On the limit distribution of discrete random variables
It is well-known that the possible limit distributions of triangular arrays \((X_{nk})\) (n\(\geq 1\), \(1\leq k\leq k_ n)\) of infinitesimal random variables with independent rows are exactly the infinitely divisible distributions. In the present paper, it is shown that requiring the random variables \(X_{nk}\) to assume only two values with prescribed probabilities \(p_{nk}\) and \(1-p_{nk}\) does not narrow the class of possible limit distributions, provided that the numbers \(p_{nk}\) satisfy \(0<p_{nk}<1\) and \[ \lim_{n\to \infty}\sum_{1\leq k\leq k_ n, p_{nk}\leq \epsilon}p_{nk}=\infty \quad for\quad every\quad \epsilon >0. \] A similar result is obtained for limit distributions of sequences of independent random variables. As an application to probabilistic number theory, the class of possible limit distributions in the central limit theorem for additive arithmetic functions is determined.
scientific article; zbMATH DE number 3956128