On accompanying measures and asymptotic expansions in limit theorems for maximum of random variables

TL;DR

This paper introduces a sequence of accompanying laws to improve the convergence rate in the limit theorem for maxima of independent random variables, providing exponential power convergence instead of logarithmic, with applications to Weibull and log-Weibull distributions.

Contribution

It proposes a new sequence of accompanying laws that enhance the convergence rate in Gnedenko's limit theorem for maxima, extending the Gumbel domain of attraction.

Findings

Accompanying laws yield exponential power convergence rate.

Gumbel distribution provides only logarithmic convergence.

Applications to Weibull and log-Weibull distributions demonstrate the approach.

Abstract

A sequence of accompanying laws is suggested in the limit theorem of B. V. Gnedenko for maximums of independent random variables belonging to maximum domain of attraction of the Gumbel distribution. It is shown that this sequence gives an exponential power rate of convergence whereas the Gumbel distribution gives only a logarithmic rate. As examples, classes of Weibull and log-Weibull type distributions are considered in details. A scale for the Gumbel maximum domain of attraction is suggested as a continuation of the considered two classes.