Uniform estimates for the tail probability of maxima over finite horizons with subexponential tails

Let F be the common distribution function of the increments of a random walk {Sn, n [greater-than-or-equal] 0} with S0 = 0 and a negative drift and let {N(t), t [greater-than-or-equal] 0} be a general counting process, independent of {Sn, n [greater-than-or-equal] 0}. This article investigates the tail probability, denoted by [psi](x; t), of the maximum of SN(v) over a finite horizon 0 [less-than-or-equal] v [less-than-or-equal] t. When F is strongly subexponential, some asymptotics for [psi](x; t) are derived as x [rightward arrow] [infty infinity]. The merit is that all of the obtained asymptotics are uniform for t in a finite or infinite time interval.