The Annals of Probability

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

Marjorie G. Hahn, Peter Hahan, and Michael J. Klass

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Abstract

We identify a representation problem involving the Radon transforms of signed measures on $\mathbb{R}^d$ of finite total variation. Specifically, if $\mu$ is a pointwise translate of $v$ (i.e., if for all $\theta \in S^{d - 1}$ the projection $\mu_\theta$ is a translate of $v_\theta$), must $\mu$ be a vector translate of $v$? We obtain results in several important special cases. Relating this to limit theorems, let $X_{n1}, \cdots, X_{nk_n}$ be a u.a.n. triangular array on $\mathbb{R}^d$ and put $S_n = X_{n1} + \cdots + X_{nk_n}$. There exist vectors $v_n \in \mathbb{R}^d$ such that $\mathscr{L}(S_n - v_n) \rightarrow \gamma$ 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 $\gamma$ has the property that the only infinitely divisible laws which are pointwise translates of $\gamma$ 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.

Article information

Source
Ann. Probab., Volume 11, Number 2 (1983), 277-301.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993597

Digital Object Identifier
doi:10.1214/aop/1176993597

Mathematical Reviews number (MathSciNet)
MR690129

Zentralblatt MATH identifier
0514.60025

JSTOR
links.jstor.org

Subjects
Primary: 60E10: Characteristic functions; other transforms
Secondary: 44A15: Special transforms (Legendre, Hilbert, etc.) 60F05: Central limit and other weak theorems 92A05

Keywords
Radon transform signed measures stable laws general multivariate central limit theorem infinitely divisible laws computerized tomography radiology pointwise translation problem

Citation

Hahn, Marjorie G.; Hahan, Peter; Klass, Michael J. Pointwise Translation of the Radon Transform and the General Central Limit Problem. Ann. Probab. 11 (1983), no. 2, 277--301. doi:10.1214/aop/1176993597. https://projecteuclid.org/euclid.aop/1176993597


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