Continuous Mapping theorem

The Continuous Mapping theorem states that stochastic convergence is preserved by continuous functions.

Table of contents

The problem
The theorem
Consequences
More details
1. Convergence of ratios
2. Random matrices
Applications
Solved exercises
1. Exercise 1
References

The problem

Suppose that a sequence of random vectors converges to a random vector (in probability, in distribution or almost surely).

Now, take a transformed sequence , where is a function.

Under what conditions is also a convergent sequence?

The Continuous Mapping theorem states that stochastic convergence is preserved if is a continuous function.

The theorem

Here is a statement of the multivariate version of the Continuous Mapping theorem.

Proposition Let be a sequence of -dimensional random vectors. Let be a continuous function. Then, [eq6] where denotes convergence in probability, denotes almost sure convergence and denotes convergence in distribution.

Proof

See, e.g., Shao (2003).

Consequences

The next sections present some important consequences of the Continuous Mapping theorem.

Sums and products of sequences converging in probability

An important implication of the Continuous Mapping theorem is that arithmetic operations preserve convergence in probability.

Proposition If and . Then, [eq12]

Proof

First of all, note that convergence in probability of and of implies their joint convergence in probability (see the lecture entitled Convergence in probability), that is, their convergence as a vector: Now, the sum and the product are continuous functions of the operands. Thus, for example,is a continuous function, and, by using the Continuous Mapping theorem, we obtain [eq17] where denotes a limit in probability.

Sums and products of sequences converging almost surely

Everything that was said in the previous subsection applies, with obvious modifications, also to almost surely convergent sequences.

Proposition If and , then [eq20]

Proof

Similar to previous proof. Just replace convergence in probability with almost sure convergence.

Sums and products of sequences converging in distribution

For convergence almost surely and convergence in probability, the convergence of and individually implies their joint convergence as a vector (see the previous two proofs), but this is not the case for convergence in distribution. Therefore, to obtain preservation of convergence in distribution under arithmetic operations, we need the stronger assumption of joint convergence in distribution.

Proposition If then [eq24]

Proof

Again, similar to the proof for convergence in probability, but this time joint convergence is already in the assumptions.

More details

The following sections contain more details about the Continuous Mapping theorem.

Convergence of ratios

As a byproduct of the propositions stated above, we also have the following proposition.

Proposition If a sequence of random variables converges to , thenprovided is almost surely different from (we did not specify the kind of convergence, which can be in probability, almost surely or in distribution).

Proof

This is a consequence of the Continuous Mapping theorem and of the fact that is a continuous function for .

An immediate consequence of the previous proposition follows.

Proposition If two sequences of random variables and converge to and respectively, thenprovided is almost surely different from . Convergence can be in probability, almost surely or in distribution (but the latter requires joint convergence in distribution of and ).

Proof

This is a consequence of the fact that the ratio can be written as a productThe first operand of the product converges by assumption. The second converges because of the previous proposition. Therefore, their product converges because convergence is preserved under products.

Random matrices

The Continuous Mapping theorem applies also to random matrices because random matrices are just random vectors whose entries have been arranged into the columns of a matrix.

In particular:

if two sequences of random matrices are convergent, then also the sum and the product of their terms are convergent (provided their dimensions are such that they can be summed or multiplied);
if a sequence of square random matrices converges to a random matrix , then the sequence of inverse matrices converges to the random matrix $X^{-1}$ (provided the matrices are invertible). This is a consequence of the fact that matrix inversion is a continuous transformation.

Applications

The Continuous Mapping theorem has several important applications. For example, it is used to prove:

Slutsky's theorem;
the Delta method.

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Consider a sequence of random variables converging in distribution to a random variable having a standard normal distribution.

Consider the function which is a continuous function.

Find the limit in distribution of the sequence .

Solution

The sequenceconverges in distribution to $X^{2}$ by the Continuous Mapping theorem. But the square of a standard normal random variable has a Chi-square distribution with one degree of freedom. Therefore, the sequence converges in distribution to a Chi-square distribution with one degree of freedom.

References

Shao, J. (2007) Mathematical statistics, Springer.

How to cite

Please cite as:

Taboga, Marco (2021). "Continuous Mapping theorem", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/asymptotic-theory/continuous-mapping-theorem.