Slutsky's theorem
Theorem in probability theory From Wikipedia, the free encyclopedia
Theorem in probability theory From Wikipedia, the free encyclopedia
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]
Let be sequences of scalar/vector/matrix random elements. If converges in distribution to a random element and converges in probability to a constant , then
where denotes convergence in distribution.
Notes:
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 y−1 are continuous (for the last function to be continuous, y has to be invertible).
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