Théorème de Cramer-Wold

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

Laisser {style d'affichage {surligner {X}}_{n}=(X_{n1},des points ,X_{nk})} et {style d'affichage ;{surligner {X}}=(X_{1},des points ,X_{k})} be random vectors of dimension k. Alors {style d'affichage {surligner {X}}_{n}} converges in distribution to {style d'affichage {surligner {X}}} si et seulement si: {somme de style d'affichage _{je=1}^{k}t_{je}X_{ni}{overset {ré}{underset {nrightarrow infty }{rightarrow }}}somme _{je=1}^{k}t_{je}X_{je}.} pour chaque {style d'affichage (t_{1},des points ,t_{k})en mathbb {R} ^{k}} , C'est, if every fixed linear combination of the coordinates of {style d'affichage {surligner {X}}_{n}} converges in distribution to the correspondent linear combination of coordinates of {style d'affichage {surligner {X}}} .[1] Si {style d'affichage {surligner {X}}_{n}} takes values in {style d'affichage mathbb {R} _{+}^{k}} , then the statement is also true with {style d'affichage (t_{1},des points ,t_{k})en mathbb {R} _{+}^{k}} .[2] Footnotes ^ Billingsley 1995, p. 383 ^ Kallenberg, Olav (2002). Foundations of modern probability (2sd éd.). New York: Springer. ISBN 0-387-94957-7. OCLC 46937587. References This article incorporates material from Cramér-Wold theorem on PlanetMath, qui est sous licence Creative Commons Attribution/Share-Alike License. Billingsley, patrick (1995). Probabilité et mesure (3 éd.). John Wiley & Sons. ISBN 978-0-471-00710-4. Cramér, Harald; Wold, Herman (1936). "Some Theorems on Distribution Functions". Journal de la Société mathématique de Londres. 11 (4): 290–294. est ce que je:10.1112/jlms/s1-11.4.290. External links Project Euclid: "When is a probability measure determined by infinitely many projections?" Cet article lié à l'analyse mathématique est un bout. Vous pouvez aider Wikipédia en l'agrandissant.

Catégories: Theorems in measure theoryProbability theoremsConvergence (mathématiques)Notes d'analyse mathématique