Convergence rate for a class of supercritical superprocesses

Rongli Liu, Yan-Xia Ren and Renming Song

Stochastic Processes and their Applications, 2022, vol. 154, issue C, 286-327

Abstract: Suppose X={Xt,t≥0} is a supercritical superprocess. Let ϕ be the non-negative eigenfunction of the mean semigroup of X corresponding to the principal eigenvalue λ>0. Then Mt(ϕ)=e−λt〈ϕ,Xt〉,t≥0, is a non-negative martingale with almost sure limit M∞(ϕ). In this paper we study the rate at which Mt(ϕ)−M∞(ϕ) converges to 0 as t→∞ when the process may not have finite variance. Under some conditions on the mean semigroup, we provide sufficient and necessary conditions for the rate in the almost sure sense. Some results on the convergence rate in Lp with p∈(1,2) are also obtained.

Keywords: Supercritical superprocess; Convergence rate; Infinite variance; Spine decomposition; Principal eigenvalue; Eigenfunction (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1016/j.spa.2022.09.009

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