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Foss and Zachary (2003) and Foss, Palmowski and Zachary (2005) studied theprobability of achieving a receding boundary on a time interval of randomlength by a random walk with a heavy-tailed jump distribution. They haveproposed and developed a new approach that allows to generalise results ofAsmussen (1998) onto the case of arbitrary stopping times and a wide class ofnonlinear boundaries, and to obtain uniform results over all stopping times.  In this paper, we consider a class of branching random walks with fadingbranching and obtain results on the tail asymptotics for the maximum of abranching random walk on a time interval of random (possibly unlimited) length,as well as uniform results within a class of bounded random time intervals.

The probability of reaching a receding boundary by branching random walk with fading branching and heavy-tailed jump distribution