Vector estimators of the Monte Carlo method with a finite variance

Abstract

In this paper we study problems related to the finiteness of the variance of weighted Monte Carlo estimators for solving a system of second-kind integral equations. Modifications of the finiteness criterion for the variance of a weighted scalar estimator 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 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 mean simulation time for a single trajectory are bounded if the original solution is bounded. The ability to use branching to construct a vector estimator of the solution to a system of integral equations is also studied.