Point Process Approach to the Winner Problem

TL;DR

The paper proves a limit theorem for a triangular array of independent, non-identically distributed r.v.'s by establishing convergence of empirical point processes to a Poisson point process with intensity $\mu\times\gamma$. This result is then used to derive the asymptotic behavior of the Argmaximum, the maximum, and ladder/step processes, with explicit distributions for $A(\zeta)$ and $M(\zeta)$. The authors connect their general result to the earlier winner problem studied in d-r and show how d-r's integral limit theorem follows from their framework under standard normalizations, thereby unifying these approaches. Overall, the work extends Resnick's theorem to triangular arrays and provides a coherent method to analyze non-identically distributed tails in argmax and maxima problems with potential applications to step processes.

Abstract

We consider a limit theorem for a triangular array of point processes generated by non-identically distributed random variables, and apply the result for the analysis of the limiting behavior of the Argmaximum of independent random variables, as well as for some step processes.

Theorems & Definitions (6)

• Remark 1
• Theorem 2
• Corollary 3
• Proposition 4
• Theorem 5
• Lemma 6