In probability theory, the continuous mapping theorem states that continuous functions preserve limits even when their arguments are sequences of random variables rather than fixed numbers, extending the ordinary fact that a continuous function maps convergent sequences of real numbers into convergent sequences. Henry Mann and Abraham Wald first proved the theorem in 1943, so it is sometimes called the Mann-Wald theorem, while Denis Sargan referred to it as the general transformation theorem.
Facts
StatementContinuous functions preserve limits even when applied to sequences of random elements: if random elements Xn converge to X on a metric space and a function g is continuous except possibly on a set that X assigns probability zero, then g(Xn) converges to g(X). 1 Sources
1. Continuous mapping theorem, Wikipedia
Statement section
Let {Xn}, X be random elements defined on a metric space S. Suppose a function g: S to S' (where S' is another metric space) has the set of discontinuity points Dg such that Pr[X in Dg] = 0.
Introduction, opening paragraph
This theorem was first proved by Henry Mann and Abraham Wald in 1943
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