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

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

where ${\displaystyle {\xrightarrow {d}}}$ denotes convergence in distribution.

Notes:

1. The requirement that Y_n converges to a constant is important — if it were to converge to a non-degenerate random variable, the theorem would be no longer valid.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 ${\displaystyle X_{n}\sim {\rm {Uniform}}(0,1)}$ and ${\displaystyle Y_{n}=-X_{n}}$. The sum ${\displaystyle X_{n}+Y_{n}=0}$ for all values of n. Moreover, ${\displaystyle Y_{n}\,\xrightarrow {d} \,{\rm {Uniform}}(-1,0)}$, but ${\displaystyle X_{n}+Y_{n}}$ does not converge in distribution to ${\displaystyle X+Y}$, where ${\displaystyle X\sim {\rm {Uniform}}(0,1)}$, ${\displaystyle Y\sim {\rm {Uniform}}(-1,0)}$, and ${\displaystyle X}$ and ${\displaystyle Y}$ 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 X_n converges in distribution to X and Y_n converges in probability to a constant c, then the joint vector (X_n, Y_n) 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

• Convergence of random variables

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.