In mathematics, the Cramér–Wold theorem in measure theory states that a Borel probability measure on 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 n 1 , … , X n k ) and
X ¯ = ( X 1 , … , X k ) be random vectors of dimension k. Then X ¯ n converges in distribution to X ¯ if and only if:
∑ i = 1 k t i X n i → n → ∞ D ∑ i = 1 k t i X i . for each ( t 1 , … , t k ) ∈ 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 ¯ .
