Deterministic Causal Quantization with Representative States and Bounded Error
Abstract
We study deterministic causal quantization with pathwise error at most ε, using ω shared representatives updated from transmitted addresses alone. For a midpoint realization with first-in, first-out replacement, we compare address entropy rates with process ε-entropy, the least entropy rate of any block quantizer meeting the same constraint. From ε-separated initialization, the gap is at most two bits per sample for every stationary source with finite scalar ε-entropy, without density, independence or mixing assumptions. A causal address-tree argument extends this comparison to any synchronized quantizer with cell overlap m, giving log2 m bits. Lattice initialization with consistent boundary ownership realizes uniform grids with sharp one-bit scalar and process comparisons. Under an entropy condition and finite Fisher information rate J, the gap is at most 2Jε2/(π2ln 2); general initializations add a term of order ωε√J. The scalar grid estimate gives a nonasymptotic form of Rényi's theorem. Since a grid is causal, the price of causality is at most min{1,2Jε2/(π2ln 2)} bits per sample, versus the high-resolution squared-error loss 1/2log2(πe/6). These bounds certify ε-entropy brackets for autoregressive Laplace and Gaussian sources. For memoryless sources, the optimal causal rate equals the scalar ε-entropy. For a stationary process on finitely many 2ε-separated centers, adding independent uniform noise of width 2ε gives exact Shannon lower-bound equality at every block length, attained by storing those centers. This includes histograms with finitely many bins. Storing two non-lattice centers beats every uniform grid of the same cell width by half a bit per sample.
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BibTeX
@misc{cont2026deterministic,
title = {{Deterministic Causal Quantization with Representative States and Bounded Error}},
author = {Rama Cont and Massoud Heidari},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35933},
eprint = {2610.35933},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35933},
}
RIS
TY - UNPB
TI - Deterministic Causal Quantization with Representative States and Bounded Error
AU - Rama Cont
AU - Massoud Heidari
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.35933
UR - https://arxiv.org/abs/2610.35933
AB - We study deterministic causal quantization with pathwise error at most ε, using ω shared representatives updated from transmitted addresses alone. For a midpoint realization with first-in, first-out replacement, we compare address entropy rates with process ε-entropy, the least entropy rate of any block quantizer meeting the same constraint. From ε-separated initialization, the gap is at most two bits per sample for every stationary source with finite scalar ε-entropy, without density, independence or mixing assumptions. A causal address-tree argument extends this comparison to any synchronized quantizer with cell overlap m, giving log_2 m bits. Lattice initialization with consistent boundary ownership realizes uniform grids with sharp one-bit scalar and process comparisons. Under an entropy condition and finite Fisher information rate J, the gap is at most 2Jε^2/(π^2ln 2); general initializations add a term of order ωε√J. The scalar grid estimate gives a nonasymptotic form of Rényi's theorem. Since a grid is causal, the price of causality is at most min{1,2Jε^2/(π^2ln 2)} bits per sample, versus the high-resolution squared-error loss 1/2log_2(πe/6). These bounds certify ε-entropy brackets for autoregressive Laplace and Gaussian sources. For memoryless sources, the optimal causal rate equals the scalar ε-entropy. For a stationary process on finitely many 2ε-separated centers, adding independent uniform noise of width 2ε gives exact Shannon lower-bound equality at every block length, attained by storing those centers. This includes histograms with finitely many bins. Storing two non-lattice centers beats every uniform grid of the same cell width by half a bit per sample.
ER -