A strong law of large numbers for super-stable processes

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:124

Article

Kouritzin, Michael A.¹  Ren, Yan-Xia^2,3 

[1] Univ Alberta, Dept Math & Stat Sci, Edmonton, AB T6G 2G1, Canada

[2] Peking Univ, LMAM Sch Math Sci, Beijing 100871, Peoples R China

[3] Peking Univ, Ctr Stat Sci, Beijing 100871, Peoples R China

关键词: Super-stable process;    Super-Brownian motion;    Strong law of large numbers;    Fourier transform;    Vague convergence;    Probability measures;   

DOI  :  10.1016/j.spa.2013.08.009

来源: Elsevier

【 摘 要 】

Let be Lebesgue measure and X = (X-1, t >= 0; P mu) be a supercritical, super-stable process corresponding to the operator 7 (-Delta)(alpha/2) u + beta u eta u(2) on Rd with constants beta, eta > 0 and a alpha is an element of (0, 2]. Put W-1 (theta) = e((theta)) r (-1 theta) X-t(e(-i theta)), which for each small theta is an a.s. convergent complex-valued martingale with limit a W(theta) say. We establish for any starting finite measure mu satisfying f(R)(d) vertical bar x vertical bar mu(dx) < infinity that t(d)/alpha W(0) c alpha W (0)l P mu-a.s. in a topology, termed the shallow topology, strictly stronger than the vague ePt topology yet weaker than the weak topology, where c(alpha) > 0 is a known constant. This result can be thought of as an extension to a class of superprocesses of Watanabe's strong law of large numbers for branching Markov processes. (C) 2013 Elsevier B.V. All rights reserved.

【 授权许可】

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