# Invariance principle for additive functionals of Markov chains

## TL;DR

This paper establishes conditions under which additive functionals of Markov chains converge in distribution, extending Dynkin's theorem and providing practical criteria based on transition probabilities.

## Contribution

It introduces a general sufficient condition for the convergence of additive functionals of Markov chains, linking it to Dynkin's theorem and transition probabilities.

## Key findings

- Provides a new convergence criterion for additive functionals
- Extends Dynkin's theorem to a broader setting
- Offers practical conditions based on transition probabilities

## Abstract

We consider a sequence of additive functionals {\phi_n}, set on a sequence of Markov chains {X_n} that weakly converges to a Markov process X. We give sufficient condition for such a sequence to converge in distribution, formulated in terms of the characteristics of the additive functionals, and related to the Dynkin's theorem on the convergence of W-functionals. As an application of the main theorem, the general sufficient condition for convergence of additive functionals in terms of transition probabilities of the chains X_n is proved.