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The Distribution of Argmaximum or a Winner Problem

Youri Davydov, Vladimir Rotar

Abstract

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.

The Distribution of Argmaximum or a Winner Problem

Abstract

We consider a limit theorem for the distribution of a r.v. where s are independent continuous non-negative random variables. The r.v.'s , may be interpreted as the gains of players in a game, and the r.v. itself as the number of a ``winner". In the case of i.i.d.r.v.'s, the distribution of is, clearly, uniform on while when the s are non-identically distributed, the problem requires some calculations.
Paper Structure (9 sections, 3 theorems, 41 equations)

This paper contains 9 sections, 3 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. A local limit theorem
  4. An integral limit theorem
  5. Proofs
  6. A basic formula
  7. Proof of Theorem \ref{['th1']}
  8. On regularity of coefficients $c_i$ (condition (\ref{['alpha']}))
  9. Conclusive Remarks

Key Result

Theorem 1

Set Then, under the above conditions, uniformly in $i$

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3