Cramér–Wold theorem

In mathematics, the Cramér–Wold theorem^[1][2] or the Cramér–Wold device^[3][4] is a theorem in measure theory and which states that a Borel probability measure on ${\displaystyle \mathbb {R} ^{k}}$ is uniquely determined by the totality of its one-dimensional projections.^[5][6][7] 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.^[8]

Let

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

and

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

be random vectors of dimension k. Then ${\displaystyle {X}_{n}}$ converges in distribution to ${\displaystyle {X}}$ 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 ${\displaystyle {X}_{n}}$ converges in distribution to the correspondent linear combination of coordinates of ${\displaystyle {X}}$.^[9]

If ${\displaystyle {X}_{n}}$ takes values in ${\displaystyle \mathbb {R} _{+}^{k}}$, then the statement is also true with ${\displaystyle (t_{1},\dots ,t_{k})\in \mathbb {R} _{+}^{k}}$.^[10]