Asymptotic expansion of the log-likelihood function based on stopping times defined on a Markov process

Consider the parameter space Θ which is an open subset of ℝ ^k, k≧1, and for each θ∈Θ, let the r.v.′s Y _n, n=0, 1, ... be defined on the probability space (X, A, P _θ) and take values in a Borel set S of a Euclidean space. It is assumed that the process {Y _n }, n≧0, is Markovian satisfying certain suitable regularity conditions. For each n≧1, let υ _n be a stopping time defined on this process and have some desirable properties. For 0 < τ _n → ∞ as n→∞, set {Mathematical expression} h _n →h ∈R ^k, and consider the log-likelihood function {Mathematical expression} of the probability measure {Mathematical expression} with respect to the probability measure {Mathematical expression}. Here {Mathematical expression} is the restriction of P _θ to the σ-field induced by the r.v.′s Y ₀, Y ₁, ..., {Mathematical expression}. The main purpose of this paper is to obtain an asymptotic expansion of {Mathematical expression} in the probability sense. The asymptotic distribution of {Mathematical expression}, as well as that of another r.v. closely related to it, is obtained under both {Mathematical expression} and {Mathematical expression}.