The Nagaev-Guivarc'h method via the Keller-Liverani theorem

TL;DR

This paper extends the Nagaev-Guivarc'h method using the Keller-Liverani theorem to establish advanced limit theorems for ergodic Markov chains, including multi-dimensional local limit, Edgeworth, and Berry-Esseen theorems.

Contribution

It proves new multi-dimensional limit theorems and Edgeworth expansions for Markov chains using spectral perturbation techniques, broadening the method's applicability.

Findings

Established multi-dimensional local limit theorem.

Proved first-order Edgeworth expansion.

Derived multi-dimensional Berry-Esseen theorem.

Abstract

The Nagaev-Guivarc'h method, via the perturbation operator theorem of Keller and Liverani, has been exploited in recent papers to establish local limit and Berry-Essen type 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. This paper outlines this method and extends it by proving a multi-dimensional local limit theorem, a first-order Edgeworth expansion, and a multi-dimensional Berry-Esseen type theorem in the sense of Prohorov metric. When applied to uniformly or geometrically ergodic chains and to iterative Lipschitz models, the above cited limit theorems hold under moment conditions similar, or close, to those of the i.i.d. case.