## Abstract

Suppose that X = (Xt, t > 0; Pµ) is a supercritical superprocess in a locally compact separable metric space E. Let ϕ0 be a positive eigenfunction corresponding to the first eigenvalue λ0 of the generator of the mean semigroup of X. Then Mt := e^{−λ}^{0t}〈ϕ0, Xt〉 is a positive martingale. Let M∞ be the limit of Mt. It is known that M∞ is non-degenerate iff the Llog L condition is satisfied. When the Llog L condition may not be satisfied, Ren et al. (2017) recently proved that there exist a non-negative function γt on [0, ∞) and a non-degenerate random variable W such that for any finite nonzero Borel measure µ on E, limt→∞ γt〈ϕ0, Xt〉 = W, a.s.-Pµ. In this paper, we mainly investigate properties of W. We prove that W has strictly positive density on (0, ∞). We also investigate the small value and tail probability problems of W.

Translated title of the contribution | On properties of a class of strong limits forsupercritical superprocesses |
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Original language | Chinese (Traditional) |

Pages (from-to) | 485-504 |

Number of pages | 20 |

Journal | Scientia Sinica Mathematica |

Volume | 49 |

Issue number | 3 |

DOIs | |

State | Published - 2019 |

## Keywords

- absolute continuity
- non-degenerate strong limit
- small value probability
- supercritical
- superprocesses
- tail probability

## ASJC Scopus subject areas

- General Mathematics