Lower Deviation for the Supremum of the Support of Super-Brownian Motion

Yan-Xia Ren (), Renming Song () and Rui Zhang ()
Additional contact information
Yan-Xia Ren: Peking University
Renming Song: University of Illinois at Urbana-Champaign
Rui Zhang: Capital Normal University

Journal of Theoretical Probability, 2024, vol. 37, issue 2, 1079-1123

Abstract: Abstract We study the asymptotic behavior of the supremum $$M_t$$ M t of the support of a supercritical super-Brownian motion. In our recent paper (Ren et al. in Stoch Proc Appl 137:1–34, 2021), we showed that, under some conditions, $$M_t-m(t)$$ M t - m ( t ) converges in distribution to a randomly shifted Gumbel random variable, where $$m(t)=c_0t-c_1\log t$$ m ( t ) = c 0 t - c 1 log t . In the same paper, we also studied the upper large deviation of $$M_t$$ M t , i.e., the asymptotic behavior of $$\mathbb {P} (M_t>\delta c_0t) $$ P ( M t > δ c 0 t ) for $$\delta \ge 1$$ δ ≥ 1 . In this paper, we study the lower large deviation of $$M_t$$ M t , i.e., the asymptotic behavior of $$\mathbb {P} (M_t\le \delta c_0t|\mathcal {S}) $$ P ( M t ≤ δ c 0 t | S ) for $$\delta

Keywords: Super-Brownian motion; Supremum of support; Lower large deviation; 60F10; 60J68 (search for similar items in EconPapers)
Date: 2024
References: Add references at CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10959-023-01292-3 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:37:y:2024:i:2:d:10.1007_s10959-023-01292-3

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-023-01292-3

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().