On the Fluctuations of the Single-Letter $d$-Tilted Sum for Binary Markov Sources

We study the source-side single-letter $d$-tilted sum for a stationary binary Markov chain under Hamming distortion, induced by the single-letter Blahut--Arimoto operating point computed from the stationary marginal $\pi$. We show that this quantity inherits the same algebraic structure as in the memoryless (i.i.d.) case: the centered sum $J_n(D)-n\mu_D$ is exactly an affine function of the chain's occupation count $N_n$, and consequently all centered cumulants are independent of the distortion level $D$. The exact finite-$n$ distribution therefore follows immediately from known results on occupation counts of two-state Markov chains. The genuinely new contributions of this note are (i) a closed-form expression for the finite-$n$ variance that includes the autocorrelation factor due to memory, and (ii) the transfer-matrix representation of the cumulant generating function. The connection, if any, between this source-side quantity and the operational finite-blocklength rate-distortion function remains open.