Stochastic Integration Rules for Infinite Regions

Stochastic integration rules are derived for infinite integration intervals, generalizing rules developed by Siegel and O’Brien [SIAM J. Sci. Statist. Comput., 6 (1985), pp. 169–181] for finite intervals. Then random orthogonal transformations of rules for integrals over the surface of the unit m-sphere are used to produce stochastic rules for these integrals. The two types of rules are combined to produce stochastic rules for multidimensional integrals over infinite regions with Normal or Student-t weights. Example results are presented to illustrate the effectiveness of the new rules.