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 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]

1. $X_n \ \xrightarrow{d}\ X \quad\Rightarrow\quad g(X_n)\ \xrightarrow{d}\ g(X);$
2. $X_n \ \xrightarrow{p}\ X \quad\Rightarrow\quad g(X_n)\ \xrightarrow{p}\ g(X);$
3. $X_n \ \xrightarrow{\!\!as\!\!}\ X \quad\Rightarrow\quad g(X_n)\ \xrightarrow{\!\!as\!\!}\ g(X).$

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 $X_n\xrightarrow{d}X$ 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 pre-image 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 pre-image 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

$\limsup_{n\to\infty} \Pr \big(g(X_n)\in F\big) \leq \Pr \big(g(X) \in F\big),$

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

$B_\delta = \big\{x\in S \mid x\notin D_g:\ \exists y\in S:\ |x-y|<\delta,\, |g(x)-g(y)|>\varepsilon\big\}.$

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 lim_δ → 0B_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

$\Pr\big(\big|g(X_n)-g(X)\big|>\varepsilon\big) \leq \Pr\big(|X_n-X|\geq\delta\big) + \Pr(X\in B_\delta) + \Pr(X\in D_g).$

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 {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

$\lim_{n\to\infty}\Pr \big(\big|g(X_n)-g(X)\big|>\varepsilon\big) = 0,$

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

$\begin{align} \operatorname{Pr}\Big(\lim_{n\to\infty}g(X_n) = g(X)\Big) &\geq \operatorname{Pr}\Big(\lim_{n\to\infty}g(X_n) = g(X),\ X\notin D_g\Big) \\ &\geq \operatorname{Pr}\Big(\lim_{n\to\infty}X_n = X,\ X\notin D_g\Big) \\ &\geq \operatorname{Pr}\Big(\lim_{n\to\infty}X_n = X\Big) - \operatorname{Pr}(X\in D_g) = 1-0 = 1. \end{align}$

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

See also

• Slutsky’s theorem
• Portmanteau theorem

References

1. Amemiya 1985, p. 88
2. Van der Vaart 1998, Theorem 2.3, page 7
3. Billingsley 1969, p. 31, Corollary 1
4. Billingsley 1999, p. 21, Theorem 2.7

Further reading

• Amemiya, Takeshi (1985). Advanced Econometrics. Cambridge, MA: Harvard University Press. ISBN 0-674-00560-0.
• Billingsley, Patrick (1969). Convergence of Probability Measures. John Wiley & Sons. ISBN 0-471-07242-7.
• Billingsley, Patrick (1999). Convergence of Probability Measures (2nd ed.). John Wiley & Sons. ISBN 0-471-19745-9.
• Mann, H. B.; Wald, A. (1943). "On Stochastic Limit and Order Relationships". Annals of Mathematical Statistics 14 (3): 217–226. doi:10.1214/aoms/1177731415. JSTOR 2235800.
• Van der Vaart, A. W. (1998). Asymptotic statistics. New York: Cambridge University Press. ISBN 0-521-49603-9.