Nonstationary Gauss-Markov Processes: Parameter Estimation and Dispersion

This paper provides a precise error analysis for the maximum likelihood estimate $\hat{a}_{\text{ML}}(u_1^n)$ of parameter $a$ given samples $u_1^n = (u_1, \ldots, u_n)'$ drawn from nonstationary Gauss-Markov process $U_i U_{i-1} + Z_i,~i\geq 1$, where $U_0 0$, $a> and $Z_i$'s are independent Gaussian random variables with zero mean variance $\sigma^2$. We show tight nonasymptotic exponentially decaying bound on tail probability estimation error. Unlike previous works, our is already sample size order hundreds. apply new to find dispersion lossy compression sources. that by same integral formula we derived previously asymptotically stationary sources, i.e., $|a| < 1$. New ideas in case include separately bounding eigenvalue (which scales exponentially) other eigenvalues bounded constants depend only $a$) covariance matrix source sequence, techniques derivation bound.