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Occupation time fluctuations of Poisson and equilibrium branching systems in critical and large dimensions

Miłoś P.

Probability and Mathematical Statistics

2008

Vol. 28, Fasc. 2

235--256

Limit theorems are presented for the rescaled occupation time fluctuation process of a critical finite variance branching particle system in Rd with symmetric α-stable motion starting off from either a standard Poisson random field or the equilibrium distribution for critical d = 2α and large d > 2α dimensions. The limit processes are generalised Wiener processes. The obtained convergence is in space-time and finite-dimensional distributions sense. Under the additional assumption on the branching law we obtain functional convergence.

Occupation time fluctuations of Poisson and equilibrium finite variance branching systems

Miłoś P.

Probability and Mathematical Statistics

2007

Vol. 27, Fasc. 2

181--203

Functional limit theorems are presented for the rescaled occupation time fluctuation process of a critical finite variance branching particle system in R dwith symmetric а-stable motion starting off from either a standard Poisson random field or from the equilibrium distribution for intermediate dimensions a < d < 2a. The limit processes are determined by sub-fractional and fractional Brownian motions, respectively.