Abstract
The problem of demonstrating the limiting normality of posterior distributions arising from stochastic processes and allied results are reconsidered.We present a fairly general set of conditions for asymptotic posterior normality which covers a wide class of problems. The single and multiparameter cases are both treated. One important difference between the conditions presented here and those of other authors (for example, Heyde and Johnstone (1979), hereafter referred to as H-J) is the use of a shrinking neighbourhood for asymptotic continuity of information, whereas in H-J (1979) a fixed neighbourhood is taken. Some examples to illustrate the importance of this study are considered in detail and this embraces diverse areas of application.Apart from a sequence of constants that is used to measure the order of Fisher's observed information, the uniformity requirement (which may be dropped for some ergodic models) allows us to dispense with further conditions regarding moments of the first and second derivatives of log-likelihood. We obtain the fundamental Bernstein Von Mises theorem in the asymptotic theory of Bayesian inference for stochastic processes. As an application of the theorem, we obtain asymptotic properties of Bayes' estimators for a suitable class of loss functions and show that the maximum likelihood estimator and Bayes estimator are asymptoticaly equivalent.Apart from obtaining some sufficient conditions under which one canobtain asymptotic posterior normality for evolutive processes, such as non-homogeneous Poisson processes and non-homogeneous birth processess, we also present and discuss some sufficient conditions for generalised linear models and other ergodic processes. We also discuss some relationships between our conditions and those imposed by earlier authors.Finally, we discuss the rate of convergence of posterior distributions to the normal distribution, some open problems and scope for further research.