A Strong Law of Large Numbers for Super-stable Processes.

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  • 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 View the MathML source, which for each smallθ is an a.s. convergent complex-valued martingale with limit View the MathML source say. We establish for any starting finite measure μ satisfying View the MathML source that View the MathML source-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.

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    Article (Published)
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    © 2014 Stochastic Processes and their Applications. This version of this article is open access and can be downloaded and shared. The original author(s) and source must be cited.
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    • M. A. Kouritzin and Y-X. Ren. (2014), " A Strong Law of Large Numbers for Super-stable Processes '', Stochastic Processes and their Applications, 124, pp. 505-521.