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Recursive estimation of distributional fix-points

Part of: Sequential methods Stochastic analysis Applications Special processes

Published online by Cambridge University Press:  14 July 2016

Paul Embrechts  and
Harro Walk
Paul Embrechts*
Affiliation:
ETH Zürich
Harro Walk*
Affiliation:
Universität Stuttgart
*
∗Postal address: Department of Mathematics, ETH, CH-8092 Zürich, Switzerland. Email address: embrecht@math.ethz.ch
∗∗Postal address: Mathematisches Institut A, Universität Stuttgart, 70550 Stuttgart, Germany. Email address: walk@mathematik.uni-stuttgart.de
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Abstract

In various stochastic models the random equation of implicit renewal theory appears where the real random variable S and the stochastic process Ψ with index space and state space R are independent. By use of stochastic approximation the distribution function of S is recursively estimated on the basis of independent or ergodic copies of Ψ. Under integrability assumptions almost sure L1-convergence is proved. The choice of gains in the recursion is discussed. Applications are given to insurance mathematics (perpetuities) and queueing theory (stationary waiting and queueing times).

Keywords

Random equationdistributional fix-pointrecursive estimationstochastic approximationinsurancestochastic discountingperpetuityqueueing theoryG/G/1stationary waiting timestationary queueing time

MSC classification

Primary: 62L20: Stochastic approximation
Secondary: 60H25: Random operators and equations 62P05: Applications to actuarial sciences and financial mathematics 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 1 , March 2000 , pp. 73 - 87
Copyright
Copyright © 2000 by The Applied Probability Trust 

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