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).
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).