Treffer: The Dispersion of the Gauss-Markov Source

The Gauss-Markov source produces $U_i = aU_{i-1} + Z_i$ for $i\geq 1$, where $U_0 = 0$, $|a|0$, 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 $R(n,d,��)$ approaches the rate-distortion function $\mathbb{R}(d)$ as $R(n,d,��) = \mathbb{R}(d) + \sqrt{\frac{V(d)}{n}}Q^{-1}(��) + o\left(\frac{1}{\sqrt{n}}\right)$, where $V(d)$ is the dispersion, $��\in (0,1)$ 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 $0 < d\leq \frac{��^2}{(1+|a|)^2}$, $R(n,d,��)$ 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.
30 pages, 8 figures, shorter version published in the proceedings of IEEE ISIT 2018