Continuous mapping theorem

In probability theory, continuous mapping theorem states that continuous functions retain their limit-preserving properties in the scope 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 deterministic sequence {x_n} with a sequence of random variables {X_n}, and 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 therefore sometimes is 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′ 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 ^[5]

Convergence 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 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
5. This proof has been adopted from (van der Vaart 1998, Theorem 2.3)