On the law of terminal value of additive martingales in a remarkable branching stable process

DOI10.1016/J.SPA.2023.01.005arXiv2209.11473MaRDI QIDQ6411694

Author name not available (Why is that?)

Publication date: 23 September 2022

Abstract: We give an explicit description of the law of terminal value

W

of additive martingales in a remarkable branching stable process. We show that the right tail probability of the terminal value decays exponentially fast and the left tail probability follows that

- l o g m a t h b b P (W < x) s i m f r a c 12 (l o g x)^{2}

as

x i g h t a r r o w 0 +

. These are in sharp contrast with results in the literature such as Liu (2000, 2001) and Buraczewski (2009). We further show that the law of

W

is self-decomposable, and therefore, possesses a unimodal density. We specify the asymptotic behavior at

0

and at

+ i n f t y

of the latter.

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