Estimating the Mixing Coefficients of Geometrically Ergodic Markov Processes

TL;DR

This paper introduces methods to estimate the mixing coefficients of geometrically ergodic Markov processes from a single sample, providing convergence rates under smoothness conditions and bounds for finite state spaces.

Contribution

The paper develops novel estimation techniques for mixing coefficients with proven convergence rates and error bounds, applicable under smoothness assumptions and in finite state spaces.

Findings

Convergence rate of order O(log(n) n^{-[s]/(2[s]+2)}) under smoothness conditions.

High-probability bounds on estimation error.

Error rate of order O(log(n) n^{-1/2}) for finite state spaces.

Abstract

We propose methods to estimate the individual $\beta$-mixing coefficients of a real-valued geometrically ergodic Markov process from a single sample-path $X_0,X_1, \dots,X_n$. Under standard smoothness conditions on the densities, namely, that the joint density of the pair $(X_0,X_m)$ for each $m$ lies in a Besov space $B^s_{1,\infty}(\mathbb R^2)$ for some known $s>0$, we obtain a rate of convergence of order $\mathcal{O}(\log(n) n^{-[s]/(2[s]+2)})$ for the expected error of our estimator in this case\footnote{We use $[s]$ to denote the integer part of the decomposition $s=[s]+\{s\}$ of $s \in (0,\infty)$ into an integer term and a {\em strictly positive} remainder term $\{s\} \in (0,1]$.}. We complement this result with a high-probability bound on the estimation error, and further obtain analogues of these bounds in the case where the state-space is finite. Naturally no density assumptions are required in this setting; the expected error rate is shown to be of order $\mathcal O(\log(n) n^{-1/2})$.