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May, 1983 Pointwise Translation of the Radon Transform and the General Central Limit Problem
Marjorie G. Hahn, Peter Hahan, Michael J. Klass
Ann. Probab. 11(2): 277-301 (May, 1983). DOI: 10.1214/aop/1176993597


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.


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Marjorie G. Hahn. Peter Hahan. Michael J. Klass. "Pointwise Translation of the Radon Transform and the General Central Limit Problem." Ann. Probab. 11 (2) 277 - 301, May, 1983.


Published: May, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0514.60025
MathSciNet: MR690129
Digital Object Identifier: 10.1214/aop/1176993597

Primary: 60E10
Secondary: 44A15 , 60F05 , 92A05

Keywords: computerized tomography , general multivariate central limit theorem , Infinitely divisible laws , pointwise translation problem , radiology , Radon transform , signed measures , Stable laws

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 2 • May, 1983
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