Functional convergence of stochastic integrals with application to statistical inference (Q765875)

The main purpose of this paper is to supply some generic conditions which guarantee the weak convergence of some stochastic sums or integrals of the type \[ \iint_{0<s<t<1} f_n(\beta,s,t)\,dU_n(s)\,dV_n(t) \] towards \[ \int_{0<s<t<1} f(\beta,s,t)\,dB_s dW_t, \] provided the martingale differences \((U_n,V_n)\) converge in law to the Brownian \((B,W)\) and the smooth \(f_n(\beta,s,t)\) converge to the smooth \(f(\beta,s,t)\) (the smoothness is with respect to the parameter \(\beta\)). Typically, they consider martingale differences \((X_n,Y_n)\) which are bounded in \(L^2\) and such that \[ {1\over\sqrt{n}} \sum^{[ns]}_{i=0} (X_i,Y_i) \] converges in law to \((B_s,W_s)\), and sums of the type \[ {1\over n} \sum_{0\leq i<j\leq n} f_n\Biggl(\beta,{i-1\over n}, {j\over n}\Biggr) X_i Y_j, \] which, under convenient technical conditions, converge in law to \[ \int_{0<s<t\leq 1} f(\beta, s,t)\,dB_s\,dW_t. \] This type of result is then applied to some examples and, in particular, to several examples of weak convergence of (joint) likelihoods.