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 P[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

Convergence in distribution

Convergence in probability

Convergence 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.
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• 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