# 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 {Xn}, {Yn} be sequences of scalar/vector/matrix random elements.

If Xn converges in distribution to a random element X;

and Yn 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 “Yn converges in probability to a constant c” may be replaced with “Yn converges in distribution to a constant c” — these two requirements are equivalent according to this property.
2. 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converges to a constant is important — if it were to converge to a non-degenerate random variable, the theorem would be no longer valid.
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 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−1y as continuous (for the last function to be continuous, x has to be invertible).

## 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.