An infinite dimensional central limit theorem for correlated martingales

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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 languageEnglish (US)
Pages (from-to)167-196
Number of pages30
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Volume40
Issue number2
DOIs
StatePublished - Mar 2004

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Martingale
Central limit theorem
Density Theorem
Functional Central Limit Theorem
Gaussian Random Field
Spaces of Continuous Functions
Empirical Distribution
Weak Convergence
Explicit Formula
Trajectory
Class
Empirical distribution
Weak convergence
Random field

Keywords

  • Central limit theorem
  • Fluctuations from hydrodynamic limit
  • Gaussian random field

ASJC Scopus subject areas

  • Statistics and Probability

Cite this

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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.",
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