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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 ≥ 0} with S0 = 0 and a negative drift and let {N(t), t ≥ 0} be a general counting process, independent of {Sn, n ≥ 0}. This article investigates the tail probability, denoted by ψ(x; t), of the maximum of SN(v) over a finite horizon 0 ≤ v ≤ t. When F is strongly subexponential, some asymptotics for ψ(x; t) are derived as x → ∞. The merit is that all of the obtained asymptotics are uniform for t in a finite or infinite time interval.