A Study of One Null Array of Random Variables

Denote \(U_1,U_2,\dots,U_n\) a sequence of independent uniformly distributed random variables. We study limit behavior of the sum $$\sum_{i=1}^m U_i^m$$ when \(m\to\infty\). It is interesting in itself and allows, although not too easily, an explicit form of the limiting infinitely divisible distribution function. It also has interesting application in the theory of large number of rare events (LNRE, see Khmaladze, Statistical Analysis of Large Number of Rare Events, 1989), which will be discussed in adjoint publication.