The Seneta-Heyde scaling for supercritical super-Brownian motion

Haojie Hou; Yan Xia Ren; Renming Song

doi:10.1214/22-AIHP1358

The Seneta-Heyde scaling for supercritical super-Brownian motion

Haojie Hou
, Yan Xia Ren
, Renming Song

Research output: Contribution to journal › Article › peer-review

Abstract

We consider the additive martingale Wt (λ) and the derivative martingale ∂Wt (λ) for one-dimensional supercritical super-Brownian motions with general branching mechanism. In the critical case λ = λ0, we prove that √tWt (λ0) converges in probability to a positive limit, which is a constant multiple of the almost sure limit ∂W∞(λ0) of the derivative martingale ∂Wt (λ0). We also prove that, on the survival event, lim supt→∞ √tWt (λ0)=∞almost surely.

Original language	English (US)
Pages (from-to)	1387-1417
Number of pages	31
Journal	Annales de l'institut Henri Poincare (B) Probability and Statistics
Volume	60
Issue number	2
DOIs	https://doi.org/10.1214/22-AIHP1358
State	Published - May 2024

Keywords

Additive martingale
Derivative martingale
Seneta-Heyde scaling
Skeleton decomposition
Spine decomposition
Super-Brownian motion

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

Online availability

10.1214/22-AIHP1358

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title = "The Seneta-Heyde scaling for supercritical super-Brownian motion",

abstract = "We consider the additive martingale Wt (λ) and the derivative martingale ∂Wt (λ) for one-dimensional supercritical super-Brownian motions with general branching mechanism. In the critical case λ = λ0, we prove that √tWt (λ0) converges in probability to a positive limit, which is a constant multiple of the almost sure limit ∂W∞(λ0) of the derivative martingale ∂Wt (λ0). We also prove that, on the survival event, lim supt→∞ √tWt (λ0)=∞almost surely.",

keywords = "Additive martingale, Derivative martingale, Seneta-Heyde scaling, Skeleton decomposition, Spine decomposition, Super-Brownian motion",

author = "Haojie Hou and Ren, \{Yan Xia\} and Renming Song",

note = "We thank the referee for helpful comments and suggestions on the first version of this paper. We also thank Professor Xinxin Chen for helping us translating the abstract into French. Part of the research for this paper was done while the third-named author was visiting Jiangsu Normal University, where he was partially supported by a grant from the National Natural Science Foundation of China (11931004) and by the Priority Academic Program Development of Jiangsu Higher Education Institutions. The research of this project is supported by the National Key R\&D Program of China (No. 2020YFA0712900). The second author was supported by NSFC (Grant Nos. 11671017, 11731009 and 12231002) and LMEQF. The third author was supported in part by a grant from the Simons Foundation (\#429343, Renming Song).",

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doi = "10.1214/22-AIHP1358",

language = "English (US)",

volume = "60",

pages = "1387--1417",

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T1 - The Seneta-Heyde scaling for supercritical super-Brownian motion

AU - Hou, Haojie

AU - Ren, Yan Xia

AU - Song, Renming

N1 - We thank the referee for helpful comments and suggestions on the first version of this paper. We also thank Professor Xinxin Chen for helping us translating the abstract into French. Part of the research for this paper was done while the third-named author was visiting Jiangsu Normal University, where he was partially supported by a grant from the National Natural Science Foundation of China (11931004) and by the Priority Academic Program Development of Jiangsu Higher Education Institutions. The research of this project is supported by the National Key R&D Program of China (No. 2020YFA0712900). The second author was supported by NSFC (Grant Nos. 11671017, 11731009 and 12231002) and LMEQF. The third author was supported in part by a grant from the Simons Foundation (#429343, Renming Song).

PY - 2024/5

Y1 - 2024/5

N2 - We consider the additive martingale Wt (λ) and the derivative martingale ∂Wt (λ) for one-dimensional supercritical super-Brownian motions with general branching mechanism. In the critical case λ = λ0, we prove that √tWt (λ0) converges in probability to a positive limit, which is a constant multiple of the almost sure limit ∂W∞(λ0) of the derivative martingale ∂Wt (λ0). We also prove that, on the survival event, lim supt→∞ √tWt (λ0)=∞almost surely.

AB - We consider the additive martingale Wt (λ) and the derivative martingale ∂Wt (λ) for one-dimensional supercritical super-Brownian motions with general branching mechanism. In the critical case λ = λ0, we prove that √tWt (λ0) converges in probability to a positive limit, which is a constant multiple of the almost sure limit ∂W∞(λ0) of the derivative martingale ∂Wt (λ0). We also prove that, on the survival event, lim supt→∞ √tWt (λ0)=∞almost surely.

KW - Additive martingale

KW - Derivative martingale

KW - Seneta-Heyde scaling

KW - Skeleton decomposition

KW - Spine decomposition

KW - Super-Brownian motion

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JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

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The Seneta-Heyde scaling for supercritical super-Brownian motion

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