On the Variance of First Passage Times in the Exponential Case

Abstract

For i.i.d. random variables x_1,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: s_n > nf(b/n)}, b ≧ 0, where s_n = x₁ +…+ x_n 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 s_T − Tf(b/T) is exponentially distributed and independent of T extending a result which is known when f(x) = x.