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 Classification
Statement FormCharacterization Theorem 1 Connections
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Source Continuous mapping theorem, Wikipedia
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Source Continuous mapping theorem, Wikipedia
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1. Continuous mapping theorem, Wikipedia
Statement
Let {Xn}, X be random elements defined on a metric space S.
Introduction, opening paragraph
This theorem was first proved by Henry Mann and Abraham Wald in 1943
In Branch: Probability and Statistics, Lead sentence
In probability theory, the continuous mapping theorem states that continuous functions preserve limits even if their arguments are
Proved By: Abraham Wald, Lead paragraph
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
In Group: Probability Theory, Lead section
In probability theory, the continuous mapping theorem states that continuous
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