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Asymptotic behaviour of Wiener-Hopf factors of a random walk

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  • Veraverbeke, N.

Abstract

For a random walk governed by a general distribution function F on (-[infinity], +[infinity]), we establish the exponential and subexponential asymptotic behaviour of the corresponding right Wiener-Hopf factor F+. The results apply to classes of distribution functions in recent publications: the subexponential class and a related (exponential) class [gamma]. Given the behaviour of F+, the Wiener-Hopf identity is used, to obtain the behaviour of F. To reverse the argument, we derive a new identity, similar in form to the first one. The results for F+ are then fruitfully applied to give a full description of the tail behaviour of the maximum of the randon walk. Also they provide new proofs for recent theorems on the tail of the waiting-time distribution in the GI/G/1 queue.

Suggested Citation

  • Veraverbeke, N., 1977. "Asymptotic behaviour of Wiener-Hopf factors of a random walk," Stochastic Processes and their Applications, Elsevier, vol. 5(1), pages 27-37, February.
    • Handle: RePEc:eee:spapps:v:5:y:1977:i:1:p:27-37
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