The average shape of a fluctuation: universality in excursions of stochastic processes

TL;DR

This paper investigates the universal scaling behavior of the average shape of fluctuations in stochastic processes, revealing that the shape function is largely independent of specific details and encodes correlation information, with applications to magnetic noise.

Contribution

It demonstrates a universal scaling law for the average fluctuation shape across various stochastic processes, highlighting the role of correlations and independence from increment distributions.

Findings

Scaling law <x(t)-x(0)>_T = T^α f(t/T) holds for many processes.

The shape function f(s) is largely independent of increment distribution.

Results are relevant for understanding Barkhausen noise in magnetic systems.

Abstract

We study the average shape of a fluctuation of a time series x(t), that is the average value <x(t)-x(0)>_T before x(t) first returns, at time T, to its initial value x(0). For large classes of stochastic processes we find that a scaling law of the form <x(t) - x(0)>_T = T^\alpha f(t/T) is obeyed. The scaling function f(s) is to a large extent independent of the details of the single increment distribution, while it encodes relevant statistical information on the presence and nature of temporal correlations in the process. We discuss the relevance of these results for Barkhausen noise in magnetic systems.