The Distribution of Argmaximum or a Winner Problem

We consider a limit theorem for the distribution of a r.v. $Y_n:=argmax\{X_i,i=1,\mathrm{.}\mathrm{.}\mathrm{.},n\},$ where $X_i^{\mathrm{\prime }}$s are independent continuous non-negative random variables. The r.v.'s $\{X_i,i=1,\mathrm{.}\mathrm{.}\mathrm{.},n\}$, 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 $\{1,\mathrm{.}\mathrm{.}\mathrm{.},n\},$ while when the $X^{\mathrm{\prime }}$s are non-identically distributed, the problem requires some calculations.