Cramér–Wold theorem

In mathematics, the Cramér–Wold theorem or the Cramér–Wold device is a theorem in measure theory and which states that a Borel probability measure on ${\mathbb{ℝ}}^k$ 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, who published the result in 1936.

Let

$X_n=(X_{n1},\dots ,X_{nk})$

and

$X=(X_1,\dots ,X_k)$

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

$\sum _{i=1}^k{t}_i{X}_{ni}\stackrel{D}{\underset{n\to \mathrm{\infty }}{\to }}\sum _{i=1}^k{t}_i{X}_i.$

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

If $X_n$ takes values in ${\mathbb{ℝ}}_{+}^k$, then the statement is also true with $(t_1,\dots ,t_k)\in {\mathbb{ℝ}}_{+}^k$.