Physical random numbers are not as widely used in Monte Carlo integration as pseudo-random numbers are. They are inconvenient for many reasons. If we want to generate them on the fly, then they may be slow. When we want reproducible results from them, we need a lot of storage. This paper shows that we may construct N=n(n−1)/2 pairwise independent random vectors from n independent ones, by summing them modulo 1 in pairs. As a consequence, the storage and speed problems of physical random numbers can be greatly mitigated. The new vectors lead to Monte Carlo averages with the same mean and variance as if we had used N independent vectors. The asymptotic distribution of the sample mean has a surprising feature: it is always symmetric, but never Gaussian. This follows by writing the sample mean as a degenerate U-statistic whose kernel is a left-circulant matrix. Because of the symmetry, a small number B of replicates can be used to get confidence intervals based on the central limit theorem.
"Recycling physical random numbers." Electron. J. Statist. 3 1531 - 1541, 2009. https://doi.org/10.1214/09-EJS541