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 R k {\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, who published the result in 1936. Let
X n = ( X n 1 , … , X n k ) {\displaystyle {X}_{n}=(X_{n1},\dots ,X_{nk})}
and
X = ( X 1 , … , X k ) {\displaystyle \;{X}=(X_{1},\dots ,X_{k})}
be random vectors of dimension k. Then X n {\displaystyle {X}_{n}} converges in distribution to X {\displaystyle {X}} if and only if:
∑ i = 1 k t i X n i → n → ∞ D ∑ i = 1 k t i X i . {\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 ( t 1 , … , t k ) ∈ R k {\displaystyle (t_{1},\dots ,t_{k})\in \mathbb {R} ^{k}} , that is, if every fixed linear combination of the coordinates of X n {\displaystyle {X}_{n}} converges in distribution to the correspondent linear combination of coordinates of X {\displaystyle {X}} . If X n {\displaystyle {X}_{n}} takes values in R + k {\displaystyle \mathbb {R} _{+}^{k}} , then the statement is also true with ( t 1 , … , t k ) ∈ R + k {\displaystyle (t_{1},\dots ,t_{k})\in \mathbb {R} _{+}^{k}} .
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