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Abstrakty
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.
Czasopismo
Rocznik
Tom
Strony
235--256
Opis fizyczny
Bibliogr. 13 poz.
Twórcy
autor
- Insitute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, Warsaw, Poland
Bibliografia
- [1] P. Billingsley, Convergence of Probability Measures, Wiley, New York 1968.
- [2] M. Birkner and I. Zähle, Functional central limit theorems for the occupation time of the origin for branching random walks in d 3, Weierstraß Insitut für Angewandte Analysis und Stochastik, Berlin, preprint No. 1011 (2005).
- [3] M. Birkner and I. Zähle, A functional CLT for the occupation time of state-dependent branching random walk, Ann. Probab. 35 (6) (2007), pp. 2063-2090.
- [4] T. Bojdecki, L. G. Gorostiza and S. Ramaswamy, Convergence of S0-valued processes and space time random fields, J. Funct. Anal. 66 (1986), pp. 21-41.
- [5] T. Bojdecki, L. G. Gorostiza and A. Talarczyk, Sub-fractional Brownian motion and its relation to occupation times, Statist. Probab. Lett. 69 (2004), pp. 405-419.
- [6] T. Bojdecki, L. G. Gorostiza and A. Talarczyk, A long range dependence stable process and an infinite variance branching system, Ann. Probab. 35 (2) (2007), pp. 500-527.
- [7] T. Bojdecki, L. G. Gorostiza and A. Talarczyk, Occupation time fluctuations of an infinite variance branching system in large dimensions, Bernoulli 13 (1) (2007), pp. 20-39.
- [8] T. Bojdecki, L. G. Gorostiza and A. Talarczyk, Limit theorems for occupation time fluctuations of branching systems I: Long-range dependence, Stochastic. Process. Appl. 116 (2006), pp. 1-18.
- [9] T. Bojdecki, L. G. Gorostiza and A. Talarczyk, Limit theorems for occupation time fluctuations of branching systems II: Critical and large dimensions functional, Stochastic Process. Appl. 116 (2006), pp. 19-35.
- [10] L. G. Gorostiza and A. Wakolbinger, Persistence criteria for a class of critical branching particle systems in continuous time, Ann. Probab. 19 (1991), pp. 266-288.
- [11] I. Iscoe, A weighted occupation time for a class of measure-valued branching processes, Probab. Theory Related Fields 71 (1986), pp. 85-116.
- [12] P. Miłoś, Occupation time fluctuations of Poisson and equilibrium finite variance branching systems, Probab. Math. Statist. 27 (2007), pp. 181-203.
- [13] I. Mitoma, Tightness of probabilities on C ([0; 1]; S0) and D([0; 1] ; S0), Ann. Probab. 11 (1983), pp. 989-999.
Typ dokumentu
Bibliografia
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