Scaling Limit, Noise, Stability

TL;DR

This paper explores the scaling limits of functions of random variables, distinguishing between classical and nonclassical noises, and introduces a new framework for analyzing their stability and spectral properties.

Contribution

It presents a novel framework for understanding the scaling limits of nonlinear functions of random variables and characterizes the stability of classical versus nonclassical noises.

Findings

Classical noises are stable under scaling.

Nonclassical noises are not stable.

A new framework for analyzing noise stability and spectral measures is proposed.

Abstract

Linear functions of many independent random variables lead to classical noises (white, Poisson, and their combinations) in the scaling limit. Some singular stochastic flows and some models of oriented percolation involve very nonlinear functions and lead to nonclassical noises. Two examples are examined, Warren's `noise made by a Poisson snake' and the author's `Brownian web as a black noise'. Classical noises are stable, nonclassical are not. A new framework for the scaling limit is proposed. Old and new results are presented about noises, stability, and spectral measures.