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 X0,X1, dots,Xn. Under standard smoothness conditions on the densities, namely, that the joint density of the pair (X0,Xm) for each m lies in a Besov space Bs1,∞(mathbb 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 casefootnoteWe use [s] to denote the integer part of the decomposition s=[s]+s of s ∈ (0,∞) into an integer term and a em 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 mathcal O(log(n) n-1/2).