Uniformly Efficient Importance Sampling for the Tail Distribution of Sums of Random Variables

Successful efficient rare-event simulation typically involves using importance sampling tailored to a specific rare event. However, in applications one may be interested in simultaneous estimation of many probabilities or even an entire distribution. In this paper, we address this issue in a simple but fundamental setting. Specifically, we consider the problem of efficient estimation of the probabilities P(S_n ≥ na) for large n, for all a lying in an interval 𝒜, where S_n denotes the sum of n independent, identically distributed light-tailed random variables. Importance sampling based on exponential twisting is known to produce asymptotically efficient estimates when 𝒜 reduces to a single point. We show, however, that this procedure fails to be asymptotically efficient throughout 𝒜 when 𝒜 contains more than one point. We analyze the best performance that can be achieved using a discrete mixture of exponentially twisted distributions, and then present a method using a continuous mixture. We show that a continuous mixture of exponentially twisted probabilities and a discrete mixture with a sufficiently large number of components produce asymptotically efficient estimates for all a ∈ 𝒜 simultaneously.