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

Abstract

LetSn=X1+ · · · +Xnbe a random walk with negative drift μ < 0, letF(x) =P(Xk≦x),v(u) =inf{n:Sn>u} and assume that for some γ > 0is a proper distribution with finite meanVarious limit theorems for functionals ofX1,· · ·,Xv(u)are derived subject to conditioning upon {v(u)< ∞} withularge, showing similar behaviour as if theXiwere i.i.d. with distributionFor example, the deviation of the empirical distribution function fromproperly normalised, is shown to have a limit inD, and an approximation forby means of Brownian bridge is derived. Similar results hold for risk reserve processes in the time up to ruin and theGI/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 theGI/G/1 queue. For exampleuniformly inN, withWNthe waiting time of the Nth customer.