# Concentration of Markov chains with bounded moments

## TL;DR

This paper extends concentration inequalities for finite state Markov chains to cases where the function has bounded moments rather than being bounded, providing dimension-independent bounds and answering a question by Kargin.

## Contribution

It introduces new concentration inequalities assuming only bounded moments of the function, generalizing Gillman's bounds and addressing an open question by Kargin.

## Key findings

- Derived moment-based concentration inequalities for Markov chains
- Generalized bounds to $L_p$-valued functions, including Hilbert spaces
- Provided dimension-independent concentration bounds

## Abstract

Let $\{W_t\}_{t=1}^{\infty}$ be a finite state stationary Markov chain, and suppose that $f$ is a real-valued function on the state space. If $f$ is bounded, then Gillman's expander Chernoff bound (1993) provides concentration estimates for the random variable $f(W_1)+\cdots+f(W_n)$ that depend on the spectral gap of the Markov chain and the assumed bound on $f$. Here we obtain analogous inequalities assuming only that the $q$'th moment of $f$ is bounded for some $q \geq 2$. Our proof relies on reasoning that differs substantially from the proofs of Gillman's theorem that are available in the literature, and it generalizes to yield dimension-independent bounds for mappings $f$ that take values in an $L_p(\mu)$ for some $p\ge 2$, thus answering (even in the Hilbertian special case $p=2$) a question of Kargin (2007).