Central limit theorem for Markov processes with spectral gap in the Wasserstein metric

Abstract

Suppose that {X t , t ≥ 0} is a non-stationary Markov process, taking values in a Polish metric space E. We prove the law of large numbers and central limit theorem for an additive functional of the form T 0 ψ(X s )ds, provided that the dual transition probability semigroup, defined on measures, is strongly contractive in an appropriate Wasserstein metric. Function ψ is assumed to be Lipschitz on E.

An invariance principle for the law of the iterated logarithm for some Markov chains

“…In particular, µ * ∈ M 1,1 . The proof was given for Markov processes with continuous time in [12] but it still remains valid in discrete case.…”

“…On the other hand, Guivarc'h and Hardy [7] proved the CLT for a class of Markov chains associated with the transfer operator having spectral gap. Recently Komorowski and Walczuk studied Markov processes with the transfer operator having spectral gap in the Wasserstein metric and proved the CLT in the non-stationary case (see [12]). Other interesting results under similar assumptions were obtained by S. Kuksin and A. Shirikyan (see [13,20]).…”

Limit theorems for some Markov chains

“…This construction appears in [6]. Note that, for every n ∈ N and x, y ∈ X, C n h 1 ,...,hn ((x, y), ·), constructed as in (11), is the n-th marginal of C ∞ h 1 ,h 2 ... ((x, y), ·). Additionally, {C n h 1 ,...,hn ((x, y), ·) : x, y ∈ X} fulfills the role of coupling for…”

“…If we have the coupling measure already constructed, the proof of the CLT is brief and less technical than typical proofs based on Gordin's martingale approximation. What led us to this intriguing solution was an unsuccessful attempt to follow the pattern given by Komorowski and Walczuk [11]. It is worth mentioning here that an auxiliary model, described by some nonhomogenous Markov chain, is needed to take adventage of coupling methods.…”

The Central Limit Theorem for Random Dynamical Systems

“…In the same spirit, the Central Limit Theorem was proven by Hille, Horbacz, Szarek and Wojewódka [13] for a stochastic model for an autoregulated gene. Komorowski and Walczuk studied Markov processes with the transfer operator having spectral gap in the Wasserstein metric and proved the CLT in the non-stationary case [25].…”