A zero-one law for linear transformations of Levy noise

August, 2009
Report Number: 
Steven N. Evans

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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