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]

References

1. Samanta, M. (1989-04-01). "Non-parametric estimation of conditional quantiles". Statistics & Probability Letters. 7 (5): 407–412. doi:10.1016/0167-7152(89)90095-3. ISSN 0167-7152.
2. Cuesta-Albertos, Juan Antonio; Fraiman, Ricardo; Ransford, Thomas (2007). "A Sharp Form of the Cramér–Wold Theorem". Journal of Theoretical Probability. 20 (2): 201–209. doi:10.1007/s10959-007-0060-7. ISSN 0894-9840.
3. Mueller, Jonas W; Jaakkola, Tommi (2015). "Principal Differences Analysis: Interpretable Characterization of Differences between Distributions". Advances in Neural Information Processing Systems. 28. Curran Associates, Inc.
4. Berger, David; Lindner, Alexander (2022-05-01). "A Cramér–Wold device for infinite divisibility of Zd-valued distributions". Bernoulli. 28 (2). doi:10.3150/21-BEJ1386. ISSN 1350-7265.
5. "Cramér-Wold theorem". planetmath.org.
6. Billingsley, Patrick (1995). Probability and Measure (3 ed.). John Wiley & Sons. ISBN 978-0-471-00710-4.
7. Bélisle, Claude; Massé, Jean-Claude; Ransford, Thomas (1997). "When is a probability measure determined by infinitely many projections?". The Annals of Probability. 25 (2). doi:10.1214/aop/1024404418. ISSN 0091-1798.
8. Cramér, H.; Wold, H. (1936). "Some Theorems on Distribution Functions". Journal of the London Mathematical Society. s1-11 (4): 290–294. doi:10.1112/jlms/s1-11.4.290.
9. Billingsley 1995, p. 383
10. Kallenberg, Olav (2002). Foundations of modern probability (2nd ed.). New York: Springer. ISBN 0-387-94957-7. OCLC 46937587.