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

Part of: Stochastic processes Distribution theory - Probability Limit theorems

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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References

Adler, R.J. (1993). Superprocess local and intersection local times and their corresponding particle pictures. Seminar on Stochastic Processes 1992. Birkhäuser, Boston.Google Scholar
Adler, R.J., and Epstein, R. (1987). Some central limit-theorems for Markov paths and some properties of Gaussian random-fields. Stoch. Proc. Appl. 24, 157–202.Google Scholar
Adler, R.J., Feldman, R., and Lewin, M. (1991). Intersection local times for infinite systems of planar Brownian motions and the Brownian density process. Ann. Prob. 19, 192–220.CrossRefGoogle Scholar
Adler, R.J., and Rosen, J.S. (1993). Intersection local times of all orders for Brownian and stable density processes–construction, renormalization and limit laws. Ann. Prob. 21, 1073–1123.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.Google Scholar
Bojdecki, T., and Gorostiza, L.G. (1995). Self-intersection local time for Gaussian S′ (Rd)-processes: existence, path continuity and examples. Stoch. Proc. Appl. 60, 191–226.Google Scholar
Feldman, R.E., and Rachev, S.T. (1993). U-statistics of random-size samples and limit theorems for systems of Markovian particles with non-Poisson initial distributions. Ann. Prob. 21, 1927–1945.Google Scholar
Holley, R., and Stroock, D. (1978). Generalized Ornstein–Uhlenbeck processes and infinite particle branching Brownian motions. Publ. RIMS Kyoto Univ. 14, 741–788.Google Scholar
Itô, K. (1983). Distribution valued processes arising from independent Brownian motions. Math. Z. 182, 17–33.Google Scholar
Martin-Löf, A. (1976). Limit theorems for the motion of a Poisson system of independent Markovian particles with high density. Z. Wahrscheinlichkeitsth. 34, 205–223.Google Scholar
Walsh, J.B. (1986). An Introduction to Stochastic Partial Differential Equations. (Springer Lecture Notes in Mathematics 1180.) Springer, Berlin, pp. 265–439.Google Scholar