On samples of independent random vectors in spaces of infinitely increasing dimension

Abstract

We study the asymptotic behavior of a set of random vectors ξ_i, i = 1,..., m, whose coordinates are independent and identically distributed in a space of infinitely increasing dimension. We investigate the asymptotics of the distribution of the random vectors, the consistency of the sets M _m ⁽ⁿ⁾ = ξ₁,..., ξ_m and X _n ^λ = x ∈ X _n: ρ(x) ≤ λn, and the mutual location of pairs of vectors.