A Zero–one Law for Linear Transformations of Lévy Noise
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چکیده
A Lévy noise on Rd assigns a random real “mass” Π(B) to each Borel subset B of Rd with finite Lebesgue measure. The distribution of Π(B) only depends on the Lebesgue measure of B, and if B1, . . . , Bn is a finite collection of pairwise disjoint sets, then the random variables Π(B1), . . . , Π(Bn) are independent with Π(B1 ∪ · · · ∪Bn) = Π(B1) + · · ·+ Π(Bn) almost surely. In particular, the distribution of Π ◦ g is the same as that of Π when g is a bijective transformation of Rd that preserves Lebesgue measure. It follows from the Hewitt–Savage zero–one law that any event which is almost surely invariant under the mappings Π 7→ Π ◦ g for every Lebesgue measure preserving bijection g of Rd 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 ≥ 2, the Lévy 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
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تاریخ انتشار 2009