Hájek-Inagaki representation theorem, under a general stochastic processes framework, based on stopping times

Let the random variables (r.v.'s) X0,X1,... be defined on the probability space and take values in , where S is a measurable subset of a Euclidean space and is the [sigma]-field of Borel subsets of S, and suppose that they form a general stochastic process. It is assumed that all finite dimensional joint distributions of the underlying r.v.'s have known functional form except that they depend on the parameter [theta], a member of an open subset [Theta] of , k>=1. What is available to us is a random number of r.v.'s X0,X1,...,X[nu]n, where [nu]n is a stopping time as specified below. On the basis of these r.v.'s, a sequence of so-called regular estimates of [theta] is considered, which properly normalized converges in distribution to a probability measure . Then the main theorem in this paper is the Hájek-Inagaki convolution representation of . The proof of this theorem rests heavily on results previously established in the framework described here. These results include asymptotic expansions-in the probability sense-of log-likelihoods, their asymptotic distributions, the asymptotic distribution of a random vector closely related to the log-likelihoods, and a certain exponential approximation. Relevant references are given in the text.