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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.

##### MSC:
 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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