A necessary and sufficient condition for the existence of the limiting probability of a tie for first place
Suppose that the scores of \(n\) players are unbounded, independent, non- negative integer-valued random variables equal in distribution to \(X\). \(P(T_ n)\) denotes the probability of a tie \(T_ n\) for the highest score. The authors show that as \(n \to \infty\), \(\lim P(T_ n)\) exists if and only if \(P(X = j)/P(X > j) \to 0\) as \(j \to \infty\). In this case \(P(T_ n) \to 0\).
scientific article; zbMATH DE number 768766