probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

, 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 ''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 with a sequence of random variables , 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 Abraham Wald (; hu, Wald Ábrahám, yi, אברהם וואַלד; – ) was a Jewish Hungarian mathematician who contributed to decision theory, geometry, and econometrics and founded the field of statistical sequential analysis. One o ...

in 1943, and it is therefore sometimes called the Mann–Wald theorem. Meanwhile, Denis Sargan refers to it as the general transformation theorem.

Statement

Let , ''X'' be random elements defined on a

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...

''S''. Suppose a function (where ''S′'' is another metric space) has the set of discontinuity points ''D_g'' such that . Then :

X_n \ \xrightarrow\ X \quad & \Rightarrow\quad g(X_n)\ \xrightarrow\ g(X). \end

where the superscripts, "d", "p", and "a.s." denote convergence in distribution,

convergence in probability In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to ...

, and almost sure convergence respectively.

Proof

This proof has been adopted from

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\xrightarrowX

is equivalent to :

\mathbb E f(X_n) \to \mathbb E f(X)

for every bounded continuous functional ''f''. So it suffices to prove that

\mathbb E f(g(X_n)) \to \mathbb E f(g(X))

for every bounded continuous functional ''f''. Note that

F = f \circ g

is itself a bounded continuous functional. And so the claim follows from the statement above.

Convergence in probability

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

B_\delta = \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'' → 0}''B_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 . 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_\Pr \big(\big, g(X_n)-g(X)\big, >\varepsilon\big) = 0,

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

Almost sure convergence

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

\lim_X_n(\omega) = X(\omega) \quad\Rightarrow\quad \lim_g(X_n(\omega)) = g(X(\omega))

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

\begin
  \Pr\left(\lim_g(X_n) = g(X)\right)
  &\geq \Pr\left(\lim_g(X_n) = g(X),\ X\notin D_g\right) \\
  &\geq \Pr\left(\lim_X_n = X,\ X\notin D_g\right)  = 1,
  \end

because the intersection of two almost sure events is almost sure. By definition, we conclude that ''g''(''X_n'') converges to ''g''(''X'') almost surely.

References