The Dispersion of the Gauss-Markov Source
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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∈ (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≤σ^2/(1+|a|)^2, R(n,d,ϵ) 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.
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