Local limit theorems for Markov chains with trend component of linear growth

TL;DR

This paper develops a modified parametrix method to establish local limit theorems for Markov chains with a linearly increasing drift component, addressing challenges posed by unbounded drift terms.

Contribution

It introduces a novel modification of the parametrix method to handle unbounded drift in Markov chains converging to diffusions.

Findings

Successfully derived local limit theorems for Markov chains with linear drift

Extended the applicability of the parametrix method to unbounded drift scenarios

Provided theoretical framework for analyzing Markov chains with trend components

Abstract

We consider a sequence of Markov chains weakly convergent to a diffusion. We suppose that a drift term contains a linearly increasing component. The usual parametrix method fails because of this unbounded drift term. We show how to modify the parametrix method to obtain local limit theorems for this case.