 6.29.
Let h v be the eigenfunction for operator T v defined in (5.25), then ∆ t (Π T,t − Π T, ) h v = 12 πi Z | z − | = ξ z − z − P t,v ) − ∆ t ( T t,v − T v ) h v dz, (6.30) where ξ > was defined in Proposition 5.4.Proof. Notice, that ( z − T v ) − h v = z − h v ,( z − T t,v ) − − ( z − T v ) − = ( z − T t,v ) − ( T t,v − T v )( z − T v ) − , and by the definition (5.1) ( z − P t,v ) − ∆ t = ∆ t ( z − T t,v ) − , Then ∆ t (Π T,t − Π T, ) h v = 12 πi Z | z − | = ξ ∆ t ( z − T t,v ) − ( T t,v − T v )( z − T v ) − h v dz = 12 πi Z | z − | = ξ z − t ( z − T t,v ) − ( T t,v − T v ) h v dz = 12 πi Z | z − | = ξ z − z − P t,v ) − ∆ t ( T t,v − T v ) h v dz, which completes the proof of (6.30). (cid:3) Proof of Theorem 6.3.
For every f ∈ B ρ,ǫ,λ and | t | ≤ k ∆ t f k ρ,ǫ,λ ≤ | f | ρ + | t | ǫ [ f ] ǫ,λ , (6.31)In view of (6.30), Proposition 5.4, inequalities (6.31), (6.11) and (6.12) with the function h v wehave k ∆ t (Π T,t − Π T, ) h v k ρ,ǫ,λ ≤≤ πξ Z | z − | = ξ (cid:13)(cid:13) ( z − P t,v ) − ∆ t ( T t,v − T v ) h v (cid:13)(cid:13) ρ,ǫ,λ dz ≤ D ( | ∆ t ( T t,v − T v ) h v | ρ + [∆ t ( T t,v − T v ) h v ] ǫ,λ ) ≤ D ( | ( T t,v − T v ) h v | ρ + t ǫ [( T t,v − T v ) h v ] ǫ,λ ) ≤ D ( C t δ + t ǫ C t δ − ǫ ) ≤ Ct δ , for every 0 ≤ t ≤ t and it completes the proof of (6.4). Now (6.6) follows. In order to prove (6.5)apply inequality (6.6) and (5.28) to obtain | (∆ t Π T,t h v )( x ) − | ≤ | Π T,t ( h v )( tx ) − Π T, ( h v )( tx ) | + | h v ( tx ) − |≤ t δ (cid:18) C (1 + | x | ) ρ + 21 − κ ( δ ) | x | δ (cid:19) . Above inequality implies (6.5) and the proof is finished. (cid:3)
Rate of convergence of eigenvalues and fractional expansions.
In this section we studythe fractional expansions (6.9) when t →
0. Their behavior is strongly related to the asymptoticsof the stationary measure ν at infinity. While Theorem 6.37 is proved then the limit Theorem 1.15follows as in [4]. First we establish (6.8).The proof goes along the same lines as in [4] with the function h v playing the role of ˆ η v there.Therefore, the details have been omitted. We shall use radial coordinates in R d i.e. every point isexpressed as tv where t > v ∈ S d − . Condition 6.32.
Assume that < ǫ < , λ > , λ + 2 ǫ < ρ = 2 λ and λ + ǫ < α as in Proposition5.4 and additionally • If < α ≤ , take any < β < such that ρ + 2 β < α . • If < α ≤ , take any λ > such that ρ = 2 λ < and ρ + 1 < α . Proposition 6.33.
Let h v be the eigenfunction of operator T v defined in (5.25). If < α < ,then lim t → t α Z R d (cid:16) e it h v,x i − (cid:17) (Π T,t ( h v )( tx ) − Π T, ( h v )( tx )) ν ( dx ) = 0 . (6.34) If α = 2 , then lim t → t | log t | Z R d (cid:16) e it h v,x i − (cid:17) (Π T,t ( h v )( tx ) − Π T, ( h v )( tx )) ν ( dx ) = 0 . (6.35) Proof.
In estimations below in view of Condition 6.32 we have to use appropriate parameters ǫ, λ, ρ, δ and η which are determined by α . • If 0 < α ≤
1, we take δ = α − β > ρ + β > ǫ and η = 2 β . • If 1 < α ≤
2, we take δ = 1 > ǫ and η = 1. EAVY TAIL PHENOMENON AND CONVERGENCE TO STABLE LAWS 21
In view of (6.6), we have (cid:12)(cid:12)(cid:12)(cid:12) t α Z R d (cid:16) e it h v,x i − (cid:17) (Π T,t ( h v )( tx ) − Π T, ( h v )( tx )) ν ( dx ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (6.36) ≤ C t η + δ − α Z R d | x | η (1 + | x | ) ρ ν ( dx ) ≤ C t η + δ − α . for every 0 < t ≤ t ≤
1. Notice that, if • < α ≤
1, then η + δ − α = 2 β + α − β − α = β > ρ + η = ρ + 2 β < α . • < α ≤
2, then η + δ − α = 1 + 1 − α = 2 − α ≥ ρ + η = ρ + 1 < α .This justifies inequalities in (6.36) and completes the proof of (6.34) and (6.35). (cid:3) Theorem 6.37.
