Estimating the Mixing Coefficients of Geometrically Ergodic Markov Processes

We propose methods to estimate the individual -mixing coefficients of a real-valued geometrically ergodic Markov process from a single sample-path X₀, X₁, , 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, (R²) for some known s>0, we obtain a rate of convergence of order O ( (n) n^-s/ (2s+2) ) for the expected error of our estimator in this caseWe 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 ( (n) n^-1/2).