Weak convergence of martingales with random indices to infinitely divisible laws
type: Item
instance of: scholarly article
title: Weak convergence of martingales with random indices to infinitely divisible laws
zbMATH Open document ID: 0531.60021
DOI: 10.1007/BF01960559
author: K. S. Kubacki
author: Dominik Szynal
published in: Acta Mathematica Hungarica
publication date: 1983-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G42
zbMATH DE Number: 3842869
zbMATH Keywords: weak convergence
zbMATH Keywords: infinitely divisible law
zbMATH Keywords: random indices
Wikidata QID: Q114852620
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3842869