Slutsky's theorem

In probability theory, Slutsky’s theorem^[1] extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables.

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

• $X_n + Y_n \ \xrightarrow{d}\ X + c ;$
• $X_nY_n \ \xrightarrow{d}\ cX ;$
• $X_n/Y_n \ \xrightarrow{d}\ X/c,$   provided that c is invertible,

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

Notes:

1. In the statement of the theorem, the condition “Y_n converges in probability to a constant c” may be replaced with “Y_n converges in distribution to a constant c” — these two requirements are equivalent according to this property.
2. 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converges to a constant is important — if it were to converge to a non-degenerate random variable, the theorem would be no longer valid.
3. The theorem remains valid if we replace all convergences in distribution with convergences in probability (due to this property).

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⁻¹y as continuous (for the last function to be continuous, x has to be invertible).

References

1. Grimmett 2001, Exercise 7.2.5
2. Slutsky 1925
3. Slutsky's theorem is also called Cramér’s theorem according to Remark 11.1 (page 249) of Allan Gut. A Graduate Course in Probability. Springer Verlag. 2005.

• Grimmett, G.; Stirzaker, D. (2001), Probability and Random Processes (3rd ed.), Oxford
• Gut, Allan (2005), Probability: a graduate course, Springer-Verlag, ISBN 0-387-22833-0
• Slutsky, E. (1925), "Über stochastische Asymptoten und Grenzwerte", Metron (in German) 5 (3): 3–89, JFM 51.0380.03