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

2026-03-08

AI Summary

This study investigates the fluctuation properties of single-letter $d$-tilted information in the finite-blocklength rate-distortion theory for binary Markov sources. By establishing an exact affine mapping between the $d$-tilted information sum and the state occupancy counts of the underlying Markov chain, the work reveals an intrinsic connection between these two quantities for the first time. The analysis leverages tools from occupancy count statistics, transition matrix methods, and cumulant generating functions. Key contributions include deriving a closed-form expression for the variance and the exact distribution of the $d$-tilted sum at finite blocklengths, proving that all central cumulants are independent of the distortion level, and providing the limiting cumulant generating function—thereby offering a novel analytical framework for finite-length rate-distortion analysis.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Web Mining and Content Analysis: Content-based information diffusion
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

The $d$-tilted information has been found to be a useful quantity in finite-blocklength rate-distortion theory for memoryless sources. We study the source-side $d$-tilted sum induced by the single-letter Blahut--Arimoto operating point for a stationary binary Markov source under Hamming distortion; this is a source-side quantity distinct from the $n$-letter operational $d$-tilted information. We show that the centered block sum $J_n(D) - n\mu_D$ is exactly an affine image of the occupation count $N_n = \sum_{t=1}^n \mathbf{1}\{X_t = 1\}$ of the Markov chain. As consequences, all centered cumulants are independent of the distortion level~$D$, the finite-$n$ variance admits a closed form, and the exact finite-$n$ distribution and limiting cumulant generating function are given by a $2 \times 2$ transfer matrix.

Problem

• d-tilted information
• binary Markov source
• rate-distortion theory
• finite-blocklength
• occupation count

Innovation

• d-tilted information
• binary Markov source
• finite-blocklength rate-distortion
• occupation count
• transfer matrix