# Nonstationary Gauss-Markov Processes: Parameter Estimation and Dispersion

## TL;DR

This paper analyzes the maximum likelihood estimation error for a nonstationary Gauss-Markov process, providing tight nonasymptotic bounds and applying these results to determine the source dispersion in lossy compression.

## Contribution

It introduces a tight nonasymptotic error bound for parameter estimation in nonstationary Gauss-Markov processes and extends dispersion analysis to the nonstationary case.

## Key findings

- Bound on estimation error decays exponentially and is tight for hundreds of samples.
- Dispersion formula for nonstationary sources matches that of stationary sources under certain conditions.
- New eigenvalue bounding techniques for covariance matrices in nonstationary processes.

## Abstract

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