  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.
