In probability theory, the continuous mapping theorem states that continuous functions are limitpreserving 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 x_{n} → x then g(x_{n}) → g(x). The continuous mapping theorem states that this will also be true if we replace the deterministic sequence {x_{n}} with a sequence of random variables {X_{n}}, 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.^{[1]}
Statement
Let {X_{n}}, 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 D_{g} such that Pr[X ∈ D_{g}] = 0. Then^{[2]}^{[3]}^{[4]}

X_n \ \xrightarrow{d}\ X \quad\Rightarrow\quad g(X_n)\ \xrightarrow{d}\ g(X);

X_n \ \xrightarrow{p}\ X \quad\Rightarrow\quad g(X_n)\ \xrightarrow{p}\ g(X);

X_n \ \xrightarrow{\!\!as\!\!}\ X \quad\Rightarrow\quad g(X_n)\ \xrightarrow{\!\!as\!\!}\ g(X).
Proof
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.
Convergence in distribution
We will need a particular statement from the portmanteau theorem: that convergence in distribution X_n\xrightarrow{d}X is equivalent to

\limsup_{n\to\infty}\operatorname{Pr}(X_n \in F) \leq \operatorname{Pr}(X\in F) \text{ for every closed set } F.
Fix an arbitrary closed set F⊂S′. Denote by g^{−1}(F) the preimage of F under the mapping g: the set of all points x∈S such that g(x)∈F. Consider a sequence {x_{k}} such that g(x_{k})∈F and x_{k}→x. Then this sequence lies in g^{−1}(F), and its limit point x belongs to the closure of this set, g^{−1}(F) (by definition of the closure). The point x may be either:

a continuity point of g, in which case g(x_{k})→g(x), and hence g(x)∈F because F is a closed set, and therefore in this case x belongs to the preimage of F, or

a discontinuity point of g, so that x∈D_{g}.
Thus the following relationship holds:

\overline{g^{1}(F)} \ \subset\ g^{1}(F) \cup D_g\ .
Consider the event {g(X_{n})∈F}. The probability of this event can be estimated as

\operatorname{Pr}\big(g(X_n)\in F\big) = \operatorname{Pr}\big(X_n\in g^{1}(F)\big) \leq \operatorname{Pr}\big(X_n\in \overline{g^{1}(F)}\big),
and by the portmanteau theorem the limsup of the last expression is less than or equal to Pr(X∈g^{−1}(F)). Using the formula we derived in the previous paragraph, this can be written as

\begin{align} & \operatorname{Pr}\big(X\in \overline{g^{1}(F)}\big) \leq \operatorname{Pr}\big(X\in g^{1}(F)\cup D_g\big) \leq \\ & \operatorname{Pr}\big(X \in g^{1}(F)\big) + \operatorname{Pr}(X\in D_g) = \operatorname{Pr}\big(g(X) \in F\big) + 0. \end{align}
On plugging this back into the original expression, it can be seen that

\limsup_{n\to\infty} \operatorname{Pr}\big(g(X_n)\in F\big) \leq \operatorname{Pr}\big(g(X) \in F\big),
which, by the portmanteau theorem, implies that g(X_{n}) converges to g(X) in distribution.
Convergence in probability
Fix an arbitrary ε>0. Then for any δ>0 consider the set B_{δ} defined as

B_\delta = \big\{x\in S\ \big\ x\notin D_g:\ \exists y\in S:\ xy<\delta,\, g(x)g(y)>\varepsilon\big\}.
This is the set of continuity points x of the function g(·) for which it is possible to find, within the δneighborhood of x, a point which maps outside the εneighborhood of g(x). By definition of continuity, this set shrinks as δ goes to zero, so that lim_{δ→0}B_{δ} = ∅.
Now suppose that g(X) − g(X_{n}) > ε. This implies that at least one of the following is true: either X−X_{n}≥δ, or X∈D_{g}, or X∈B_{δ}. In terms of probabilities this can be written as

\operatorname{Pr}\big(\bigg(X_n)g(X)\big>\varepsilon\big) \leq \operatorname{Pr}\big(X_nX\geq\delta\big) + \operatorname{Pr}(X\in B_\delta) + \operatorname{Pr}(X\in D_g).
On the righthand side, the first term converges to zero as n → ∞ for any fixed δ, by the definition of convergence in probability of the sequence {X_{n}}. The second term converges to zero as δ → 0, since the set B_{δ} shrinks to an empty set. And the last term is identically equal to zero by assumption of the theorem. Therefore the conclusion is that

\lim_{n\to\infty}\operatorname{Pr}\big(\bigg(X_n)g(X)\big>\varepsilon\big) = 0,
which means that g(X_{n}) converges to g(X) in probability.
Convergence almost surely
By definition of the continuity of the function g(·),

\lim_{n\to\infty}X_n(\omega) = X(\omega) \quad\Rightarrow\quad \lim_{n\to\infty}g(X_n(\omega)) = g(X(\omega))
at each point X(ω) where g(·) is continuous. Therefore

\begin{align} \operatorname{Pr}\Big(\lim_{n\to\infty}g(X_n) = g(X)\Big) &\geq \operatorname{Pr}\Big(\lim_{n\to\infty}g(X_n) = g(X),\ X\notin D_g\Big) \\ &\geq \operatorname{Pr}\Big(\lim_{n\to\infty}X_n = X,\ X\notin D_g\Big) \\ &\geq \operatorname{Pr}\Big(\lim_{n\to\infty}X_n = X\Big)  \operatorname{Pr}(X\in D_g) = 10 = 1. \end{align}
By definition, we conclude that g(X_{n}) converges to g(X) almost surely.
See also
References
Literature
Notes
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.