The Gauss-Markov source produces $U_i = aU_{ i-1} + Z_i$ for $i\geq 1$, where $U_0 = 0$, $|a|<1$ and $Z_i\sim\mathcal{ N}(0, \sigma^2)$ 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 \emph{ first} finite blocklength result for lossy compression of \emph{ sources with memory}. We prove that the finite blocklength rate-distortion function $R(n,d,\epsilon)$ approaches the rate-distortion function $\mathbb{ R}(d)$ as $R(n,d,\epsilon) = \mathbb{ R}(d) + \sqrt{ \frac{ V(d)}{ n}}Q^{ -1}(\epsilon) + o\left(\frac{ 1}{ \sqrt{ n}}\right)$, where $V(d)$ is the dispersion, $\epsilon \in (0,1)$ 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 $0 < d\leq \frac{ \sigma^2}{ (1+|a|)^2}$, $R(n,d,\epsilon)$ of the Gauss-Markov source coincides with that of $Z_k$, 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.
Categories: cs.IT math.IT