The Dispersion of the Gauss–Markov Source

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)+√(V(d)/n) Q
^-1
(ϵ)+0(1/√n),where V(d) is the dispersion,ϵ ∈ (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+|a|
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 observation.