Pointwise Translation of the Radon Transform and the General Central Limit Problem

• Published in: The Annals of probability 1983-05, Vol.11 (2), p.277-301
• Main Authors: Hahn, Marjorie G., Hahan, Peter, Klass, Michael J.
• Format: Article
• Language: English
• Subjects: 44A15; 60E10; 60F05; 92A05; Banach space; computerized tomography; Density; Eigenfunctions; Fourier transformations; general multivariate central limit theorem; Hyperplanes; infinitely divisible laws; Linear transformations; Mathematical vectors; Mathematics; pointwise translation problem; radiology; Radon; Radon transform; signed measures; Spectral index; stable laws
• ISSN: 0091-1798; 2168-894X
• DOI: 10.1214/aop/1176993597

We identify a representation problem involving the Radon transforms of signed measures on Rdof finite total variation. Specifically, if μ is a pointwise translate of v (i.e., if for all θ ∈ Sd - 1the projection μθis a translate of vθ), must μ be a vector translate of v? We obtain results in several important special cases. Relating this to limit theorems, let Xn1, ⋯, Xnkn be a u.a.n. triangular array on Rdand put Sn= Xn1+ ⋯ + Xnkn . There exist vectors vn∈ Rdsuch that L(Sn- vn) → γ iff (I) a tail probability condition, (II) a truncated variance condition, and (III) a centering condition hold. We find that condition (III) is superfluous in that (I) and (II) always imply (III) iff the limit law γ has the property that the only infinitely divisible laws which are pointwise translates of γ are vector translates. Not all infinitely divisible laws have this property. We characterize those which do. A physical interpretation of the pointwise translation problem in terms of the parallel beam x-ray transform is also discussed.