Stable central limit theorems for super ornstein-uhlenbeck processes

Yan Xia Ren, Renming Song, Zhenyao Sun, Jianjie Zhao

Research output: Contribution to journalArticlepeer-review


In this paper, we study the asymptotic behavior of a supercritical (ξ, ψ)-superprocess (Xt)t≥0 whose underlying spatial motion ξ is an Ornstein-Uhlenbeck process on Rd with generator L =12σ2 ∆ − bx · ∇ where σ, b > 0; and whose branching mechanism ψ satisfies Grey’s condition and a perturbation condition which guarantees that, when z → 0, ψ(z) = −αz + ηz1+β (1 + o(1)) with α > 0, η > 0 and β ∈ (0, 1). Some law of large numbers and (1 + β)-stable central limit theorems are established for (Xt(f))t≥0, where the function f is assumed to be of polynomial growth. A phase transition arises for the central limit theorems in the sense that the forms of the central limit theorem are different in three different regimes corresponding to the branching rate being relatively small, large or critical at a balanced value.

Original languageEnglish (US)
Article number141
JournalElectronic Journal of Probability
StatePublished - 2019


  • Branching rate regime
  • Central limit theorem
  • Law of large numbers
  • Ornstein-Uhlenbeck processes
  • Stable distribution
  • Superprocesses

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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