Cramér–Wold theorem

In mathematics, the Cramér–Wold theorem in measure theory 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.

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:


 * $$ \sum_{i=1}^k t_iX_{ni} \overset{D}{\underset{n\rightarrow\infty}{\rightarrow}} \sum_{i=1}^k t_iX_i. $$

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 $$ (t_1,\dots,t_k)\in \mathbb{R}_+^k $$.