Asymptotic Posterior Normality for Stochastic Processes Revisited

The problem of demonstrating the limiting normality of a posterior distribution arising from some stochastic process is revisited. It is shown that certain processes of practical interest are not covered by conditions currently available in the literature. In this paper we present a fairly general set of conditions for asymptotic posterior normality which cover a wide class of problems. The theory is applied to a nonhomogeneous Poisson process and birth process. case of independent and identically distributed (i.i.d.) observations, we mention Le Cam (1953, 1958), Freedman (1963), Lindley (1965), Bickel and Yahav (1969), Walker (1969), Johnson (1970), Dawid (1970), Strasser (1976) and Hartigan (1983). Markov processes, and stochastic processes more generally, are discussed in Borwanker et al (1971), Moore (1976), Heyde and Johnstone (1979), Basawa and Rao (1980) and Chen (1985). The present paper is in the spirit of Heyde and Johnstone (1979), hereafter referred to as H-J, where asymptotic posterior normality is obtained for general stochastic processes under certain conditions. These conditions however do not cover some cases of practical interest; they fail for example for certain nonhomogeneous Poisson processes which are of interest in reliability. We prove here a general result on asymptotic posterior normality which covers a wide class of problems, in particular the type of nonhomogeneous process not covered in the literature. In Section 2 we state our regularity conditions and the main result. We have specialized to the case of a single parameter for reasons of clarity and comparison with existing results. Under the regularity conditions imposed by H-J, it appears that asymptotic posterior normality holds under weaker conditions than those needed for asymptotic normality of the maximum likelihood (ML) estimator. However, a weakening of H-J's conditions, in order to cover a broader range of applications, necessitates the introduction of other conditions which also guarantee the asymptotic normality of the ML estimator. From this point of view, then, it appears that further conditions are needed for posterior normality, since nonlocal, as well as