Tian, Peida and Kostina, Victoria
(2019)
*The Dispersion of the Gauss-Markov Source.*
IEEE Transactions on Information Theory, 65
(10).
pp. 6355-6384.
ISSN 0018-9448.
https://resolver.caltech.edu/CaltechAUTHORS:20190610-093125985

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## Abstract

The Gauss-Markov source produces U_i = aU_(i–1) + Z_i for i ≥ 1, where U_0 = 0, |a| < 1 and Z_i ~ N(0; σ^2) are i.i.d. Gaussian random variables. We consider lossy compression of a block of n samples of the Gauss-Markov source under squared error distortion. We obtain the Gaussian approximation for the Gauss-Markov source with excess-distortion criterion for any distortion d > 0, and we show that the dispersion has a reverse waterfilling representation. This is the first finite blocklength result for lossy compression of sources with memory. We prove that the finite blocklength rate-distortion function R(n; d; ε) approaches the rate-distortion function R(d) as R(n; d; ε) = R(d)+ √ V(d)/n Q–1(ε)+o(1√n), where V (d) is the dispersion, ε ε 2 (0; 1) is the excess-distortion probability, and Q^(-1) is the inverse Q-function. We give a reverse waterfilling integral representation for the dispersion V (d), which parallels that of the rate-distortion functions for Gaussian processes. Remarkably, for all 0 < d ≥ σ^2 (1+|σ|)^2, R(n; d; ε) of the Gauss-Markov source coincides with that of Z_i, the i.i.d. Gaussian noise driving the process, up to the second-order term. Among novel technical tools developed in this paper is a sharp approximation of the eigenvalues of the covariance matrix of n samples of the Gauss-Markov source, and a construction of a typical set using the maximum likelihood estimate of the parameter a based on n observations.

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Additional Information: | © 2019 IEEE. Manuscript received May 14, 2018; revised November 17, 2018 and April 11, 2019; accepted May 11, 2019. Date of publication May 29, 2019; date of current version September 13, 2019. This work was supported by the National Science Foundation (NSF) under Grant CCF-1566567 and Grant CCF-1751356. This paper was presented at the 2018 IEEE International Symposium on Information Theory [1]. We would like to thank the associate editor Dr. Shun Watanabe and the anonymous reviewers for their insightful comments that are reflected in the final version. | ||||||||||||

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Subject Keywords: | Lossy source coding, Gauss-Markov source, dispersion, finite blocklength regime, rate-distortion theory, sources with memory, achievability, converse, autoregressive processes, covering in probability spaces, parameter estimation | ||||||||||||

Issue or Number: | 10 | ||||||||||||

Record Number: | CaltechAUTHORS:20190610-093125985 | ||||||||||||

Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20190610-093125985 | ||||||||||||

Official Citation: | P. Tian and V. Kostina, "The Dispersion of the Gauss–Markov Source," in IEEE Transactions on Information Theory, vol. 65, no. 10, pp. 6355-6384, Oct. 2019. doi: 10.1109/TIT.2019.2919718 | ||||||||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||||

ID Code: | 96232 | ||||||||||||

Collection: | CaltechAUTHORS | ||||||||||||

Deposited By: | Tony Diaz | ||||||||||||

Deposited On: | 10 Jun 2019 17:03 | ||||||||||||

Last Modified: | 03 Oct 2019 21:20 |

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