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.

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

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

Notes:

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

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