Stable limits for Markov chains via the Principle of Conditioning

Abstract

We study limit theorems for partial sums of instantaneous functions of a homogeneous Markov chain on a general state space. The summands are heavy-tailed and the limits are stable distributions.We show that if the transition operator of the chain is operator uniformly integrable and the chain is ρ-mixing, then the limit is the same as if the summands were independent.We provide an example of a Markov chain that is operator uniformly integrable (and admits a spectral gap) while it is not hyperbounded.What makes our assumptions working is a new, efficient version of the Principle of Conditioning.