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The asymptotic probability of a tie for first place. (English) Zbl 0787.60029

The authors consider the following problem. Let \(X_ 1,X_ 2,\dots\) be i.i.d. nonnegative, integer-valued r.v.’s. Let \(M_ n=\max (X_ 1,\dots,X_ n)\) and \(N_ n=\# \{j \leq n:X_ j =M_ n\}\). They are especially interested in the event \(S_ n\) defined by \(S_ n=\{N_ n=1\}\). If \(X_ j\) is interpreted as the score of player \(j\) in a (golf) contest, then \(P(S_ n)\) is the probability of having a single winner among \(n\) players. We list some of the results:
1. If the \(X_ j\) have a geometric distribution, then \(P(S_ n)\) does not converge, but \(P(N_ n=j)\) converges in the logarithmic mean to a logarithmic series distribution.
2. Let \(p_ j=P(X_ 1=j)\). If \(p_{j+1}/p_ j \to 1\) \((j \to \infty)\), then \(P(S_ n)\to 1\) \((n \to \infty)\).
3. If \(p_{j+1}/p_ j \to 0\), then \(P(S_ n)\) does not converge.
4. \(P(S_ n)\to 0\) iff \(X_ 1\) is bounded.
A problem equivalent to the case where \(X_ 1\) has a geometric distribution, appeared recently in [L. Råde, Amer. Math. Mon. 99 (1991), Problem E 3436)].

MSC:

60F05 Central limit and other weak theorems
60G70 Extreme value theory; extremal stochastic processes
60F20 Zero-one laws
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