🤖 AI Summary
This paper addresses the estimation of β-mixing coefficients for geometrically ergodic Markov processes, aiming to quantify temporal dependence from a single observed trajectory. We propose a nonparametric estimation framework grounded in kernel density estimation and statistical learning theory, integrating tools from Besov space analysis, mixing process theory, and concentration inequalities. Our main contributions are threefold: (1) Under Besov-type regularity assumptions on the stationary density, we establish the first convergence rate with an explicit logarithmic correction factor; (2) For finite-state-space Markov chains, we achieve the optimal expected error rate $O(log n / sqrt{n})$ and corresponding high-probability bounds—without requiring any density boundedness or smoothness assumptions; (3) In continuous state spaces, we derive an expected error rate of $O(log n cdot n^{-s/(2s+2)})$, where $s > 0$ denotes the Besov smoothness index, along with matching high-probability guarantees.
📝 Abstract
We propose methods to estimate the individual $eta$-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 casefootnote{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 $mathcal O(log(n) n^{-1/2})$.