Probability distribution of the maximum of a smooth temporal signal

TL;DR

This paper derives an approximate distribution for the maximum of a smooth stationary temporal signal and applies it to compute the persistence exponent for Gaussian processes, linking it explicitly to the correlation function.

Contribution

It introduces an approximate method to calculate the maximum distribution of smooth signals and explicitly relates it to the correlation function for Gaussian processes.

Findings

Derived an explicit formula for the maximum distribution of Gaussian signals.

Computed the persistence exponent in terms of the correlation function.

Provided a practical approach for analyzing maximum statistics of smooth signals.

Abstract

We present an approximate calculation for the distribution of the maximum of a smooth stationary temporal signal X(t). As an application, we compute the persistence exponent associated to the probability that the process remains below a non-zero level M. When X(t) is a Gaussian process, our results are expressed explicitly in terms of the two-time correlation function, f(t)=<X(0)X(t)>.