Functional convergence of stochastic integrals with application to statistical inference

Assuming that {(Un,Vn)}{(Un,Vn)} is a sequence of càdlàg   processes converging in distribution to (U,V)(U,V) in the Skorohod topology, conditions are given under which {∬fn(β,u,v)dUndVn}{∬fn(β,u,v)dUndVn} converges weakly to ∬f(β,x,y)dUdV∬f(β,x,y)dUdV in the space C(R)C(R), where fn(β,u,v)fn(β,u,v) is a sequence of “smooth” functions converging to f(β,u,v)f(β,u,v). Integrals of this form arise as the objective function for inference about a parameter ββ in a stochastic model. Convergence of these integrals play a key role in describing the asymptotics of the estimator of ββ which optimizes the objective function. We illustrate this with a moving average process.