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

TL;DR

This paper analyzes the fluctuations of a specific sum related to binary Markov sources under distortion, revealing algebraic structures and providing new formulas for variance and cumulant generating functions.

Contribution

It introduces a closed-form variance expression including autocorrelation effects and a transfer-matrix representation of the cumulant generating function for the source-side sum.

Findings

Centered sum is an affine function of occupation count N_n

All centered cumulants are independent of distortion level D

Finite-n variance includes autocorrelation factor due to memory

Abstract

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.