Continuous mapping theorem

In probability theory, the continuous mapping theorem states that continuous functions preserve limits even if their arguments are sequences of random variables. A continuous function, in Heine's definition, is such a function that maps convergent sequences into convergent sequences: if xn → x then g(xn) → g(x). The continuous mapping theorem states that this will also be true if we replace the deterministic sequence {xn} with a sequence of random variables {Xn}, and replace the standard notion of convergence of real numbers “→” with one of the types of convergence of random variables.

Not to be confused with the contraction mapping theorem.

This theorem was first proved by Henry Mann and Abraham Wald in 1943, and it is therefore sometimes called the Mann–Wald theorem. Meanwhile, Denis Sargan refers to it as the general transformation theorem.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.