On the Law of Large Numbers and Convergence Rates for the Random Projections

The aim of this paper is to establish the Marcinkiewicz-Zygmund (MZ) type law of large numbers for the randomly weighted sums with weights chosen randomly, uniformly over the unit sphere in ${\mathbb{R}}^n$. We also establish a theorem that describes the rate of convergence in the law of large numbers for these weighted sums.