The Distribution of Argmaximum or a Winner Problem
Abstract
We consider a limit theorem for the distribution of a r.v. Yn:=argmax \Xi, i= 1,..., n\, where Xi's are independent continuous non-negative random variables. The r.v.'s \Xi, i=1,..., n\, may be interpreted as the gains of n players in a game, and the r.v. Yn itself as the number of a ``winner". In the case of i.i.d.r.v.'s, the distribution of Yn is, clearly, uniform on \1,..., n\, while when the X's are non-identically distributed, the problem requires some calculations.
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