Slutsky's theorem
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 Cramér.[3]
Statement
Let [math]\displaystyle{ X_n, Y_n }[/math] be sequences of scalar/vector/matrix random elements. If [math]\displaystyle{ X_n }[/math] converges in distribution to a random element [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y_n }[/math] converges in probability to a constant [math]\displaystyle{ c }[/math], then
- [math]\displaystyle{ X_n + Y_n \ \xrightarrow{d}\ X + c ; }[/math]
- [math]\displaystyle{ X_nY_n \ \xrightarrow{d}\ Xc ; }[/math]
- [math]\displaystyle{ X_n/Y_n \ \xrightarrow{d}\ X/c, }[/math] provided that c is invertible,
where [math]\displaystyle{ \xrightarrow{d} }[/math] 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converges to a constant is important — if it were to converge to a non-degenerate random variable, the theorem would be no longer valid.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 [math]\displaystyle{ X_n \sim {\rm Uniform}(0,1) }[/math] and [math]\displaystyle{ Y_n = -X_n }[/math]. The sum [math]\displaystyle{ X_n + Y_n = 0 }[/math] for all values of n. Moreover, [math]\displaystyle{ Y_n \, \xrightarrow{d} \, {\rm Uniform}(-1,0) }[/math], but [math]\displaystyle{ X_n + Y_n }[/math] does not converge in distribution to [math]\displaystyle{ X + Y }[/math], where [math]\displaystyle{ X \sim {\rm Uniform}(0,1) }[/math], [math]\displaystyle{ Y \sim {\rm Uniform}(-1,0) }[/math], and [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] 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) (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
References
- ↑ Goldberger, Arthur S. (1964). Econometric Theory. New York: Wiley. pp. 117–120. https://archive.org/details/econometrictheor0000gold.
- ↑ "Über stochastische Asymptoten und Grenzwerte" (in de). Metron 5 (3): 3–89. 1925.
- ↑ 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)". Advanced Probability and Statistical Inference I (BIOS 760). University of North Carolina at Chapel Hill. Slide 59. https://www.bios.unc.edu/~dzeng/BIOS760/ChapC_Slide.pdf#page=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. https://books.google.com/books?id=QyIW8WUIyzcC&pg=PA92.
Original source: https://en.wikipedia.org/wiki/Slutsky's theorem.
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