Sampling from Gauss Rules

Approximating multidimensional integrals via product quadrature rules becomes increasingly intractable as the dimension increases. Hammersley [Ann. New York Acad. Sci., 86 (1960), pp. 844–874] suggested sampling from product quadrature rules and Tsuda [Numer. Math., 20 (1973), pp. 377–391] considered this method using Fejer rules. In this paper we consider this approach using Gauss rules. Results are obtained concerning the variance of this form of sampling relative to sampling from the continuous distributions represented by the weight functions. It is shown that this approach can lead to variance reduction and its use is discussed in several examples.