The Use of the Scalar Monte Carlo Estimators for the Optimization of the Corresponding Vector Weight Algorithms

Abstract

In this paper problems related to the finiteness of the variance of weighted Monte Carlo estimators for the estimation of the solution of the system of second-kind linear integral equations are studied. We present some modifications of the finiteness criterion for the variance of a weighted scalar estimator that are based on the construction of an appropriate system of linear integral equations with majorant kernels. A majorant criterion of the finiteness of a vector weighted estimator is given. This criterion uses a comparison of the vector weight estimator with a scalar estimator obtained by the method of randomization. It is shown that for the corresponding randomized algorithm the variance of the scalar estimator and the average simulation time for a single trajectory are bounded if the original solution of the system is bounded. Finally, some remarks about vector weight collision estimator with the use of branching the trajectory into random independent number of branches are presented.