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On the Almost Sure Central Limit Theorem for Vector Martingales: Convergence of Moments and Statistical Applications

Part of: Probability theory on algebraic and topological structures Miscellaneous applications of functional analysis

Published online by Cambridge University Press:  14 July 2016

Bernard Bercu ,
Peggy Cénac  and
Guy Fayolle
Bernard Bercu*
Affiliation:
Université Bordeaux 1 and INRIA Bordeaux Sud-Ouest
Peggy Cénac*
Affiliation:
Université de Bourgogne
Guy Fayolle*
Affiliation:
INRIA Paris-Rocquencourt
*
∗Postal address: Université Bordeaux 1, Institut de Mathématiques de Bordeaux, UMR 5251, France. Email address: bernard.bercu@math.u-bordeaux1.fr
∗∗Postal address: Université de Bourgogne, Institut de Mathématiques de Bourgogne, UMR 5584, 9 rue Alain Savary, BP 47870, 21078 Dijon Cedex, France. Email address: peggy.cenac@u-bourgogne.fr
∗∗∗Postal address: INRIA CR Paris-Rocquencourt, Domaine de Voluceau, BP 105, 78153 Le Chesnay Cedex, France. Email address: guy.fayolle@inria.fr
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Abstract

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We investigate the almost sure asymptotic properties of vector martingale transforms. Assuming some appropriate regularity conditions both on the increasing process and on the moments of the martingale, we prove that normalized moments of any even order converge in the almost sure central limit theorem for martingales. A conjecture about almost sure upper bounds under wider hypotheses is formulated. The theoretical results are supported by examples borrowed from statistical applications, including linear autoregressive models and branching processes with immigration, for which new asymptotic properties are established on estimation and prediction errors.

Keywords

Almost-sure central limit theoremvector martingalemomentstochastic regression

MSC classification

Secondary: 60B12: Limit theorems for vector-valued random variables (infinite-dimensional case) 46N30: Applications in probability theory and statistics
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 1 , March 2009 , pp. 151 - 169
Copyright
Copyright © Applied Probability Trust 2009 

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