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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A calculus proof of the Cramér–Wold theorem
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by Russell Lyons and Kevin Zumbrun PDF
Proc. Amer. Math. Soc. 146 (2018), 1331-1334 Request permission

Abstract:

We present a short, elementary proof not involving Fourier transforms of the theorem of Cramér and Wold that a Borel probability measure is determined by its values on half-spaces.
References
  • Patrick Billingsley, Probability and measure, 3rd ed., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication. MR 1324786
  • H. Cramér and H. Wold, Some Theorems on Distribution Functions, J. London Math. Soc. 11 (1936), no. 4, 290–294. MR 1574927, DOI 10.1112/jlms/s1-11.4.290
  • M. W. Crofton, On the theory of local probability, applied to straight lines drawn at random in a plane; the methods used being also extended to the proof of certain new theorems in the integral calculus, Philos. Trans. Royal Soc. London 158 (1868), 181–199.
  • Fritz John, Plane waves and spherical means applied to partial differential equations, Interscience Publishers, New York-London, 1955. MR 0075429
  • Russell Lyons, Distance covariance in metric spaces, Ann. Probab. 41 (2013), no. 5, 3284–3305. MR 3127883, DOI 10.1214/12-AOP803
  • David Pollard, A user’s guide to measure theoretic probability, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 8, Cambridge University Press, Cambridge, 2002. MR 1873379
  • Johann Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Computed tomography (Cincinnati, Ohio, 1982) Proc. Sympos. Appl. Math., vol. 27, Amer. Math. Soc., Providence, R.I., 1982, pp. 71–86 (German). MR 692055
  • A. Rényi, On projections of probability distributions, Acta Math. Acad. Sci. Hungar. 3 (1952), 131–142 (English, with Russian summary). MR 53422, DOI 10.1007/bf02022515
  • Rolf Schneider and Wolfgang Weil, Stochastic and integral geometry, Probability and its Applications (New York), Springer-Verlag, Berlin, 2008. MR 2455326, DOI 10.1007/978-3-540-78859-1
  • Guenther Walther, On a conjecture concerning a theorem of Cramér and Wold, J. Multivariate Anal. 63 (1997), no. 2, 313–319. MR 1484318, DOI 10.1006/jmva.1997.1705
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Additional Information
  • Russell Lyons
  • Affiliation: Department of Mathematics, 831 East 3rd Street, Indiana University, Bloomington, Indiana 47405-7106
  • MR Author ID: 196888
  • Email: rdlyons@indiana.edu
  • Kevin Zumbrun
  • Affiliation: Department of Mathematics, 831 East 3rd Street, Indiana University, Bloomington, Indiana 47405-7106
  • MR Author ID: 330192
  • Email: kzumbrun@indiana.edu
  • Received by editor(s): July 11, 2016
  • Received by editor(s) in revised form: April 22, 2017
  • Published electronically: October 12, 2017
  • Additional Notes: This research was partially supported by NSF grants DMS-1007244 and DMS-1400555
  • Communicated by: Mark M. Meerschaert
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 146 (2018), 1331-1334
  • MSC (2010): Primary 60E10; Secondary 44A12, 53C65
  • DOI: https://doi.org/10.1090/proc/13794
  • MathSciNet review: 3750244