On a problem in probability theory

Abstract

For continuous random variables, we study a problem similar to that considered earlier by one of the authors for discrete random variables. Let numbers

$$N > 0, E > 0, 0 \leqslant \lambda _1 \leqslant \lambda _2 \leqslant \cdots \leqslant \lambda _s $$

be given. Consider a random vector x = (x ₁, …, x _s), uniformly distributed on the set

$$x_j \geqslant 0, j = 1, \ldots ,s; \sum\limits_{j = 1}^s {x_j = N} , \sum\limits_{j = 1}^s {\lambda _j x_j \leqslant E} .$$

We study the weak limit of x as s → ∞.