Normal approximation for functions of hidden Markov models

TL;DR

This paper extends Stein's method to derive normal approximation rates for functions of hidden Markov models, accounting for dependencies and providing convergence rates with additional logarithmic factors.

Contribution

It introduces a generalized perturbative approach for dependent data, specifically for hidden Markov models, expanding the applicability of Stein's method.

Findings

Rates of convergence are established for functions of hidden Markov models.

An extra logarithmic factor appears in the convergence rate compared to independent cases.

Applications to stochastic geometry demonstrate the method's effectiveness.

Abstract

The generalized perturbative approach is an all purpose variant of Stein's method used to obtain rates of normal approximation. Originally developed for functions of independent random variables this method is here extended to functions of the realization of a hidden Markov model. In this dependent setting, rates of convergence are provided in some applications to stochastic geometry, leading, in each instance, to an extra log-factor vis a vis the rate in the independent case.