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

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

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

Notes:

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to a constant is important — if it were to converge to a non-degenerate random variable, the theorem would be no longer valid.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

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.

Slutsky's theorem

Statement

Proof

See also

References

Further reading