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:

$\sum _{i=1}^{k}t_{i}X_{ni}{\overset {D}{\underset {n\rightarrow \infty }{\rightarrow }}}\sum _{i=1}^{k}t_{i}X_{i}.$

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 $(t_{1},\dots ,t_{k})\in \mathbb {R} _{+}^{k}$.