A local limit theorem for densities of the additive component of a finite Markov Additive Process

TL;DR

This paper establishes a local limit theorem for the densities of the additive component in finite Markov Additive Processes, extending classical results to a Markov-dependent setting with applications to local times of jump processes.

Contribution

It provides a new local limit theorem for densities of additive components in finite Markov Additive Processes under standard conditions, with explicit convergence rates.

Findings

Proves a local limit theorem for the density of scaled additive components.

Identifies conditions under which the theorem holds, matching i.i.d. case rates.

Includes an application to local times of finite jump processes.

Abstract

In this paper, we are concerned with centered Markov Additive Processes $\{(X_t,Y_t)\}_{t\in\T}$ where the driving Markov process $\{X_t\}_{t\in\T}$ has a finite state space. Under suitable conditions, we provide a local limit theorem for the density of the absolutely continuous part of the probability distribution of $t^{-1/2}Y_t$ given $X_0$. The rate of convergence and the moment condition are the expected ones with respect to the i.i.d case. An application to the joint distribution of local times of a finite jump process is sketched.