Cramér–Wold Theorem

In mathematics, the Cramér–Wold theorem in measure theory states that a Borel probability measure on $\mathbb^k$ 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
:$\overline_n = (X_,\dots,X_)$
and
:$\; \overline = (X_1,\dots,X_k)$
be random vectors of dimension ''k''. Then $\overline_n$ converges in distribution to $\overline$ if and only if:

:$\sum_^k t_iX_ \overset \sum_^k t_iX_i.$

for each $(t_1,\dots,t_k)\in \mathbb^k$, that is, if every fixed linear combination of the coordinates of $\overline_n$ converges in distribution to the correspondent linear combination of coordinates of $\overline$.

If $\overline_n$ takes values in $\mathbb_+^k$, then the statement is also true with $(t_1,\dots,t_k)\in \mathbb_+^k$.

Footnotes

References

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External links

* Project Euclid
"When is a probability measure determined by infinitely many projections?"

Theorems in measure theory
Probability theorems
Convergence (mathematics)

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