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

First available in Project Euclid: 19 April 2007

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

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


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.

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