Weak convergence of martingales with random indices to infinitely divisible laws (Q788379)

Let \((X_{nk})\) (\(k\geq 1,n\geq 1)\) be a doubly infinite array of random variables defined on a common probability space (\(\Omega\),\({\mathcal F},P)\). Assume that \((X_{nk})\) is adapted to an array (\({\mathcal F}_{nk})\) of row-wise increasing sub-\(\sigma\)-fields of \({\mathcal F}\), i.e. \(X_{nk}\) is \({\mathcal F}_{nk}\)-measurable and \({\mathcal F}_{nk-1}\subset {\mathcal F}_{nk}\). \({\mathcal F}_{n0}\) need not be the trivial \(\sigma\)-field \(\{ \emptyset,\Omega \}\). Further, let \(\{N_ n,n\geq 1\}\) be a sequence of positive integer-valued rvs defined on the same probability space (\(\Omega\),\({\mathcal F},P)\). Let us denote \(S_{nN_ n}=X_{n1}+...+X_{nN_ n},\) and \(\phi_{nk}(t)=E(\exp \{itX_{nk}\}| {\mathcal F}_{n,k-1})\), \(f_{nN_ n}(t)=\prod_{1\leq k\leq N_ n}\phi_{nk}(t).\) We give necessary and sufficient conditions for the convergence in probability, as \(n\to \infty\), of \(f_{nN_ n}(t)\) to a characteristic function of an infinitely divisible law with finite variance. From this result we obtain sufficient conditions for \(S_{nN_ n}\) to converge weakly to infinitely divisible laws.