# Gumbel and Fr\'echet convergence of the maxima of independent random walks

## TL;DR

This paper develops asymptotic theory for the maxima of independent random walks, showing their convergence to Gumbel or Fréchet distributions, based on large deviation principles.

## Contribution

It establishes the convergence of the maximum of iid random walks to extreme value distributions, extending existing results with precise large deviation techniques.

## Key findings

- Maximum of iid random walks converges to Gumbel or Fréchet distributions.
- Convergence results depend on large deviation principles for sums of independent variables.
- Provides a rigorous foundation for extreme value analysis of random walks.

## Abstract

We consider point process convergence for sequences of iid random walks. The objective is to derive asymptotic theory for the largest extremes of these random walks. We show convergence of the maximum random walk to the Gumbel or the Fr\'echet distributions. The proofs heavily depend on precise large deviation results for sums of independent random variables with a finite moment generating function or with a subexponential distribution.