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

Let $\{Y_i\}_{i=1}^{\infty}$ be a stationary reversible Markov chain with state space $[N]$, let $(X, \| \cdot \|)$ be a real-valued Banach space and let $f_1, \ldots, f_n: [N] \rightarrow X$ be functions with mean $0$ such that $\|f_i(v)\| \leq 1$ for all $i$ and $v$. We prove bounds on the expected value of and deviation bounds for the random variable $\|f_1(Y_1)+\cdots+f_n(Y_n)\|$. 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 $\ell_2 \rightarrow \ell_2$ 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.