A perturbation analysis of some Markov chains models with time-varying parameters

TL;DR

This paper investigates the differentiability of invariant distributions in locally stationary Markov models with time-varying parameters, providing new conditions applicable even when exponential moments are not finite.

Contribution

It introduces an alternative notion of derivative process for Markov models and extends perturbation analysis to cases lacking finite exponential moments.

Findings

Conditions for differentiability of invariant measures are established.

Results are applied to integer-valued autoregressive, categorical, and threshold autoregressive processes.

The approach broadens perturbation analysis to more general Markov models.

Abstract

We study some regularity properties in locally stationary Markov models which are fundamental for controlling the bias of nonparametric kernel estimators. In particular, we provide an alternative to the standard notion of derivative process developed in the literature and that can be used for studying a wide class of Markov processes. To this end, for some families of V-geometrically ergodic Markov kernels indexed by a real parameter u, we give conditions under which the invariant probability distribution is differentiable with respect to u, in the sense of signed measures. Our results also complete the existing literature for the perturbation analysis of Markov chains, in particular when exponential moments are not finite. Our conditions are checked on several original examples of locally stationary processes such as integer-valued autoregressive processes, categorical time series or threshold autoregressive processes.