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.
Publié le : 1983-05-14
Classification:
Radon transform,
signed measures,
stable laws,
general multivariate central limit theorem,
infinitely divisible laws,
computerized tomography,
radiology,
pointwise translation problem,
60E10,
44A15,
60F05,
92A05
@article{1176993597,
author = {Hahn, Marjorie G. and Hahan, Peter and Klass, Michael J.},
title = {Pointwise Translation of the Radon Transform and the General Central Limit Problem},
journal = {Ann. Probab.},
volume = {11},
number = {4},
year = {1983},
pages = { 277-301},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993597}
}
Hahn, Marjorie G.; Hahan, Peter; Klass, Michael J. Pointwise Translation of the Radon Transform and the General Central Limit Problem. Ann. Probab., Tome 11 (1983) no. 4, pp. 277-301. http://gdmltest.u-ga.fr/item/1176993597/