The Distribution of Argmaximum or a Winner Problem

Abstract: We consider a limit theorem for the distribution of a r.v.

Y_{n} : = a r g m a x 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.

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