Functional Convergence to Stable Lévy Motions for Iterated Random Lipschitz Mappings

Sana Louhichi,Emmanuel Rio

doi:10.1214/ejp.v16-965

Abstract

It is known that, in the dependent case, partial sums processes which are elements of $D([0,1])$ (the space of right-continuous functions on $[0,1]$ with left limits) do not always converge weakly in the $J_1$-topology sense. The purpose of our paper is to study this convergence in $D([0,1])$ equipped with the $M_1$-topology, which is weaker than the $J_1$ one. We prove that if the jumps of the partial sum process are associated then a functional limit theorem holds in $D([0,1])$ equipped with the $M_1$-topology, as soon as the convergence of the finite-dimensional distributions holds. We apply our result to some stochastically monotone Markov chains arising from the family of iterated Lipschitz models.

Highlights

Let (Xn)n∈Z be a sequence of real-valued random variables
The sequence of processes (ξn(t))t∈[0,1] obeys a functional limit theorem (FLT) if there exists a proper stochastic process (Y (t))t∈[0,1] with sample paths in the space D := D([0, 1], R) of all rightcontinuous R-valued functions with left limits defined on [0, 1], such that ξn(·) =⇒ Y (·) in D equipped with some topology
Convergence in D equipped with the J1-topology does not allow more than one jump in a very little time. This limit condition does not ensure the convergence of partial sums processes constructed from any dependent random variables (Xi)i∈Z in D equipped with the J1-topology

Summary

Introduction

Let (Xn)n∈Z be a sequence of real-valued random variables. Let S0 = 0 and Sn = X1 + . . . + Xn be the associated partial sums and let (ξn(t))t∈[0,1] be the normalized partial sum process,. Convergence in D equipped with the J1-topology does not allow more than one jump in a very little time This limit condition does not ensure the convergence of partial sums processes constructed from any dependent random variables (Xi)i∈Z in D equipped with the J1-topology. Our first new result, which was announced in Louhichi and Rio (2011), is that, if the jumps of the process (ξn(t))t∈[0,1] are associated a FLT holds in D equipped with the M1-topology as soon as the convergence of the finite-dimensional distributions holds. Be a strictly stationary sequence of associated real-valued random variables. Let us note that for a strictly stationary sequence of associated real-valued random variables with finite second moment and finite series of covariances, the functional convergence towards a Brownian motion was already proved by Newman and Wright (1981). The paper ends with an Appendix discussing the association property for the stochastically monotone Markov chains

Limit theorems for iterated Lipschitz models

Proof of Theorem 1

Proof of Proposition 2

Proof of Theorem 2

Proof of Proposition 4

Proof of Lemma 7