The first aim of this paper is to show how to present a random variable with the beta distribution (of the first kind) as a finite or infinite product of independent random variable's (r.v.)'s Xk, where k[set membership, variant][1,2,...,n] or . Such a presentation of an r.v. with the gamma distribution in the form of an infinite product of r.v.'s was used by Lu and Richards [Lu, I-Li, Richards, D., 1993. Random discriminants. Ann. Statist. 21 (4) 1992-2000] to define the square of the Vandermonde determinant with random elements. Next, the convergence of the series [summation operator]1[infinity](1/2k)ln Xk to ln X with probability one has been proved. Finally, the Mieshalkin-Rogozin theorem [Mieshalkin, D., Rogozin, B.A., 1963. Ocenka razstajania miedu funkcjami raspredelenia po blizosti ich charakteristiczeskich funkcji i jeje primenenie k centralnoj predielnoj teoremie. (in Predelnyje teoremy verojatnostej. Taszkent. Akad Nauk Uzbec. SSR)], modified in this paper, has been applied to the beta distribution.
Mellin transform Knar formula Marcinkiewicz-Zygmund and Loeve theorems Convergence with probability one Stochastic equality Infinite convolutions Mieshalkin-Rogozin theorem
