Asymptotic Rate of Discrimination for Markov Processes

Abstract

Simple hypotheses $H_P$ and $H_Q$, specifying two distinct positive transition densities $p(x \mid y)$ and $q \mid y)$ and initial densities $p(x)$ and $q(x)$ with respect to a finite Lebesque-Stieltjes measure, are assumed for a discrete time parameter Markov process. Let $R_n$ be the likelihood ratio static based on the first $n + 1$ observations of the process, and consider the class of sequences of likelihood ratio tests $T(a, \alpha) = \{\lbrack R_n > n^\alpha a\rbrack: n = 0, 1, 2, \cdots\}$ generated by letting $a$ and $\alpha$ vary over the real numbers. Under certain regularity assumptions on $K_t(x, y) = p^{1-t} (x \mid y)q^t (x \mid y)$ and the initial densities $p$ and $q$, the subclass of consistent sequences is determined, and the limiting rates at which the error probabilities tend to zero for tests in this subclass are found. A definition of the best asymptotic rate for distinguishing between $H_P$ and $H_Q$ is made for the class of consistent tests. This "asymptotic rate of discrimination" is evaluated and is shown to be attained by a certain subclass of these tests. Some applications and extensions of the theory to infinite Lebesgue-Stieltjes measures are given.