A long range dependence stable process and an infinite variance branching system

• 成果类型: Article
• 署名作者: Bojdecki, Tomasz; Gorostiza, Luis G.; Talarczyk, Anna
• 署名单位: University of Warsaw; University of Warsaw
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117906000000737
• 发表日期: 2007
• 页码: 500-527
• 关键词: occupation time fluctuations; divisible processes; particle-systems; LIMIT-THEOREMS; point

摘要

We prove a functional limit theorem for the resealed occupation time fluctuations of a (d, alpha, beta)-branching particle system [particles moving in R-d according to a symmetric alpha-stable Levy process, branching law in the domain of attraction of a (1+ beta)-stable law, 0 < beta < 1, uniform Poisson initial state] in the case of intermediate dimensions, alpha / beta < d < alpha (1+ beta) / beta. The limit is a process of the form K gimel xi, where K is a constant, gimel is the Lebesgue measure on R-d, and xi = (xi t) (t >= 0) is a (1+ beta)-stable process which has long range dependence. For alpha < 2, there are two long range dependence regimes, one for beta > d / (d + alpha), which coincides with the case of finite variance branching (beta = 1), and another one for beta <= d / (d + alpha), where the long range dependence depends on the value of beta. The long range dependence is characterized by a dependence exponent kappa which describes the asymptotic behavior of the codifference of increments of xi on intervals far apart, and which is d / alpha for the first case (and for alpha = 2) and (1 + beta - d / (d + alpha))d / alpha for the second one. The convergence proofs use techniques of S'(R-d)-valued processes.