Continuous mapping theorem

In probability theory, the continuous mapping theorem states that continuous functions retain their limit-preserving properties in the sense of the convergence of random variables. A continuous function, in Heine’s definition, is such a function which 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. ${\displaystyle X_{n}\ {\xrightarrow {d}}\ X\quad \Rightarrow \quad g(X_{n})\ {\xrightarrow {d}}\ g(X);}$ 
2. ${\displaystyle X_{n}\ {\xrightarrow {p}}\ X\quad \Rightarrow \quad g(X_{n})\ {\xrightarrow {p}}\ g(X);}$ 
3. ${\displaystyle X_{n}\ {\xrightarrow {\!\!as\!\!}}\ X\quad \Rightarrow \quad g(X_{n})\ {\xrightarrow {\!\!as\!\!}}\ g(X).}$ 

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 Euclidian.

Convergence in distribution

We will need a particular statement from the portmanteau theorem: that convergence in distribution ${\displaystyle X_{n}{\xrightarrow {d}}X}$  is equivalent to

${\displaystyle \limsup _{n\to \infty }\operatorname {Pr} (X_{n}\in F)\leq \operatorname {Pr} (X\in F){\text{ for every closed set }}F.}$ 

Fix arbitrary closed set F⊂S′. Denote g⁻¹(F) its pre-image 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 point x belongs to the closure of this set, g⁻¹(F) (by definition of the closure). Point x may be either the continuity point of g, in which case g(x_k)→g(x), which must lie in F because F is a closed set, and therefore x belongs to the pre-image of F; or x may be the discontinuity point of g: x∈D_g. Thus we established the following relationship:

${\displaystyle {\overline {g^{-1}(F)}}\ \subset \ g^{-1}(F)\cup D_{g}\ .}$ 

Now let’s consider the event {g(X_n)∈F}. The probability of this event can be estimated as

${\displaystyle \operatorname {Pr} {\big (}g(X_{n})\in F{\big )}=\operatorname {Pr} {\big (}X_{n}\in g^{-1}(F){\big )}\leq \operatorname {Pr} {\big (}X_{n}\in {\overline {g^{-1}(F)}}{\big )},}$ 

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

${\displaystyle {\begin{aligned}&\operatorname {Pr} {\big (}X\in {\overline {g^{-1}(F)}}{\big )}\leq \operatorname {Pr} {\big (}X\in g^{-1}(F)\cup D_{g}{\big )}\leq \\&\operatorname {Pr} {\big (}X\in g^{-1}(F){\big )}+\operatorname {Pr} (X\in D_{g})=\operatorname {Pr} {\big (}g(X)\in F{\big )}+0.\end{aligned}}}$ 

Plugging back into the original expression, we see that

${\displaystyle \limsup _{n\to \infty }\operatorname {Pr} {\big (}g(X_{n})\in F{\big )}\leq \operatorname {Pr} {\big (}g(X)\in F{\big )},}$ 

which by the portmanteau theorem implies that g(X_n) converges to g(X) in distribution.

Convergence in probability

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

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

This is the set of continuity points x of 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

${\displaystyle \operatorname {Pr} {\big (}{\big |}g(X_{n})-g(X){\big |}>\varepsilon {\big )}\leq \operatorname {Pr} {\big (}|X_{n}-X|\geq \delta {\big )}+\operatorname {Pr} (X\in B_{\delta })+\operatorname {Pr} (X\in D_{g})}$ 

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 we can conclude that

${\displaystyle \lim _{n\to \infty }\operatorname {Pr} {\big (}{\big |}g(X_{n})-g(X){\big |}>\varepsilon {\big )}=0,}$ 

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

Convergence almost surely

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

${\displaystyle \lim _{n\to \infty }X_{n}(\omega )=X(\omega )\quad \Rightarrow \quad \lim _{n\to \infty }g(X_{n}(\omega ))=g(X(\omega ))}$ 

at each point X(ω) where g(·) is continuous. Therefore

${\displaystyle {\begin{aligned}\operatorname {Pr} {\Big (}\lim _{n\to \infty }g(X_{n})=g(X){\Big )}&\geq \operatorname {Pr} {\Big (}\lim _{n\to \infty }g(X_{n})=g(X),\ X_{n}\notin D_{g}{\Big )}\\&\geq \operatorname {Pr} {\Big (}\lim _{n\to \infty }X_{n}=X,\ X\notin D_{g}{\Big )}\\&\geq \operatorname {Pr} {\Big (}\lim _{n\to \infty }X_{n}=X{\Big )}-\operatorname {Pr} (X\in D_{g})=1-0=1.\end{aligned}}}$ 

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

References

See also

• Slutsky’s theorem
• Portmanteau theorem

Literature

• Amemiya, Takeshi (1985). Advanced Econometrics. Cambridge, MA: Harvard University Press. ISBN 0674005600. {{cite book}}: Unknown parameter |lcc= ignored (help)
• Billingsley, Patrick (1969). Convergence of Probability Measures. John Wiley & Sons. ISBN 0471072427.
• Billingsley, Patrick (1999). Convergence of Probability Measures (2nd ed.). John Wiley & Sons. ISBN 0471197459.
• Template:Cite article
• van der Vaart, A. W. (1998). Asymptotic statistics. New York: Cambridge University Press. ISBN 9780521496032. {{cite book}}: Unknown parameter |lcc= ignored (help)

Notes

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