Gumbel convergence of the maximum of convoluted half-normally distributed random variables

TL;DR

This paper proves the convergence of the maximum of certain convoluted half-normal distributed variables to the Gumbel distribution, extending classical results to non-symmetric, non-stable cases related to high-frequency jump tests.

Contribution

It establishes Gumbel convergence for maxima of sums and differences of half-normal variables, a novel extension beyond symmetric stable distributions.

Findings

Gumbel convergence shown for sums of half-normal variables

Monte Carlo simulations validate theoretical results

Tail behaviors relate to the normal distribution case

Abstract

In this note, we establish the convergence in distribution of the maxima of i.i.d. random variables to the Gumbel distribution with the associated normalizing sequences for several examples that are related to the normal distribution. Motivated by tests for jumps in high-frequency data, our main interest is in the half-normal distribution and the sum or difference of two independent half-normally distributed random variables. Since the half-normal distribution is neither stable nor symmetric, these examples are non-obvious generalizations. It is shown that the sum and difference of two independent half-normally distributed random variables and other examples yield distributions with tail behaviours that relate to the normal case. It turns out that the Gumbel convergence for all such distributions can be proved following similar steps. We illustrate the results in Monte Carlo simulations.