Rare numbers (Q1187530)

Let \(X_ 1,X_ 2,\dots\) be i.i.d. positive integer valued random variables. Let \(Y_ n\) be the number of positive integers that occur exactly once in the sequence \(X_ 1,X_ 2,\dots,X_ n\). The paper studies the asymptotic behaviour of \(Y_ n\), \(EY_ n\) and \(P(Y_ n\neq 0)\) for \(n\to\infty\). Examples of theorems: conditions sufficient for \(n^{-\theta}Y_ n\to 0\) a.s., with \(0<\theta\leq 1\), bounds and exact values for \(\liminf EY_ n\) and \(\limsup EY_ n\), a condition for \(Y_ n\to\infty\) in probability, bounds and exact values for \(\liminf P(Y_ n=0)\) and \(\limsup P(Y_ n=0)\) and a condition under which \(\sum n^{- 2}EY_ n<\infty\).