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Continuous mapping theorem

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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.

This theorem was first proved by Henry Mann and Abraham Wald in 1943,cite-ref-1[1] and it is therefore sometimes called the Mann–Wald theorem.cite-ref-2[2] Meanwhile, Denis Sargan refers to it as the general transformation theorem.cite-ref-3[3]

Contents

• Statement

• Proof

• Convergence in distribution

• Convergence in probability

• Almost sure convergence

• See also

• References

Statement

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. Thencite-ref-4[4]cite-ref-5[5]

X n → d X ⇒ ⇒ g ( X n ) → d g ( X ) ; X n → p X ⇒ ⇒ g ( X n ) → p g ( X ) ; X n → a.s. X ⇒ ⇒ g ( X n ) → a.s. g ( X ) . {\displaystyle {\begin{aligned}X_{n}\ {\xrightarrow {\text{d}}}\ X\quad &\Rightarrow \quad g(X_{n})\ {\xrightarrow {\text{d}}}\ g(X);\\[6pt]X_{n}\ {\xrightarrow {\text{p}}}\ X\quad &\Rightarrow \quad g(X_{n})\ {\xrightarrow {\text{p}}}\ g(X);\\[6pt]X_{n}\ {\xrightarrow {\!\!{\text{a.s.}}\!\!}}\ X\quad &\Rightarrow \quad g(X_{n})\ {\xrightarrow {\!\!{\text{a.s.}}\!\!}}\ g(X).\end{aligned}}}

where the superscripts, "d", "p", and "a.s." denote convergence in distribution, convergence in probability, and almost sure convergence respectively.

Proof

This proof has been adopted from (

van der Vaart 1998

, Theorem 2.3)

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 → d X {\displaystyle X_{n}{\xrightarrow {d}}X} is equivalent to

E f ( X n ) → → E f ( X ) {\displaystyle \mathbb {E} f(X_{n})\to \mathbb {E} f(X)} for every bounded continuous functional f.

So it suffices to prove that E f ( g ( X n ) ) → → E f ( g ( X ) ) {\displaystyle \mathbb {E} f(g(X_{n}))\to \mathbb {E} f(g(X))} for every bounded continuous functional f. For simplicity we assume g continuous. Note that F = f ∘ ∘ g {\displaystyle F=f\circ g} is itself a bounded continuous functional. And so the claim follows from the statement above. The general case is slightly more technical.

Convergence in probability

Fix an arbitrary ε > 0. Then for any δ > 0 consider the set Bδ defined as

B δ δ = { x ∈ ∈ S ∣ ∣ x ∉ ∉ D g : ∃ ∃ y ∈ ∈ S : | x − − y | < δ δ , | g ( x ) − − g ( y ) | > ε ε } . {\displaystyle B_{\delta }={\big \{}x\in S\mid x\notin D_{g}:\ \exists y\in S:\ |x-y|<\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 δ-neighborhood of x, a point which maps outside the ε-neighborhood of g(x). By definition of continuity, this set shrinks as δ goes to zero, so that limδ → 0Bδ = ∅.

Now suppose that |g(X) − g(Xn)| > ε. This implies that at least one of the following is true: either |X−Xn| ≥ δ, or X ∈ Dg, or X∈Bδ. In terms of probabilities this can be written as

Pr ( | g ( X n ) − − g ( X ) | > ε ε ) ≤ ≤ Pr ( | X n − − X | ≥ ≥ δ δ ) + Pr ( X ∈ ∈ B δ δ ) + Pr ( X ∈ ∈ D g ) . {\displaystyle \Pr {\big (}{\big |}g(X_{n})-g(X){\big |}>\varepsilon {\big )}\leq \Pr {\big (}|X_{n}-X|\geq \delta {\big )}+\Pr(X\in B_{\delta })+\Pr(X\in D_{g}).}

On the right-hand side, the first term converges to zero as n → ∞ for any fixed δ, by the definition of convergence in probability of the sequence {Xn}. The second term converges to zero as δ → 0, since the set Bδ 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 → → ∞ ∞ Pr ( | g ( X n ) − − g ( X ) | > ε ε ) = 0 , {\displaystyle \lim _{n\to \infty }\Pr {\big (}{\big |}g(X_{n})-g(X){\big |}>\varepsilon {\big )}=0,}

which means that g(Xn) converges to g(X) in probability.

Almost sure convergence

By definition of the continuity of the function g(·),

lim n → → ∞ ∞ X n ( ω ω ) = X ( ω ω ) ⇒ ⇒ lim n → → ∞ ∞ g ( X n ( ω ω ) ) = g ( X ( ω ω ) ) {\displaystyle \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,

Pr ( lim n → → ∞ ∞ g ( X n ) = g ( X ) ) ≥ ≥ Pr ( lim n → → ∞ ∞ g ( X n ) = g ( X ) , X ∉ ∉ D g ) ≥ ≥ Pr ( lim n → → ∞ ∞ X n = X , X ∉ ∉ D g ) = 1 , {\displaystyle {\begin{aligned}\Pr \left(\lim _{n\to \infty }g(X_{n})=g(X)\right)&\geq \Pr \left(\lim _{n\to \infty }g(X_{n})=g(X),\ X\notin D_{g}\right)\\&\geq \Pr \left(\lim _{n\to \infty }X_{n}=X,\ X\notin D_{g}\right)=1,\end{aligned}}}

because the intersection of two almost sure events is almost sure.

By definition, we conclude that g(Xn) converges to g(X) almost surely.