Estimating the Mixing Coefficients of Geometrically Ergodic Markov Processes

We propose methods to estimate the β-mixing coefficients of a real-valued geometrically ergodic Markov process from a single sample-path X₀, X₁, . . . , X_n−1. Under standard smoothness conditions on the densities, namely, that the joint density of the pair (X₀, X_m) for each m lies in a Besov space B^s_1,∞ (R²) for some known s > 0, we obtain a rate of convergence of order O(log(n)n^{−[s]/(2[s]+2)}) for the expected error of our estimator in this case; we use [s] to denote the integer part of the decomposition s = [s] + {s} of s ∈ (0, ∞) into an integer term and a strictly positive remainder term {s} ∈ (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 O(log(n)n^−1/2). The theoretical results are complemented with empirical evaluations.