Additional material on local limit theorem for finite Additive Markov Processes [HL13]

TL;DR

This paper extends the local limit theorem for finite additive Markov processes by providing additional material, applications to renewal processes, and a uniform version with respect to transition matrices.

Contribution

It offers new applications, detailed assumptions verification, and a uniform version of the existing local limit theorem for finite additive Markov processes.

Findings

Application to Renewal Markov Processes

Verification of assumptions for joint distribution of local times

Uniform version with respect to transition matrices

Abstract

This paper proposes additional material to the main statements of [HL13] which are are recalled in Section~2. In particular an application of [Theorem~2.2,HL13] to Renewal Markov Processes is provided in Section~4 and a detailed checking of the assumptions of [Theorem~2.2,HL13] for the joint distribution of local times of a finite jump process is reported in Section~5. A uniform version of [Theorem~2.2,HL13] with respect to a compact set of transition matrices is given in Section~6 (see [Remark 2.4,HL13]). The basic material on the semigroup of Fourier matrices and the spectral approach used in [HL13] is recalled in Section~3 in order to obtain a good understanding of the properties involved in this uniform version.