An infinite dimensional central limit theorem for correlated martingales

Ilie Grigorescu

doi:10.1016/j.anihpb.2003.03.001

An infinite dimensional central limit theorem for correlated martingales

Ilie Grigorescu

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

The paper derives a functional central limit theorem for the empirical distributions of a system of strongly correlated continuous martingales at the level of the full trajectory space. We provide a general class of functionals for which the weak convergence to a centered Gaussian random field takes place. An explicit formula for the covariance is established and a characterization of the limit is given in terms of an inductive system of SPDEs. We also show a density theorem for a Sobolev-type class of functionals on the space of continuous functions.

Original language	English (US)
Pages (from-to)	167-196
Number of pages	30
Journal	Annales de l'institut Henri Poincare (B) Probability and Statistics
Volume	40
Issue number	2
DOIs	https://doi.org/10.1016/j.anihpb.2003.03.001
State	Published - Mar 2004

Keywords

Central limit theorem
Fluctuations from hydrodynamic limit
Gaussian random field

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

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title = "An infinite dimensional central limit theorem for correlated martingales",

abstract = "The paper derives a functional central limit theorem for the empirical distributions of a system of strongly correlated continuous martingales at the level of the full trajectory space. We provide a general class of functionals for which the weak convergence to a centered Gaussian random field takes place. An explicit formula for the covariance is established and a characterization of the limit is given in terms of an inductive system of SPDEs. We also show a density theorem for a Sobolev-type class of functionals on the space of continuous functions.",

keywords = "Central limit theorem, Fluctuations from hydrodynamic limit, Gaussian random field",

author = "Ilie Grigorescu",

note = "Funding Information: E-mail address: igrigore@math.miami.edu (I. Grigorescu). 1 Research supported in part by McMaster University and The Fields Institute, Ontario, Canada, the University of Utah and the University of Miami.",

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N2 - The paper derives a functional central limit theorem for the empirical distributions of a system of strongly correlated continuous martingales at the level of the full trajectory space. We provide a general class of functionals for which the weak convergence to a centered Gaussian random field takes place. An explicit formula for the covariance is established and a characterization of the limit is given in terms of an inductive system of SPDEs. We also show a density theorem for a Sobolev-type class of functionals on the space of continuous functions.

AB - The paper derives a functional central limit theorem for the empirical distributions of a system of strongly correlated continuous martingales at the level of the full trajectory space. We provide a general class of functionals for which the weak convergence to a centered Gaussian random field takes place. An explicit formula for the covariance is established and a characterization of the limit is given in terms of an inductive system of SPDEs. We also show a density theorem for a Sobolev-type class of functionals on the space of continuous functions.

KW - Central limit theorem

KW - Fluctuations from hydrodynamic limit

KW - Gaussian random field

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SN - 0246-0203

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An infinite dimensional central limit theorem for correlated martingales

Abstract

Keywords

ASJC Scopus subject areas

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