# 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]}

## Contents

## 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

- provided that
*c*is invertible,

where denotes convergence in distribution.

**Notes:**

- 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. - The requirement that
*Y*converges to a constant is important — if it were to converge to a non-degenerate random variable, the theorem would be no longer valid._{n} - The theorem remains valid if we replace all convergences in distribution with convergences in probability (due to this property).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*converges in probability to a constant

_{n}*c*, then the joint vector (

*X*) converges in distribution to (

_{n}, Y_{n}*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*^{−1}*y* as continuous (for the last function to be continuous, *x* 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.

## Further reading

- Grimmett, G.; Stirzaker, D. (2001).
*Probability and Random Processes*(3rd ed.). Oxford.