Estimating the Mixing Coefficients of Geometrically Ergodic Markov Processes

TL;DR

The paper develops nonparametric methods to estimate the beta-mixing coefficients of stationary geometrically ergodic Markov processes from a single sample path. It introduces a kernel density estimator-based approach for real-valued state spaces under Besov smoothness and complementary empirical-frequency methods for finite state spaces, delivering nonasymptotic error bounds: $O(\\log(n) n^{- [s]/(2[s]+2)})$ in the real-valued case and $O(\\log(n) n^{-1/2})$ in the finite-state case, along with high-probability guarantees. The estimators are model-agnostic, demonstrated to be robust under model misspecification, and supported by empirical evaluations. The results enable data-driven estimation of dependence measures that appear in concentration inequalities and other finite-time guarantees for dependent data.

Abstract

We propose methods to estimate the individual $β$-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 $\mathcal O(\log(n) n^{-1/2})$.

Figures (2)

• Figure 1: The plot on the left illustrates the performance of our KDE-based estimators, while the plot on the right demonstrates the performance of the ACF-based estimators for the estimation of $\beta(m)$, $m = 1,3,5$ when samples are generated according to AR(1). The horizontal lines represent the true values of $\beta$, with the top solid line (red) corresponding to $m=1$, the middle solid line (green) to $m=3$, and the bottom solid line (blue) to $m=5$. The mean estimates (averaged over 20 rounds) are plotted for each $m$ value, and the shaded areas indicate the uncertainty in these estimates. As $n$ increases, the estimates clearly converge to their respective true $\beta$ values. In the right plot, the results for the ACF-based estimator are shown. The ACF-based estimator takes advantage of the knowledge that the process is AR(1) and, more specifically, assumes Gaussian densities. This assumption leads to faster convergence in this particular setting, as the model assumption is accurate.
• Figure 2: Performance of our estimators (top) and the ACF-based estimator (bottom) when the sample is a transformation of an AR(1) process.

Theorems & Definitions (11)

• Remark 1
• Theorem 3
• Remark 4
• Theorem 5
• Theorem 6
• Lemma 7
• Lemma 8: berbee1979random
• proof : Proof of Lemma \ref{['lem:ourberbee']}
• proof : Proof of Theorem \ref{['lem:expected_beta_error']}
• proof : Proof of Theorem \ref{['prop:beta_d']}