Asmussen-Kroese [1] Monte Carlo estimators of P(Sn > u) and P(SN > u) are known to work well in rare event settings when Sn is the sum of n i.i.d. heavy-tailed random variables, and N is a non-negative integer-valued random variable independent of the Xi. In this paper we show how to improve the Asmussen-Kroese estimators of both probabilities when the Xi are non-negative. We also apply our ideas to estimate the quantity E[(SN u)+].
