In probability theory, Slutsky's theorem extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables.
The theorem was named after Eugen Slutsky. Slutsky's theorem is also attributed to Harald Cramér.
Statement
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
- provided that c is invertible,
where 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converges to a constant is important — if it were to converge to a non-degenerate random variable, the theorem would be no longer valid.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 and . The sum for all values of n. Moreover, , but does not converge in distribution to , where , , and and are independent.
- 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) (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 are continuous (for the last function to be continuous, y has to be invertible).
See also
References
- Goldberger, Arthur S. (1964). Econometric Theory. New York: Wiley. pp. 117–120.
- 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. 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
- Casella, 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.