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.

The theorem was named after Eugen Slutsky. Slutsky's theorem is also attributed to Harald Cramér.

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converges to a constant is important — if it were to converge to a non-degenerate random variable, the theorem would be no longer valid.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.
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

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.