Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the GI/G/1 queue
- 1 March 1982
- journal article
- research article
- Published by Cambridge University Press (CUP) in Advances in Applied Probability
- Vol. 14 (01) , 143-170
- https://doi.org/10.1017/s0001867800036740
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.Keywords
This publication has 22 references indexed in Scilit:
- Conditioned limit theorems for random walks with negative driftProbability Theory and Related Fields, 1980
- Central Limit Theorems for Empirical MeasuresThe Annals of Probability, 1978
- On the maximum and absorption time of left-continuous random walkJournal of Applied Probability, 1978
- On a Functional Central Limit Theorem for Random Walks Conditioned to Stay PositiveThe Annals of Probability, 1976
- Bounds on the delay distribution inGI/G/1 queuesJournal of Applied Probability, 1974
- Some examples and results in the theory of mixing and random-sum central limit theoremsPeriodica Mathematica Hungarica, 1973
- Functional central limit theorems for processes with positive drift and their inversesProbability Theory and Related Fields, 1972
- Weak convergence of first passage time processesJournal of Applied Probability, 1971
- Weak convergence of stochastic processes defined on semi-infinite time intervalsProceedings of the American Mathematical Society, 1963
- Large-sample theory of sequential estimationMathematical Proceedings of the Cambridge Philosophical Society, 1952