On error rates in rare event simulation with heavy tails

For estimating P(S
_n
> x) by simulation where S
_k
= Y
₁
+...+Y
_k
with Y
₁
, ..., 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(Y
₁
, ..., 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.