On the Variance of First Passage Times in the Exponential Case

Abstract

For i.i.d. random variables x1,x2,… with positive mean, finite variance and exponential right tail distribution, asymptotic expansions up to vanishing terms will be derived for the variance of first passage times of the form T = T(b) = inf{n ≧ 1: sn > nf(b/n)}, b ≧ 0, where sn = x1 +…+ xn and f is a strictly increasing, positive and three times continuously differentiable function on (0,∞). In particular, it will be shown that the excess over the boundary sT − Tf(b/T) is exponentially distributed and independent of T extending a result which is known when f(x) = x.