When is a probability measure determined by infinitely many projections?
The well-known Cramér-Wold theorem states that a Borel probability measure \(\mu\) on \(\mathbb{R}^d\) is uniquely determined by the totality of its one-dimensional projections. Various conditions are examined under which a probability measure is determined by an infinite family \(\mathcal L\) of \((d-1)\)-dimensional subspaces of \(\mathbb{R}^d\). The question can be seen in the context of the work of \textit{A. Rényi} [Acta Math. Acad. Sci. Hung. 3, 131-142 (1952; Zbl 0048.10804)]. The authors formulate conditions on \(\mathcal L\) and the moment generating function of \(\mu\). In particular, a finite moment generating function of \(\mu\) in a neighborhood of the origin implies the theorem of Rényi. The paper concludes with a brief discussion for discrete probability measures.
scientific article; zbMATH DE number 1018408