Cramér–Wold theorem

In mathematics, the Cramér–Wold theorem in measure theory states that a Borel probability measure on ${\displaystyle \mathbb {R} ^{k}}$ 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

${\displaystyle {\overline {X}}_{n}=(X_{n1},\dots ,X_{nk})}$

and

${\displaystyle \;{\overline {X}}=(X_{1},\dots ,X_{k})}$

be random vectors of dimension k. Then ${\displaystyle {\overline {X}}_{n}}$ converges in distribution to ${\displaystyle {\overline {X}}}$ if and only if:

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

for each ${\displaystyle (t_{1},\dots ,t_{k})\in \mathbb {R} ^{k}}$, that is, if every fixed linear combination of the coordinates of ${\displaystyle {\overline {X}}_{n}}$ converges in distribution to the correspondent linear combination of coordinates of ${\displaystyle {\overline {X}}}$.[1]