Central Limit Theorem for ergodic averages of Markov chains \& the comparison of sampling algorithms for heavy-tailed distributions
Miha Brešar, Aleksandar Mijatović, Gareth Roberts
TL;DR
This work builds a discrete-time L-drift framework (V,φ,Ψ) to derive verifiable necessary conditions for the ergodic CLT of Markov chains on general state spaces, and to obtain sharp lower bounds on convergence rates and invariant-tail behavior. It then applies this theory to a wide class of heavy-tailed sampling algorithms (RWM, iv-RWM, MALA, ULA, SPS, IS), distinguishing single-jump and many-jump tail exploration regimes and showing how algorithm design and tail assumptions critically affect CLT validity and convergence speed. The results provide practical criteria for algorithm selection and underscore the limitations and biases of biased/unadjusted schemes when targeting heavy-tailed distributions, while identifying scenarios where certain methods (e.g., SPS or iv-RWM with infinite-variance proposals) yield substantial gains. Overall, the paper significantly advances the theoretical understanding of CLTs in MCMC with heavy tails and offers actionable guidance for designing and comparing sampling algorithms in high-tail settings.
Abstract
Establishing central limit theorems (CLTs) for ergodic averages of Markov chains is a fundamental problem in probability and its applications. Since the seminal work~\cite{MR834478}, a vast literature has emerged on the sufficient conditions for such CLTs. To counterbalance this, the present paper provides verifiable necessary conditions for CLTs of ergodic averages of Markov chains on general state spaces. Our theory is based on drift conditions, which also yield lower bounds on the rates of convergence to stationarity in various metrics. The validity of the ergodic CLT is of particular importance for sampling algorithms, where it underpins the error analysis of estimators in Bayesian statistics and machine learning. Although heavy-tailed sampling is of central importance in applications, the characterisation of the CLT and the convergence rates are theoretically poorly understood for almost all practically-used Markov chain Monte Carlo (MCMC) algorithms. In this setting our results provide sharp conditions on the validity of the ergodic CLT and establish convergence rates for large families of MCMC sampling algorithms for heavy-tailed targets. Our study includes a rather complete analyses for random walk Metropolis samplers (with finite- and infinite-variance proposals), Metropolis-adjusted and unadjusted Langevin algorithms and the stereographic projection sampler (as well as the independence sampler). By providing these sharp results via our practical drift conditions, our theory offers significant insights into the problems of algorithm selection and comparison for sampling heavy-tailed distributions (see short YouTube presentations~\cite{YouTube_talk} describing our \href{https://youtu.be/m2y7U4cEqy4}{\underline{theory}} and \href{https://youtu.be/w8I_oOweuko}{\underline{applications}}).