Cramér–Wold theorem

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.

Let

and

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]

If takes values in , then the statement is also true with .[2]

Footnotes

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  1. ^ Billingsley 1995, p. 383
  2. ^ Kallenberg, Olav (2002). Foundations of modern probability (2nd ed.). New York: Springer. ISBN 0-387-94957-7. OCLC 46937587.

References

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