# A zero-one law for linear transformations of Levy noise

August, 2009
Report Number:
780
Authors:
Steven N. Evans
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, \ldots, B_n\$ is a finite collection of pairwise disjoint sets, then the random variables \$\Pi(B_1), \ldots, \Pi(B_n)\$ are independent with \$\Pi(B_1 \cup \cdots \cup B_n) = \Pi(B_1) + \cdots + \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.

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