Abstract
We prove a version of McDiarmid’s bounded differences inequality for Markov chains, with constants proportional to the mixing time of the chain. We also show variance bounds and Bernstein-type inequalities for empirical averages of Markov chains. In the case of non-reversible chains, we introduce a new quantity called the “pseudo spectral gap”, and show that it plays a similar role for non-reversible chains as the spectral gap plays for reversible chains. Our techniques for proving these results are based on a coupling construction of Katalin Marton, and on spectral techniques due to Pascal Lezaud. The pseudo spectral gap generalises the multiplicative reversiblication approach of Jim Fill.
Highlights
Consider a vector of random variablesX := (X1, X2, . . . , Xn) taking values in Λ := (Λ1 × . . . × Λn), and having joint distribution P
In the non-reversible case, we introduce the “pseudo spectral gap", γps := maximum of (the spectral gap of (P ∗)kP k divided by k) for k ≥ 1, and prove similar bounds using it
The following proposition shows that for uniformly ergodic Markov chains, there exists a partition and a Marton coupling such that the size of the partition is comparable to the mixing time, and the operator norm of the coupling matrix is an absolute constant
Summary
X := (X1, X2, . . . , Xn) taking values in Λ := (Λ1 × . . . × Λn), and having joint distribution P. For more complicated functions than sums, we show a version of McDiarmid’s bounded differences inequality, with constants proportional to the mixing time of the chain. This inequality is proven by combining the martingale-type method of [4] and a coupling structure introduced by Katalin Marton.
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