Effective Berry–Esseen and concentration bounds for Markov chains with a spectral gap

Abstract

Applying quantitative perturbation theory for linear operators, we prove non-asymptotic bounds for Markov chains whose transition kernel has a spectral gap in an arbitrary Banach algebra of functions X . The main results are concentration inequalities and Berry-Esseen bounds, obtained assuming neither reversibility nor "warm start" hypothesis: the law of the first term of the chain can be arbitrary. The spectral gap hypothesis is basically a uniform X -ergodicity hypothesis, and when X consist in regular functions this is weaker than uniform ergodicity. We show on a few examples how the flexibility in the choice of function space can be used. The constants are completely explicit and reasonable enough to make the results usable in practice, notably in MCMC methods. 1 Here and in the sequel, we write indifferently µ(f ) or f dµ for the integral of f with respect to the measure µ.bounds, both for the Law of Large Numbers (1) ("concentration inequalities") and for the CLT ("Berry-Esseen bounds").A word on effectivity In this paper, the emphasis will be on effective bounds, i.e. given an explicit sample size n, one should be able to deduce from the bound that the quantity being considered lies in some explicit interval around its limit with at least some explicit probability. In other words, the result should be non-aymptotic and all constants should be made explicit. The motivations for this are at least twofold.First, in practical applications of the Markov chain Monte-Carlo (MCMC) method, where one uses (1) to estimate the integral µ 0 (ϕ), effective results are needed to obtain proven convergence of a given precision. MCMC methods are important when the measure of interest is either unknown, or difficult to sample independently (e.g. uniform in a convex set in large dimension), but happens to be the stationary measure for an easily simulated Markov chain. The Metropolis-Hastings algorithm for example makes it possible to deal with an absolutely continuous measure whose density is only known up to the normalization constant.A second, more theoretical motivation is that the constants appearing in limit theorem depend on a number of parameters (e.g. the mixing speed of the Markov chain, the law of X 0 , etc.). When the constants are not made explicit, one may not be able to deduce from the result how the convergence speed changes when some parameter approaches the limit of the domain where the result is valid (e.g. when the spectral gap tends to 0).There are many works proving concentration inequalities and (to a lesser extent) Berry-Esseen bounds for Markov chains, under a variety of assumptions, and we will only mention a small number of them. To explain the purpose of this article, let us discuss briefly three directions. Previous works (1): total variation convergenceThe first direction is mainly motivated by MCMC; we refer to [RR + 04] for a detailed introduction to the topic.The Markov chains being considered are usually ergodic (either uniformly, which corresponds to a spectral gap on L ∞ , or geometrically);...

Concentration of Markov chains with bounded moments

“…the results of [Klo19] hold in a setting that imposes structural hypotheses on the aforementioned norm of the "observable" f which notably excludes its L q (π W ) norm (which appears in the right-hand side of the bound (1) that we prove here), but it is noted in [Klo19, Remark 2.2] that "classically one only makes moment assumptions on the observable." Corollary 1.3 addresses this question, though note that [Klo19] also covers settings that are not treated here.…”

“…Remark 1. Kloeckner investigated in [Klo19] the question of obtaining concentration bounds such as (3) with the…”