On error rates in rare event simulation with heavy tails

For estimating P(S<sub>n</sub> > x) by simulation where S<sub>k</sub> = Y<sub>1</sub>+...+Y<sub>k</sub> with Y<sub>1</sub>, ..., Y<sub>n</sub> are non-negative and heavy-tailed with distribution F, (Asmussen and Kroese 2006) suggested the estimator nF(M<sub>n-1</sub> V(x - S<sub>n-1</sub>)) where M<sub>k</sub> = max(Y<sub>1</sub>, ..., Y<sub>k</sub>). The estimator has shown to perform excellently in practice and has also nice theoretical properties. In particular, (Hartinger and Kortschak 2009) showed that the relative error goes to 0 as × → ∞. We identify here the exact rate of decay and propose some related estimators with even faster rates.

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