Let $X_i$ be a (finite-state, irreducible, aperiodic) Markov chain, not necessarily stationary. (That is, it doesn't start from the invariant distribution; I'm happy to have it be time-homogeneous if that helps.) Let $Y_i = f(X_i)$. Let $S_i = \sum_{s=1}^i Y_i$ be the partial sum sequence. Is it true that $$ \mathbb E_s [\max_{i \leq n} S_i^2] \leq C \mathbb E_s[S_n^2]? $$ Here $\mathbb E_s$ is the expectation given $X_0 = s$. The constant $C$ may potentially depend on the size of the state space, the mixing time of the chain, etc. (It's also possible that the correct functional dependence is actually more like $\leq C_1 \mathbb E_s S_n^2 + C_2$.)

Basically, I'll take any sensible bound on the maximum of the partial sum sequence for a (functional of a) non-stationary Markov chain; I'm not after the optimal bound.

A sub-question: there are lots of maximal inequalities for dependent sequences satisfying certain mixing conditions. (E.g.: Peligrad, Utev, and Wu, Proceedings of the AMS 2005; Rio, Journal of Theoretical Probability 2009; Merlevede and Peligrad, Annals of Probability 2013; the list is not meant to be in any way exhaustive.) Is there an easy way to convert a maximal inequality for a stationary mixing sequence to a maximal inequality for a (functional of a) non-stationary finite-state Markov chain?