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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.

Some remarks on the almost sure central limit theorem for independent random variables

Rychlik Z., Szuster K. S.

Probability and Mathematical Statistics

2003

Vol. 23, Fasc. 2

241--249

The purpose of this paper is the proof of an almost sure central limit theorem for subsequences. We obtain an almost sure convergence limit theorem for independent nonidentically distributed random variables. The presented results extend,to nonidentically distributed random variables, theorems given by Schatte [13].

An Empirical Functional Central Limit Theorem for weakly dependent sequences

Prieur C.

Probability and Mathematical Statistics

2002

Vol. 22, Fasc. 2

259--287

In this paper we obtain a Functional Central Limit Theorem for the empirical process of a stationary sequence under a new weak dependence condition introduced by Doukhan and Louhichi [5]. This result improves on the Empirical Functional Central Limit Theorem in Doukhan and Louhichi [5]. Our proof relies on new moment inequalities and on a Central Limit Theorem. Techniques of proofs come from Louhichi [12] and Rio [16], respectively. We also deduce a rate of convergence in a Marcinkiewicz-Zygmund Strong Law.