Poisson points, resetting, universality and the role of the last item

TL;DR

This paper investigates the universality of probability distributions of observables in stochastic processes with resets, showing that for Poisson resets, universality depends on combinatorial and Poisson properties, but not for general distributions.

Contribution

It demonstrates that universality in reset processes is guaranteed only under Poissonian resetting, highlighting the role of interval distribution in such stochastic systems.

Findings

Universality holds for Poissonian resetting due to combinatorial and Poisson properties.

For non-Poissonian resets, universality generally breaks down.

The last item in the process influences the universality behavior.

Abstract

For a stochastic process reset at random times, we discuss to what extent the probabilities of some orderings of observables associated with the intervals of time between resetting events are universal, i.e., independent of the choice of the observables, and in particular, to what extent universality depends on the choice of the distribution of these intervals. For Poissonian resetting, universality relies only on a combinatorial argument and on the statistical properties of Poisson points. For a generic distribution of time intervals between resets, universality no longer holds in general.