Assume that ψ θ satisfies assumptions 1.6, 1.7 and 1.14 for θ ∈ Θ . We de-fine S xn = P nk =1 X xk for n ∈ N . Let h v ( x ) = E (cid:16) e i h v, P ∞ k =1 ψ k ◦ ... ◦ ψ ( x ) i (cid:17) for x ∈ R d , where ψ k ( x ) = ψ θ k ( x ) , and ψ θ k ’s were defined in (H1) of assumption 1.6. Measure ν is the station-ary measure for the recursion (1.1), and Λ and σ Λ are the measures defined in Theorem 4.3. Case < α < . Let Ξ nα be the characteristic function of the random variable n − α S xn . Then forevery t > and v ∈ S d − lim n →∞ Ξ nα ( tv ) = Υ α ( tv ) = exp( t α C α ( v )) , (6.38) where C α ( v ) = lim t → k v ( t ) − t α = Z R d (cid:16) e i h v,x i − (cid:17) h v ( x )Λ( dx ) . (6.39) Case α = 1 . Let Ξ n be the characteristic function of the random variable n − S xn − nξ ( n − ) . Thenfor every t > and v ∈ S d − lim n →∞ Ξ n ( tv ) = Υ ( tv ) = exp( tC ( v ) + it h v, τ ( t ) i ) , (6.40) where C ( v ) = lim t → k v ( t ) − − i h v, ξ ( t ) i t = Z R d (cid:18)(cid:16) e i h v,x i − (cid:17) h v ( x ) − i h v, x i | x | (cid:19) Λ( dx ) , (6.41) ξ ( t ) = R R d tx | tx | ν ( dx ) and τ ( t ) = R R d (cid:16) x | tx | − x | x | (cid:17) Λ( dx ) . Case < α < . Assume that m = R R d xν ( dx ) . Let Ξ nα be the characteristic function of therandom variable n − α ( S xn − nm ) . Then for every t > and v ∈ S d − lim n →∞ Ξ nα ( tv ) = Υ α ( tv ) = exp( t α C α ( v )) , (6.42) where C α ( v ) = lim t → k v ( t ) − − i h v, tm i t α = Z R d (cid:16)(cid:16) e i h v,x i − (cid:17) h v ( x ) − i h v, x i (cid:17) Λ( dx ) . (6.43) Case α = 2 . Assume that m = R R d xν ( dx ) . Let Ξ n be the characteristic function of the randomvariable ( n log n ) − ( S xn − nm ) . Then for every t > and v ∈ S d − lim n →∞ Ξ n ( tv ) = Υ ( tv ) = exp( t C ( v )) , (6.44) where C ( v ) = lim t → k v ( t ) − − i h v, tm i t | log t | = − Z S d − (cid:0) h v, w i + 2 h v, w ih v, E ( ϕ ( w )) i (cid:1) σ Λ ( dw ) , (6.45) and ϕ ( x ) = P ∞ k =1 ψ k ◦ . . . ◦ ψ ( x ) = P ∞ k =1 M k · . . . · M x , where M k ’s were defined in (H1) ofassumption 1.6.Moreover, C α ( tv ) = t α C α ( v ) for every t > , v ∈ S d − and α ∈ (0 , ∪ (1 , . If supp σ Λ spans R d as a linear space, then ℜ C α ( v ) < for every v ∈ S d − and α ∈ (0 , .Proof. In order to obtain fractional expansions (6.39), (6.41), (6.43) and (6.45), we have to proceedas in Theorem 5.1 from [4] (see also [15]) using formula (6.2), Proposition 6.33, inequality (6.5) andconvergence (4.4) which holds for every function from family F Λ . Proof of (6.38), (6.40), (6.42)and (6.44) base on section 6 from [4] (see also [15]). Using formula h v ( tx ) = h tv ( x ) from Lemma5.26 we obtain that C α ( tv ) = t α C α ( v ) is valid for every t > v ∈ S d − and α ∈ (0 , ∪ (1 , ℜ C α ( v ) < v ∈ S d − and α ∈ (0 , (cid:3) Acknowledgements
The results of this paper are part of the author’s PhD thesis, written under the supervision ofEwa Damek at the University of Wroclaw. I wish to thank her for many stimulating conversationsand several helpful suggestions during the preparation of this paper. I would like also to thankDariusz Buraczewski for beneficial discussions and comments.The author is grateful to the referee for a very careful reading of the manuscript and usefulremarks that lead to the improvement of the presentation.
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J.Appl. Probab. M.Mirek, University of Wroclaw, Institute of Mathematics, 50-384 Wroclaw, Poland
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