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Rare-event simulation of heavy-tailed random walks by sequential importance sampling and resampling. (English) Zbl 1269.65011

Let \({ Y_n}=(y_1,\dots,y_n)\) be the random data and \(\alpha=P({Y_n}\in \Gamma_n)\) be a small probability to be estimated. A Monte Carlo (MC) method using \(m\) simulation runs to estimate \(\alpha\) by \(\hat\alpha_n\) is called \(C_n\) efficient if for any \(\varepsilon>0\) var\((\hat\alpha_n)\leq \varepsilon\alpha_n^2\) for \(m=m_n=O(C_n)\). The authors consider a general MC scheme based on an approximated Doobs \(h\)-transform which carries out importance sampling sequentially within each simulated trajectory and then resamples across all trajectories (SISR). The scheme is applied to the estimation of \({ P}(\sum_{i=1}^n X_i>b)\) and \({ P}(\max_j \sum_{i=1}^j X_i>b)\) for i.i.d. \(X_i\) with heavy tailed distributions. It is shown that SISR provides linearly efficient \(C_n=n\) estimates. Weibull and log-normal distributions are considered as examples.

MSC:

65C50 Other computational problems in probability (MSC2010)
65C05 Monte Carlo methods
60G50 Sums of independent random variables; random walks
60E05 Probability distributions: general theory
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References:

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