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Rare events simulation for heavy-tailed distributions. (English) Zbl 0958.65010
The authors consider the problem of simulation of rare events. The family \(\{ A(x)\}\) of events defined on a probability space \(\{ \Omega, F,P\}\) are rare in the sense that \(z(x)=P(A(x))\to 0\), as \(x\to\infty.\) An estimator for \(z(x)\) is a random variable \(Z(x)\) such that \(z(x)=E Z(x).\) The simulation is performed by producing \(N\) independent and identically distributed random variables \(Z_1, \ldots, Z_n\) such that \(EZ_i =z(x)\) and form the estimate of \(z(x)\) \((Z_1 +\cdots + Z_N)/N\), and a confidence interval based upon the empirical variance of the \(Z_i.\) There exist standard algorithms for the simulation for the light-tailed case that is when the distributions have the exponential moments.
The authors consider heavy-tailed distributions, for example, log-normal, Weibull with decreasing failure rate \(1-G(x)=\exp\{-x^\beta\}\) with \(0<\beta<1\), regulary varying, \(1-G(x)= L(x)/x^\alpha\), where \(L\) is a slowly varying function. The authors present two asymptotically efficient algorithms which use a conditional Monte Carlo method and order statistics, and method using importance sampling idea.

65C50 Other computational problems in probability (MSC2010)
60F10 Large deviations
65C05 Monte Carlo methods
65C60 Computational problems in statistics (MSC2010)
62G30 Order statistics; empirical distribution functions
60G50 Sums of independent random variables; random walks
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