A zero-one law for linear transformations of Levy noise

TL;DR

This paper establishes a zero-one law for invariant events of Levy noise under linear transformations, showing that non-compact groups of measure-preserving linear transformations lead to trivial invariant events.

Contribution

It proves a zero-one law for Levy noise under linear groups of transformations, characterizing when invariant events are trivial based on the group's compactness.

Findings

Invariant events are trivial for non-compact groups of linear transformations.

Levy noise's invariance under measure-preserving transformations follows a zero-one law.

The law does not hold for compact groups of transformations.

Abstract

A L\'evy noise on $\mathbb{R}^d$ assigns a random real "mass" $\Pi(B)$ to each Borel subset $B$ of $\mathbb{R}^d$ with finite Lebesgue measure. The distribution of $\Pi(B)$ only depends on the Lebesgue measure of $B$, and if $B_1, ..., B_n$ is a finite collection of pairwise disjoint sets, then the random variables $\Pi(B_1), ..., \Pi(B_n)$ are independent with $\Pi(B_1 \cup >... \cup B_n) = \Pi(B_1) + ... + \Pi(B_n)$ almost surely. In particular, the distribution of $\Pi \circ g$ is the same as that of $\Pi$ when $g$ is a bijective transformation of $\mathbb{R}^d$ that preserves Lebesgue measure. It follows from the Hewitt--Savage zero--one law that any event which is almost surely invariant under the mappings $\Pi \mapsto \Pi \circ g$ for every Lebesgue measure preserving bijection $g$ of $\mathbb{R}^d$ must have probability 0 or 1. We investigate whether certain smaller groups of Lebesgue measure preserving bijections also possess this property. We show that if $d \ge 2$, the L\'evy noise is not purely deterministic, and the group consists of linear transformations and is closed, then the invariant events all have probability 0 or 1 if and only if the group is not compact.