On the Fluctuations of the Single-Letter $d$-Tilted Sum for Binary Markov Sources
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917456121954304 |
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| author | Krishnamachari, Bhaskar |
| author_facet | Krishnamachari, Bhaskar |
| contents | 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 $π$. We show that this quantity inherits the same algebraic structure as in the memoryless (i.i.d.) case: the centered sum $J_n(D)-nμ_D$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_07435 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Fluctuations of the Single-Letter $d$-Tilted Sum for Binary Markov Sources Krishnamachari, Bhaskar Information Theory 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 $π$. We show that this quantity inherits the same algebraic structure as in the memoryless (i.i.d.) case: the centered sum $J_n(D)-nμ_D$ 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. |
| title | On the Fluctuations of the Single-Letter $d$-Tilted Sum for Binary Markov Sources |
| topic | Information Theory |
| url | https://arxiv.org/abs/2603.07435 |