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 (X,A,Pθ) and take values in (S,S), where S is a measurable subset of a Euclidean space and S is the σ-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 θ, a member of an open subset Θ of Rk, k≥1. What is available to us is a random number of r.v.'s X0,X1,…,Xνn, where νn is a stopping time as specified below. On the basis of these r.v.'s, a sequence of so-called regular estimates of θ is considered, which properly normalized converges in distribution to a probability measure L(θ). Then the main theorem in this paper is the Hájek-Inagaki convolution representation of L(θ). 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.