Multinomial ratio [Paul Erdős solves a problem] (Q1268662)

This article combines two papers in one. The first reviews a mathematical problem solved by the author and \textit{P. Erdős} [Mem. Am. Math. Soc. 6, 1-9 (1951; Zbl 0042.37601)]. It provides a probabilistic proof, and challenges the reader to find a purely analytic one. The second paper comprises reminiscences of the author of working with Paul Erdős (and others), personal memories and recollections of Erdős' viewpoints. The principal result is not new, but the presentation is an improvement over the original. The starting point are Theorems 1 and \(1^*\), which state that for a set of integers \(\{a_j\} \) the following are equivalent: (1) There are positive and negative \(a_j\), and the GCD of the differences \(a_j-a_{j'}\) is 1. (2) For any integer \(c\), for all \(n\geq M(c)\) there are integers \(x_j\geq 0\) such that\(\sum x_j=n\) and \(\sum x_j a_j=c\). (3) If \(X_k\) are i.i.d. random variables taking values in \(\{a_j\}\) (and \(P[X=a_j]>0)\), then \(P[\sum^n_1X_k=c]>0\) for all \(n\geq n(c)\). Under the additional condition that \(E(X^+)=E(X^-)<\infty\) the paper proves (Theorem 3) that as \(n\to\infty\), \(P[\sum^n_1X_k=c]\sim P[\sum^n_1X_k=c']\) for distinct integers \(c\), \(c'\). The probabilistic proof is repeated by a Fourier analysis proof for symmetric \(X\), and a purely analytic proof is posed as an open problem.