Estimating the Mixing Coefficients of Geometrically Ergodic Markov Processes

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, …,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,∞(ℝ^2) 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).