TRAPEZOIDAL STRATIFIED MONTE-CARLO INTEGRATION

Weighted integrals of random processes are approximated by the trapezoidal rule based on a stratified and symmetrized random sample of size n. The weight functions are assumed to be twice continuously differentiable. The rate of convergence to zero of the mean-square integral approximation error as the sample size increases indefinitely is considered. For random processes which are twice mean-square continuously differentiable, it is shown that the rate is n-5, just as without a random component [Math. Comp., 21 (1967), pp. 388-397]. For random processes which are a bit more than once, but not twice, mean-square continuously differentiable, the rate is shown to be n-4. In both cases, the asymptotic constant is also determined.