Slutsky's theorem

Theorem in probability theory

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 X n , Y n {\displaystyle X_{n},Y_{n}} be sequences of scalar/vector/matrix random elements. If X n {\displaystyle X_{n}} converges in distribution to a random element X {\displaystyle X} and Y n {\displaystyle Y_{n}} converges in probability to a constant c {\displaystyle c} , then

  • X n + Y n   → d   X + c ; {\displaystyle X_{n}+Y_{n}\ {\xrightarrow {d}}\ X+c;}
  • X n Y n   → d   X c ; {\displaystyle X_{n}Y_{n}\ \xrightarrow {d} \ Xc;}
  • X n / Y n   → d   X / c , {\displaystyle X_{n}/Y_{n}\ {\xrightarrow {d}}\ X/c,}   provided that c is invertible,

where → d {\displaystyle {\xrightarrow {d}}} 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converges to a constant is important — if it were to converge to a non-degenerate random variable, the theorem would be no longer valid.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 X n ∼ U n i f o r m ( 0 , 1 ) {\displaystyle X_{n}\sim {\rm {Uniform}}(0,1)} and Y n = − X n {\displaystyle Y_{n}=-X_{n}} . The sum X n + Y n = 0 {\displaystyle X_{n}+Y_{n}=0} 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)} , but X n + Y n {\displaystyle X_{n}+Y_{n}} does not converge in distribution to X + Y {\displaystyle X+Y} , where X ∼ U n i f o r m ( 0 , 1 ) {\displaystyle X\sim {\rm {Uniform}}(0,1)} , Y ∼ U n i f o r m ( − 1 , 0 ) {\displaystyle Y\sim {\rm {Uniform}}(-1,0)} , and X {\displaystyle X} and Y {\displaystyle Y} are independent.[4]
  2. 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

  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

  • 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.