Continuous mapping theorem
In probability theory, the continuous mapping theorem states that continuous functions are limit-preserving 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.
This theorem was first proved by Mann & Wald (1943), and it is therefore sometimes called the Mann–Wald theorem.
Let {Xn}, X be random elements defined on a metric space S. Suppose a function g: S→S′ (where S′ is another metric space) has the set of discontinuity points Dg such that Pr[X ∈ Dg] = 0. Then
Spaces S and S′ are equipped with certain metrics. For simplicity we will denote both of these metrics using the |x−y| notation, even though the metrics may be arbitrary and not necessarily Euclidean.
We will need a particular statement from the portmanteau theorem: that convergence in distribution is equivalent to
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