The maximum of the Gaussian $1/f^{\alpha}$-noise in the case $\alpha<1$

TL;DR

This paper proves that the normalized maximum of Gaussian $1/f^{eta}$-noise with $eta<1$ converges to the Gumbel distribution, advancing understanding of extreme values in such stochastic processes.

Contribution

It establishes the convergence in distribution of the maximum of Gaussian $1/f^{eta}$-noise for $eta<1$ to the Gumbel law, a novel result in this context.

Findings

Normalized maximum converges to Gumbel distribution

Provides rigorous proof for $eta<1$ case

Enhances understanding of extreme value behavior in $1/f^{eta}$-noise

Abstract

We prove that the appropriately normalized maximum of the Gaussian $1/f^{\alpha}$-noise with $\alpha<1$ converges in distribution to the Gumbel double-exponential law.