Abstract. -The Nagaev-Guivarc'h method, via the perturbation operator theorem of Keller and Liverani, has been exploited in recent papers to establish limit theorems for unbounded functionals of strongly ergodic Markov chains. The main difficulty of this approach is to prove Taylor expansions for the dominating eigenvalue of the Fourier kernels. The paper outlines this method and extends it by stating a multidimensional local limit theorem, a one-dimensional Berry-Esseen theorem, a first-order Edgeworth expansion, and a multidimensional Berry-Esseen type theorem in the sense of the Prohorov metric. When applied to the exponentially L 2 -convergent Markov chains, to the v-geometrically ergodic Markov chains and to the iterative Lipschitz models, the three first above cited limit theorems hold under moment conditions similar, or close (up to ε > 0), to those of the i.i.d. case.
Résumé
To cite this version:Fabienne Castell, Nadine Guillotin-Plantard, Françoise Pene, Bruno Schapira. A local limit theorem for random walks in random scenery and on randomly oriented lattices. Annals of Probability, Institute of Mathematical Statistics, 2011, pp.Vol. 39, No 6, 2079-2118.1214/10-AOP606>.
A LOCAL LIMIT THEOREM FOR RANDOM WALKS IN RANDOM SCENERY AND ON RANDOMLY ORIENTED LATTICESFABIENNE CASTELL, NADINE GUILLOTIN-PLANTARD, FRANÇ OISE PÈNE, AND BRUNO SCHAPIRA Abstract. Random walks in random scenery are processes defined by Zn := n k=1 ξX 1 +...+X k , where (X k , k ≥ 1) and (ξy, y ∈ Z) are two independent sequences of i.i.d. random variables. We assume here that their distributions belong to the normal domain of attraction of stable laws with index α ∈ (0, 2] and β ∈ (0, 2] respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved the convergence in distribution when α = 1 and as n → ∞, of n −δ Zn, for some suitable δ > 0 depending on α and β. Here we are interested in the convergence, as n → ∞, of n δ P(Zn = ⌊n δ x⌋), when x ∈ R is fixed. We also consider the case of random walks on randomly oriented lattices for which we obtain similar results.
We show how Rio's method [Probab. Theory Related Fields 104 (1996) 255-282] can be adapted to establish a rate of convergence in 1 √ n in the multidimensional central limit theorem for some stationary processes in the sense of the Kantorovich metric. We give two applications of this general result: in the case of the Knudsen gas and in the case of the Sinai billiard.
We study the persistence probability for processes with stationary increments. Our results apply to a number of examples: sums of stationary correlated random variables whose scaling limit is fractional Brownian motion; random walks in random sceneries; random processes in Brownian scenery; and the Matheron-de Marsily model in Z 2 with random orientations of the horizontal layers. Using a new approach, strongly related to the study of the range, we obtain an upper bound of the optimal order in general and improved lower bounds (compared to previous literature) for many specific processes.2000 Mathematics Subject Classification. 60F05; 60G52.
Abstract. We consider some nonuniformly hyperbolic invertible dynamical systems which are modeled by a Gibbs-Markov-Young tower. We assume a polynomial tail for the inducing time and a polynomial control of hyperbolicity, as introduced by Alves, Pinheiro and Azevedo. These systems admit a physical measure with polynomial rate of mixing. In this paper we prove that the distribution of the number of visits to a ball B(x, r) converges to a Poisson distribution as the radius r → 0 and after suitable normalization.
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