Optimal Monte Carlo integration with fixed relative precision

We consider Monte Carlo algorithms for computing an integral @[email protected][email protected] which is positive but can be arbitrarily close to 0. It is assumed that we can generate a sequence X"n of uniformly bounded random variables with expectation @q. Estimator @[email protected][email protected]@?(X"1,X"2,...,X"N) is called an (@e,@a)-approximation if it has fixed relative precision @e at a given level of confidence [email protected], that is it satisfies P(|@[email protected][email protected]|@[email protected]@q)>[email protected] for all problem instances. Such an estimator exists only if we allow the sample size N to be random and adaptively chosen. We propose an (@e,@a)-approximation for which the cost, that is the expected number of samples, satisfies [email protected]^-^1/(@[email protected]^2) for @e->0 and @a->0. The main tool in the analysis is a new exponential inequality for randomly stopped sums. We also derive a lower bound on the worst case complexity of the (@e,@a)-approximation. This bound behaves as [email protected]^-^1/(@[email protected]^2). Thus the worst case efficiency of our algorithm, understood as the ratio of the lower bound to the expected sample size EN, approaches 1 if @e->0 and @a->0. An L^2 analogue is to find @[email protected]? such that E(@[email protected][email protected])^[email protected][email protected]^[email protected]^2. We derive an algorithm with the expected cost EN~1/(@[email protected]^2) for @e->0. To this end, we prove an inequality for the mean square error of randomly stopped sums. A corresponding lower bound also behaves as 1/(@[email protected]^2). The worst case efficiency of our algorithm, in the L^2 sense, approaches 1 if @e->0.

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