Théorème de Slutsky

Théorème de Slutsky en théorie des probabilités, Slutsky's theorem extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables.[1] The theorem was named after Eugen Slutsky.[2] Slutsky's theorem is also attributed to Harald Cramér.[3] Contenu 1 Déclaration 2 Preuve 3 Voir également 4 Références 5 Further reading Statement Let {style d'affichage X_{n},O_{n}} be sequences of scalar/vector/matrix random elements. Si {style d'affichage X_{n}} converges in distribution to a random element {style d'affichage X} et {style d'affichage Y_{n}} converges in probability to a constant {displaystyle c} , alors {style d'affichage X_{n}+O_{n} {xrightarrow {ré}} X+c;} {style d'affichage X_{n}O_{n} xrightarrow {ré} Xc;} {style d'affichage X_{n}/O_{n} {xrightarrow {ré}} X/c,} provided that c is invertible, où {style d'affichage {xrightarrow {ré}}} denotes convergence in distribution.

Remarques: The requirement that Yn converges to a constant is important — if it were to converge to a non-degenerate random variable, the theorem would be no longer valid. Par exemple, laisser {style d'affichage X_{n}sim {rm {Uniform}}(0,1)} et {style d'affichage Y_{n}=-X_{n}} . La somme {style d'affichage X_{n}+O_{n}=0} for all values of n. En outre, {style d'affichage Y_{n},xrightarrow {ré} ,{rm {Uniform}}(-1,0)} , mais {style d'affichage X_{n}+O_{n}} does not converge in distribution to {displaystyle X+Y} , où {displaystyle Xsim {rm {Uniform}}(0,1)} , {displaystyle Ysim {rm {Uniform}}(-1,0)} , et {style d'affichage X} et {style d'affichage Y} are independent.[4] The theorem remains valid if we replace all convergences in distribution with convergences in probability. Proof This theorem follows from the fact that if Xn converges in distribution to X and Yn converges in probability to a constant c, then the joint vector (Xn, Yn) converges in distribution to (X, c) (vois ici).

Next we apply the continuous mapping theorem, recognizing the functions g(X,y) =x + y, g(X,y) = xy, et g(X,y) = x y−1 are continuous (for the last function to be continuous, y has to be invertible).

See also Convergence of random variables References ^ Goldberger, Arthur S.. (1964). Econometric Theory. New York: Wiley. pp. 117–120. ^ Slutsky, E. (1925). "Über stochastische Asymptoten und Grenzwerte". Metron (en allemand). 5 (3): 3–89. JFM 51.0380.03. ^ Slutsky's theorem is also called Cramér's theorem according to Remark 11.1 (page 249) of Gut, Alain (2005). Probabilité: a graduate course. Springer Verlag. ISBN 0-387-22833-0. ^ See Zeng, Donglin (Tomber 2018). "Large Sample Theory of Random Variables (lecture slides)" (PDF). Advanced Probability and Statistical Inference I (BIOS 760). University of North Carolina at Chapel Hill. Slide 59. Further reading Casella, George; Berger, Roger L. (2001). Statistical Inference. Pacific Grove: Duxbury. pp. 240–245. ISBN 0-534-24312-6. Grimmett, G.; Stirzaker, ré. (2001). Probability and Random Processes (3e éd.). Oxford. Hayashi, Fumio (2000). Econometrics. Presse de l'Université de Princeton. pp. 92–93. ISBN 0-691-01018-8. Catégories: Théorie asymptotique (statistiques)Théorèmes de probabilitéThéorèmes en statistique