A multivariate central limit theorem for continuous local martingales
type: Item
instance of: scholarly article
title: A multivariate central limit theorem for continuous local martingales
zbMATH Open document ID: 1004.60020
DOI: 10.1016/S0167-7152(00)00108-5
author: Harry van Zanten
published in: Statistics \& Probability Letters
publication date: 2003-02-06T00:00:00Z
review text: Let \(M\) be a \(d\)-dimensional continuous local martingale with \(K_t\langle M\rangle_tK^d_t \to^P\eta \eta^T\) for some non-random invertible \(d\times d\)-matrices \(K_t\) such that \(\|K_t\|\to 0\) as \(t\to\infty\) and some possibly random \(d\times d\)-matrix \(\eta\). The main result: Under these conditions, \((K_tM_t)\) converges for \(t\to\infty\) stably to a ``variance mixed'' normal distribution with mixing distribution equal to those of \(\eta\eta^T\) in the sense that \((K_tM_t)\) converges stably to \(V\) where \(V\) has the characteristic function \(u\to E\exp (-{1\over 2}u^T \eta\eta^Tu)\). This theorem forms a generalization of a similar result of the reviewer and \textit{M. Sørensen} [Bernoulli 5, No. 3, 483--493 (1999; Zbl 0943.60016)] for (in general discontinuous) square integrable martingales.
reviewed by: Uwe Küchler
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G44
zbMATH DE Number: 1546527
zbMATH Keywords: multivariate central limit theorems
zbMATH Keywords: continuous martingales
zbMATH Keywords: stable convergence
MaRDI profile type: Publication
cites work: On mixing and stability of limit theorems
cites work: Q5560061
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Wikidata QID: Q126777908
description: scientific article; zbMATH DE number 1546527