The distribution of argmaximum or a winner problem

Abstract

We consider a limit theorem for the distribution of a random variable (r.v.) An=argmaxi:1…n{Xi}, where Xi’s are independent continuous non-negative random r.v.’s. The Xi,i=1.…,n, may be interpreted as the gains of n players in a game, and the r.v. An itself as the number of a “winner”. The paper contains some limit theorems for the distribution of An as n→∞.