Geodesic
The Probability of Reaching a Receding Boundary by a Branching Random Walk with Fading Branching and Heavy-Tailed Jump Distribution
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching Processes and Related Topics, Tome 316 (2022), pp. 336-354

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Foss and Zachary (2003) and Foss, Palmowski and Zachary (2005) studied the probability of achieving a receding boundary on a time interval of random length by a random walk with a heavy-tailed jump distribution. They have proposed and developed a new approach that allows one to generalise the results of Asmussen (1998) to the case of arbitrary stopping times and to a wide class of nonlinear boundaries, and to obtain uniform results over all stopping times. In this paper, we consider a class of branching random walks with fading branching and obtain results on the tail asymptotics for the maximum of a branching random walk on a time interval of random (possibly unlimited) length, as well as uniform results within a class of bounded random time intervals.
Keywords: subexponential and strong subexponential distributions, branching random walk, receding boundary, principle of a single big jump. 
@article{TM_2022_316_a21,
     author = {P. I. Tesemnivkov and S. G. Foss},
     title = {The {Probability} of {Reaching} a {Receding} {Boundary} by a {Branching} {Random} {Walk} with {Fading} {Branching} and {Heavy-Tailed} {Jump} {Distribution}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {336--354},
     publisher = {mathdoc},
     volume = {316},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2022_316_a21/}
}
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P. I. Tesemnivkov; S. G. Foss. The Probability of Reaching a Receding Boundary by a Branching Random Walk with Fading Branching and Heavy-Tailed Jump Distribution. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching Processes and Related Topics, Tome 316 (2022), pp. 336-354. http://geodesic.mathdoc.fr/item/TM_2022_316_a21/