E¢ cient simulation for the maximum of innite horizon Gaussian processes

We consider the problem of estimating the probability that the maximum of a Gaussian process with negative mean function and indexed by positive integers reaches a high level, say b. In great generality such probability converges to zero exponentially fast in a power of b. Under mild assumptions on the marginal distributions of the process and no assumption on the correlation structure, we develop an importance sampling procedure, called Target Bridge Sampler (TBS) which is very easy to implement and takes a polynomial (in b) number of function evaluations achieve a small relative error. The procedure also yields samples that are arbitrarily close in total variation to the process given that it hits b in …nite time. We also apply our method to the problem of estimating the tail of the maximum of a superposition of a large number, n, of independent Gaussian sources. In this situation TBS achieves a …xed relative error with a bounded number of function evaluations as n % 1. A remarkable feature of TBS is that it is not based on exponential changes of measure. Our numerical experiments validate our theoretical …ndings.