TY - JOUR

T1 - Supercritical superprocesses

T2 - Proper normalization and non-degenerate strong limit

AU - Ren, Yan Xia

AU - Song, Renming

AU - Zhang, Rui

N1 - Funding Information:
This work was supported by National Natural Science Foundation of China (Grant Nos. 11671017, 11731009 and 11601354), Key Laboratory of Mathematical Economics and Quantitative Finance (Peking University), Ministry of Education, the Simons Foundation (Grant No. 429343) and Youth Innovative Research Team of Capital Normal University. The authors thank the referees for very careful reading of the paper and for very helpful comments.
Funding Information:
This work was supported by National Natural Science Foundation of China (Grant Nos. 11671017, 11731009 and 11601354), Key Laboratory of Mathematical Economics and Quantitative Finance (Peking University), Ministry of Education, the Simons Foundation (Grant No. 429343) and Youth Innovative Research Team of Capital Normal University. The authors thank the referees for very careful reading of the paper and for very helpful comments.
Publisher Copyright:
© 2019, Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2019/8/1

Y1 - 2019/8/1

N2 - 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−λ0t〈ϕ0,Xt〉 is a positive martingale. Let M∞ be the limit of Mt. It is known (see Liu et al. (2009)) that M∞ is non-degenerate if and only if the L log L condition is satisfied. In this paper we are mainly interested in the case when the L log L condition is not satisfied. We prove that, under some conditions, there exist a positive function γt on [0, ∞) and a non-degenerate random variable W such that for any finite nonzero Borel measure μ on E, Mt:=e-λ0t〈φ0,Xt〉 We also give the almost sure limit of γt〈f, Xt〉 for a class of general test functions f.

AB - 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−λ0t〈ϕ0,Xt〉 is a positive martingale. Let M∞ be the limit of Mt. It is known (see Liu et al. (2009)) that M∞ is non-degenerate if and only if the L log L condition is satisfied. In this paper we are mainly interested in the case when the L log L condition is not satisfied. We prove that, under some conditions, there exist a positive function γt on [0, ∞) and a non-degenerate random variable W such that for any finite nonzero Borel measure μ on E, Mt:=e-λ0t〈φ0,Xt〉 We also give the almost sure limit of γt〈f, Xt〉 for a class of general test functions f.

KW - 60F15

KW - 60J68

KW - Seneta-Heyde norming

KW - martingales

KW - non-degenerate strong limit

KW - superprocesses

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U2 - 10.1007/s11425-018-9402-4

DO - 10.1007/s11425-018-9402-4

M3 - Article

AN - SCOPUS:85068909376

VL - 62

SP - 1519

EP - 1552

JO - Science China Mathematics

JF - Science China Mathematics

SN - 1674-7283

IS - 8

ER -