Cramér–Wold theorem

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In mathematics, the Cramér–Wold theorem in measure theory states that a Borel probability measure on is uniquely determined by the totality of its one-dimensional projections. It is used as a method for proving joint convergence results; the theorem is named after Harald Cramér and Herman Ole Andreas Wold.



be random vectors of dimension k. Then converges in distribution to if and only if:

for each , that is, if every fixed linear combination of the coordinates of converges in distribution to the correspondent linear combination of coordinates of .[1]


  1. ^ Billingsley 1995, p. 383


  • This article incorporates material from Cramér-Wold theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
  • Billingsley, Patrick (1995). Probability and Measure (3 ed.). John Wiley & Sons. ISBN 978-0-471-00710-4.
  • Cramér, Harald; Wold, Herman (1936). "Some Theorems on Distribution Functions". Journal of the London Mathematical Society. 11 (4): 290–294. doi:10.1112/jlms/s1-11.4.290.

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