Estimating the Mixing Coefficients of Geometrically Ergodic Markov Processes

2024/02/11 by Steffen Grünewälder, Grünewälder, Steffen, Azadeh Khaleghi +1
Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML) #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Statistics Theory (math.ST) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2402.07296

openalex publication_date 2024/02/11 · openalex created_date 2024/02/14 · openalex updated_date 2026/07/28

Abstract

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, …,X_n. 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,∞(\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 case\footnoteWe 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).

Related

• Inferring the mixing properties of an ergodic process
• Estimating the Mixing Time of Ergodic Markov Chains
• Spectral Theory and Limit Theorems for Geometrically Ergodic Markov Processes
• Subgeometric ergodicity and β-mixing
• Sharp Mixing Rates for Markov Chains on General Spaces with Unbounded Random Environments