I know that for two i.i.d distributions $P$ and $Q$ the probability that $Q$ will produce a length $n$ sequence that is $\epsilon$-typical according to $P$ is bounded by $$Q(T_{P,\epsilon})\leq e^{-nD(P||Q)-n|\mathcal{X}|\epsilon}$$ where $D(P||Q)$ is the KL-dvergence and $T_{P,\epsilon}$ is the (strong) $\epsilon$-typical set of the distribution $P$, i.e. sequences with empirical distribution that is $\epsilon$-close to $P$.

Will this bound hold for $Q$ that is not i.i.d? For example, a Markov source? That is, will I have $$Q^n(T_{P^n,\epsilon})\leq e^{-D(P^n||Q^n)-n|\mathcal{X}|\epsilon}$$ where $Q^n$ is the distribution on sequences of length $n$?