A strong law of large numbers for super-stable processes

Michael A. Kouritzin and Yan-Xia Ren

Stochastic Processes and their Applications, 2014, vol. 124, issue 1, 505-521

Abstract: Let ℓ be Lebesgue measure and X=(Xt,t≥0;Pμ) be a supercritical, super-stable process corresponding to the operator −(−Δ)α/2u+βu−ηu2 on Rd with constants β,η>0 and α∈(0,2]. Put Wˆt(θ)=e(|θ|α−β)tXt(e−iθ⋅), which for each smallθ is an a.s. convergent complex-valued martingale with limit Wˆ(θ) say. We establish for any starting finite measure μ satisfying ∫Rd|x|μ(dx)<∞ that td/αXteβt→cαWˆ(0)ℓPμ-a.s. in a topology, termed the shallow topology, strictly stronger than the vague topology yet weaker than the weak topology, where cα>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.

Keywords: Super-stable process; Super-Brownian motion; Strong law of large numbers; Fourier transform; Vague convergence; Probability measures (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (3)

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DOI: 10.1016/j.spa.2013.08.009

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