Cramér–Wold theorem explained

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 _{n1} , ... ,X _{nk} ) and{X} = (X _{1,...,X} _{k)} be random vectors of dimension k. Then {X} _{n} converges in distribution to {X} if and only if:\sum ^{k} _{i =1} t _{iX} _{ni} \overset{D}{\underset{n → inf ty}{ → }} \sum ^{k} _{i =1} t _{iX} _{i.}

for each

(t _{1,...,t} _{k)\in} 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 R ^{k} _{+} , then the statement is also true with (t _{1,...,t} _{k)\in} R ^{k} _{+} .^[1]

References

• • Book: Billingsley, Patrick . Patrick Billingsley. 1995 . Probability and Measure . 3. . 978-0-471-00710-4 .
• Cramér . Harald . Wold . Herman . 1936. Some Theorems on Distribution Functions. Journal of the London Mathematical Society. 11. 4. 290–294. 10.1112/jlms/s1-11.4.290.

External links

• Project Euclid: "When is a probability measure determined by infinitely many projections?"

Notes and References

1. Book: Kallenberg, Olav. Foundations of modern probability. 2002. Springer. 0-387-94957-7. 2nd. New York. 46937587.