Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the GI/G/1 queue

Abstract

Let S n = X 1 + · · · + X n be a random walk with negative drift μ < 0, let F ( x ) = P ( X k ≦ x ), v ( u ) =inf{ n : S n > u } and assume that for some γ > 0 is a proper distribution with finite mean Various limit theorems for functionals of X 1 ,· · ·, X v ( u ) are derived subject to conditioning upon { v ( u )< ∞} with u large, showing similar behaviour as if the X i were i.i.d. with distribution For example, the deviation of the empirical distribution function from properly normalised, is shown to have a limit in D , and an approximation for by means of Brownian bridge is derived. Similar results hold for risk reserve processes in the time up to ruin and the GI/G/ 1 queue considered either within a busy cycle or in the steady state. The methods produce an alternate approach to known asymptotic formulae for ruin probabilities as well as related waiting-time approximations for the GI/G/ 1 queue. For example uniformly in N , with W N the waiting time of the Nth customer.