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Non-Gaussian Density Processes Arising from Non-Poisson Systems of Independent Brownian Motions

Part of: Limit theorems Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Raisa E. Feldman  and
Srikanth K. Iyer
Raisa E. Feldman*
Affiliation:
University of California, Santa Barbara
Srikanth K. Iyer*
Affiliation:
Indian Institute of Technology
*
∗Postal address: Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106–3110, USA
∗∗Postal address: Indian Institute of Technology, Kanpur, India
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Abstract

The Brownian density process is a Gaussian distribution-valued process. It can be defined either as a limit of a functional over a Poisson system of independent Brownian particles or as a solution of a stochastic partial differential equation with respect to Gaussian martingale measure. We show that, with an appropriate change in the initial distribution of the infinite particle system, the limiting density process is non-Gaussian and it solves a stochastic partial differential equation where the initial measure and the driving measure are non-Gaussian, possibly having infinite second moment.

Keywords

Brownian density processinfinite particle systeminvariance principlenon-Gaussian martingale measuresstochastic partial differential equation

MSC classification

Primary: 60G60: Random fields 60F17: Functional limit theorems; invariance principles
Secondary: 60F05: Central limit and other weak theorems 60E07: Infinitely divisible distributions; stable distributions
Type
Short Communications
Information
Journal of Applied Probability , Volume 35 , Issue 1 , March 1998 , pp. 213 - 220
Copyright
Copyright © Applied Probability Trust 1998 

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