Extended convergence to continuous in probability processes with independent increments
type: Item
instance of: scholarly article
title: Extended convergence to continuous in probability processes with independent increments
zbMATH Open document ID: 0596.60038
DOI: 10.1007/BF00343896
author: Adam Jakubowski
author: Leszek Slominski
publication date: 1986-01-01T00:00:00Z
review text: Criteria for Aldous-extended convergence [\textit{D. Aldous}, A concept of weak convergence for stochastic processes viewed in the Strasbourg manner. Preprint, Statist. Laboratory, Univ. Cambridge (1979)] of semimartingales to processes of the kind mentioned in the title are obtained in terms of the known assumptions in functional limit theorems. The extended in Aldous sense convergence of real stochastic processes \(X^ n\) on probabilistic basis with filtrations \(F^ n=(F^ n_ t)_{R^+}\), \(n\geq 1\), is by definition the convergence of random elements \((X^ n,Z^ n)\) in D(R\(\times {\mathcal M}_{w}(D(R)))\) by distribution. Here D(S) is the Skorokhod space of functions in a Polish space S, and \(Z^ n=Z^ n_ t\) is the regular \(F^ n_ t\)- conditional distribution of D(R)-valued random element \(X^ n.\) Assume that the known Jacod conditions of the usual functional convergence of semimartingales \((X^ n,F^ n)\) to a quasi-continuous process (X,F) with independent increments are fulfilled [\textit{J. Jacod}, Z. Wahrscheinlichkeitstheor. Verw. Geb. 63, 109-136 (1983; Zbl 0526.60065), theorem 1.21]. The main theorem: By this assumptions \((X^ n,F^ n)\) converges extendedly to (X,F). Corollaries: The assumptions of CLT's for semimartingales due to \textit{R. Liptser} and \textit{A. Shiryaev} [ibid. 59, 311-318 (1982; Zbl 0482.60030)] are sufficient and necessary for extended convergence of \((X^ n,F^ n)\) to a Gaussian local martingale. Theorem: \((X^ n,F^ n)\) converges extendedly to (X,F) iff the family \((X^ n,F^ n)\) satisfies the Aldous condition of C-tightness, and there exists a sequence of small-scaling on compacts discretisations \(\rho\) (n) of the time line such that \((X^ n_{\rho (n)},F^ n_{\rho (n)})\) fulfills the discrete analogue of the Jacod conditions.
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60J99
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60G48
zbMATH DE Number: 3960670
zbMATH Keywords: functional limit theorems
zbMATH Keywords: Jacod conditions
zbMATH Keywords: functional convergence of semimartingales
zbMATH Keywords: semimartingales
zbMATH Keywords: Gaussian local martingale
reviewed by: Evgeni Trofimov
MaRDI profile type: Publication
cites work: Stopping times and tightness
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full work available at URL: https://doi.org/10.1007/bf00343896
OpenAlex ID: W2005462170
Wikidata QID: Q131317082
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 3960670