The Distribution of Argmaximum or a Winner Problem

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