From Parameter Estimation to Dispersion of Nonstationary Gauss-Markov Processes

This paper provides a precise error analysis for the maximum likelihood estimate â (u) of the parameter a given samples u = (u 1 , … , u n )^⊤ drawn from a nonstationary Gauss-Markov process U i = aU i−1 + Z i , i ≥ 1, where a > 1, U 0 = 0, and Z i 's are independent Gaussian random variables with zero mean and variance σ^2 . We show a tight nonasymptotic exponentially decaying bound on the tail probability of the estimation error. Unlike previous works, our bound is tight already for a sample size of the order of hundreds. We apply the new estimation bound to find the dispersion for lossy compression of nonstationary Gauss-Markov sources. We show that the dispersion is given by the same integral formula derived in our previous work [1] for the (asymptotically) stationary Gauss-Markov sources, i.e., |a| < 1. New ideas in the nonstationary case include a deeper understanding of the scaling of the maximum eigenvalue of the covariance matrix of the source sequence, and new techniques in the derivation of our estimation error bound.