TY - JOUR
T1 - Supercritical superprocesses
T2 - Proper normalization and non-degenerate strong limit
AU - Ren, Yan Xia
AU - Song, Renming
AU - Zhang, Rui
N1 - 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
SN - 1674-7283
VL - 62
SP - 1519
EP - 1552
JO - Science China Mathematics
JF - Science China Mathematics
IS - 8
ER -