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

Published online by Cambridge University Press:  01 July 2016

Søren Asmussen
Søren Asmussen*
Affiliation:
University of Copenhagen
*
∗Postal address: Institute of Mathematical Statistics, 5 Universitetsparken, DK-2100 Copenhagen Ø, Denmark.
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Abstract

Let Sn = X1 + · · · + Xn be a random walk with negative drift μ < 0, let F(x) = P(Xk ≦ x), v(u) =inf{n : Sn > u} and assume that for some γ > 0 is a proper distribution with finite mean Various limit theorems for functionals of X1,· · ·, Xv(u) are derived subject to conditioning upon {v(u)< ∞} with u large, showing similar behaviour as if the Xi 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 WN the waiting time of the Nth customer.

Keywords

RANDOM WALKRISK RESERVE PROCESSGI/G/1 QUEUECONDITIONED LIMIT THEOREMFIRST-PASSAGE TIMEASSOCIATED RANDOM WALKEMPIRICAL DISTRIBUTION FUNCTIONBROWNIAN BRIDGERUIN PROBABILITYWAITING TIME
Type
Research Article
Information
Advances in Applied Probability , Volume 14 , Issue 1 , March 1982 , pp. 143 - 170
Copyright
Copyright © Applied Probability Trust 1982 

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Footnotes

Research supported in part by the tenure of a Visiting Fellowship at the Australian National University and a grant from the Danish Natural Science Research Council.

References

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