On error rates in rare event simulation with heavy tails

For estimating P(S_n > x) by simulation where S_k = Y₁+...+Y_k with Y₁, ..., Y_n are non-negative and heavy-tailed with distribution F, (Asmussen and Kroese 2006) suggested the estimator nF(M_n-1 V(x - S_n-1)) where M_k = max(Y₁, ..., Y_k). 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.