### 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) |
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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 | |

State | Published - Mar 2004 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

**An infinite dimensional central limit theorem for correlated martingales.** / Grigorescu, Ilie.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - An infinite dimensional central limit theorem for correlated martingales

AU - Grigorescu, Ilie

PY - 2004/3

Y1 - 2004/3

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

UR - http://www.scopus.com/inward/record.url?scp=1542400003&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1542400003&partnerID=8YFLogxK

U2 - 10.1016/j.anihpb.2003.03.001

DO - 10.1016/j.anihpb.2003.03.001

M3 - Article

AN - SCOPUS:1542400003

VL - 40

SP - 167

EP - 196

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

SN - 0246-0203

IS - 2

ER -