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Slutsky's theorem

From Wikipedia, the free encyclopedia

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

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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:

  1. 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converges to a constant is important — if it were to converge to a non-degenerate random variable, the theorem would be no longer valid.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]
  2. The theorem remains valid if we replace all convergences in distribution with convergences in probability.

Proof

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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 (Xc) (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

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References

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  1. ^ Goldberger, Arthur S. (1964). Econometric Theory. New York: Wiley. pp. 117–120.
  2. ^ Slutsky, E. (1925). "Über stochastische Asymptoten und Grenzwerte". Metron (in German). 5 (3): 3–89. JFM 51.0380.03.
  3. ^ 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.
  4. ^ 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

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  • 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.