Towards a Banach Space Chernoff Bound for Markov Chains via Chaining Arguments

Let $\{Y_i{\}}_{i=1}^{\mathrm{\infty }}$ be a stationary reversible Markov chain with state space $[N]$, let $(X,\mathrm{\parallel }\cdot \mathrm{\parallel })$ be a real-valued Banach space and let $f_1,\dots ,f_n:[N]\to X$ be functions with mean $0$ such that $\mathrm{\parallel }{f}_i(v)\mathrm{\parallel }\le 1$ for all $i$ and $v$. We prove bounds on the expected value of and deviation bounds for the random variable $\mathrm{\parallel }{f}_1(Y_1)+\cdots +f_n(Y_n)\mathrm{\parallel }$. For large enough $n$ that depends on the Banach space (and not $N$), these bounds behave similarly as known bounds for independent random variables. When the Banach space in question is the set of matrices equipped with the ${\mathrm{\ell }}_2\to {\mathrm{\ell }}_2$ operator norm, for large enough $n$, our bounds on the expected value improve upon known bounds and match what is known for independent random variables up to a factor in the spectral gap.