On average dimensions of particle transport estimators

We considered average dimensions of the weighted Monte Carlo algorithm for a particle transport problem with multi-scattering setting and estimated the probability of particles penetration through a layer. The average dimension {\hat{d}} of the algorithm turned out to be small so that quasi-Monte Carlo estimates of the probability converge much faster than the Monte Carlo estimates. We justified the reasons to expect that the convergence of quasi-Monte Carlo estimates continue to be faster as the thickness of the layer increases. Here we calculated {\hat{d}} without the use of the ANOVA expansion.