Let be sequences of scalar/vector/matrix random elements. If converges in distribution to a random element and converges in probability to a constant , then
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converges to a constant is important — if it were to converge to a non-degenerate random variable, the theorem would be no longer valid. For example, let and . The sum for all values of n. Moreover, , but does not converge in distribution to , where , , and and are independent.[4]
The theorem remains valid if we replace all convergences in distribution with convergences in probability.
Proofedit
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) (see here).
Next we apply the continuous mapping theorem, recognizing the functions g(x,y) = x + y, g(x,y) = xy, and g(x,y) = xy−1 are continuous (for the last function to be continuous, y has to be invertible).
^Slutsky, E. (1925). "Über stochastische Asymptoten und Grenzwerte". Metron (in German). 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, Allan (2005). Probability: a graduate course. Springer-Verlag. ISBN0-387-22833-0.
^See Zeng, Donglin (Fall 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 readingedit
Casella, George; Berger, Roger L. (2001). Statistical Inference. Pacific Grove: Duxbury. pp. 240–245. ISBN0-534-24312-6.
Grimmett, G.; Stirzaker, D. (2001). Probability and Random Processes (3rd ed.). Oxford.
slutsky, theorem, probability, theory, extends, some, properties, algebraic, operations, convergent, sequences, real, numbers, sequences, random, variables, theorem, named, after, eugen, slutsky, also, attributed, harald, cramér, contents, statement, proof, al. In probability theory 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 Cramer 3 Contents 1 Statement 2 Proof 3 See also 4 References 5 Further readingStatement editLet X n Y n displaystyle X n Y n nbsp be sequences of scalar vector matrix random elements If X n displaystyle X n nbsp converges in distribution to a random element X displaystyle X nbsp and Y n displaystyle Y n nbsp converges in probability to a constant c displaystyle c nbsp then X n Y n d X c displaystyle X n Y n xrightarrow d X c nbsp X n Y n d X c displaystyle X n Y n xrightarrow d Xc nbsp X n Y n d X c displaystyle X n Y n xrightarrow d X c nbsp provided that c is invertible where d displaystyle xrightarrow d nbsp denotes convergence in distribution Notes 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 For example let X n U n i f o r m 0 1 displaystyle X n sim rm Uniform 0 1 nbsp and Y n X n displaystyle Y n X n nbsp The sum X n Y n 0 displaystyle X n Y n 0 nbsp for all values of n Moreover Y n d U n i f o r m 1 0 displaystyle Y n xrightarrow d rm Uniform 1 0 nbsp but X n Y n displaystyle X n Y n nbsp does not converge in distribution to X Y displaystyle X Y nbsp where X U n i f o r m 0 1 displaystyle X sim rm Uniform 0 1 nbsp Y U n i f o r m 1 0 displaystyle Y sim rm Uniform 1 0 nbsp and X displaystyle X nbsp and Y displaystyle Y nbsp are independent 4 The theorem remains valid if we replace all convergences in distribution with convergences in probability Proof editThis 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 see here Next we apply the continuous mapping theorem recognizing the functions g x y x y g x y xy and g x y x y 1 are continuous for the last function to be continuous y has to be invertible See also editConvergence of random variablesReferences edit Goldberger Arthur S 1964 Econometric Theory New York Wiley pp 117 120 Slutsky E 1925 Uber stochastische Asymptoten und Grenzwerte Metron in German 5 3 3 89 JFM 51 0380 03 Slutsky s theorem is also called Cramer s theorem according to Remark 11 1 page 249 of Gut Allan 2005 Probability a graduate course Springer Verlag ISBN 0 387 22833 0 See Zeng Donglin Fall 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 editCasella George Berger Roger L 2001 Statistical Inference Pacific Grove Duxbury pp 240 245 ISBN 0 534 24312 6 Grimmett G Stirzaker D 2001 Probability and Random Processes 3rd ed Oxford Hayashi Fumio 2000 Econometrics Princeton University Press pp 92 93 ISBN 0 691 01018 8 Retrieved from https en wikipedia org w index php title Slutsky 27s theorem amp oldid 1186968185, wikipedia, wiki, book, books, library,