Limit theorems for some adaptive MCMC algorithms with subgeometric kernels: Part II

TL;DR

This paper establishes a central limit theorem for a broad class of adaptive MCMC algorithms with subgeometric kernels, including an analysis of an adaptive Langevin algorithm with heavy-tailed targets.

Contribution

It extends theoretical understanding of adaptive MCMC algorithms driven by subgeometric kernels and applies the results to analyze an adaptive Langevin algorithm.

Findings

Proves a CLT for adaptive MCMC with subgeometric kernels.

Analyzes asymptotic behavior of adaptive Langevin algorithms with heavy tails.

Provides theoretical foundation for convergence properties of adaptive algorithms.

Abstract

We prove a central limit theorem for a general class of adaptive Markov Chain Monte Carlo algorithms driven by sub-geometrically ergodic Markov kernels. We discuss in detail the special case of stochastic approximation. We use the result to analyze the asymptotic behavior of an adaptive version of the Metropolis Adjusted Langevin algorithm with a heavy tailed target density.