Cramér–Wold theorem

Cramér–Wold theorem In mathematics, the Cramér–Wold theorem in measure theory states that a Borel probability measure on {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.

Let {displaystyle {overline {X}}_{n}=(X_{n1},dots ,X_{nk})} and {displaystyle ;{overline {X}}=(X_{1},dots ,X_{k})} be random vectors of dimension k. Then {displaystyle {overline {X}}_{n}} converges in distribution to {displaystyle {overline {X}}} if and only if: {displaystyle sum _{i=1}^{k}t_{i}X_{ni}{overset {D}{underset {nrightarrow 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 {overline {X}}_{n}} converges in distribution to the correspondent linear combination of coordinates of {displaystyle {overline {X}}} .[1] If {displaystyle {overline {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}} .[2] Footnotes ^ Billingsley 1995, p. 383 ^ Kallenberg, Olav (2002). Foundations of modern probability (2nd ed.). New York: Springer. ISBN 0-387-94957-7. OCLC 46937587. References This article incorporates material from Cramér-Wold theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. Billingsley, Patrick (1995). Probability and Measure (3 ed.). John Wiley & Sons. ISBN 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. doi:10.1112/jlms/s1-11.4.290. External links Project Euclid: "When is a probability measure determined by infinitely many projections?" This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.

Categories: Theorems in measure theoryProbability theoremsConvergence (mathematics)Mathematical analysis stubs

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