Ergodic Central limit theorem for subgeometric Markov chains

Abstract

We establish verifiable necessary conditions for central limit theorems (CLTs) of ergodic averages of Markov chains on general state spaces. Our criteria are based on discrete-time $\mathbf{L}$-drift conditions that can be verified through estimates for the one-step transition kernel of the original chain. They yield an integral criterion for failure of the CLT for a class of observables that includes unbounded functions and indicators of tail events. The main technical step transfers lower bounds for excursions of the original chain to regenerative excursions of its Nummelin split chain. The same drift conditions yield lower bounds on the tails of return times and modulated excursion functionals, on invariant tails, and on rates of convergence in $f$-variation and $L^p$-Wasserstein distance. These estimates apply without an explicit formula for the invariant distribution. Applications to unadjusted Langevin algorithms show preservation of polynomial target tail indices at every fixed step size. For Euler schemes with additive infinite-variance noise, we establish restrictions on invariant tails and transience under superlinear drifts. Schemes with linear drift, multiplicative noise and logarithmic innovation tails exhibit sharp polynomial convergence exponents and exact CLT thresholds, despite the absence of positive Euclidean moments.

arXiv comment55 pages; this version has streamlined proofs for the theoretical results, with applications focusing on ULA-type algorithms (MCMC results of the first version are not included); a short YouTube presentation describes our theory and its applications to ULA-type chains with increments of finite and infinite variance, see https://youtu.be/m2y7U4cEqy4