The distribution of argmaximum or a winner problem

Abstract

We consider a limit theorem for the distribution of a random variable (r.v.) $A_n=arg{max}_{i:1\dots n}\left\{X_i\right\}$, where $X_i$’s are independent continuous non-negative random r.v.’s. The $X_i,i=1.\dots ,n$, may be interpreted as the gains of $n$ players in a game, and the r.v. $A_n$ itself as the number of a “winner”. The paper contains some limit theorems for the distribution of $A_n$ as $n\to \infty$.