  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 to generalise results of
Asmussen (1998) onto the case of arbitrary stopping times and 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.
