The Dispersion of the Gauss-Markov Source

The Gauss-Markov source produces Ui=aUi-1+ Zi for i ≥ 1, where and Zi ~ 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 ∈ (0,1) is the excess-distortion probability, and Q
^-1
is the inverse of the 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 <; σ2/ (1+|a|)
²
,R(n, d, c)of the Gauss-Markov source coincides with that of Zi, 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.