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.
A statement of the Continuous Mapping theorem follows.
Proposition (Continuous Mapping) Let be a sequence of -dimensional random vectors. Let be a continuous function. Then,where denotes convergence in probability, denotes almost sure convergence and denotes convergence in distribution.
See, e.g., Shao (2003).
The next sections present some important consequences of the Continuous Mapping theorem.
An important implication of the Continuous Mapping theorem is that arithmetic operations preserve convergence in probability.
Proposition If and . Then,
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 obtainwhere denotes a limit in probability.
Everything that was said in the previous subsection applies, with obvious modifications, also to almost surely convergent sequences.
Proposition If and , then
Similar to previous proof. Just replace convergence in probability with almost sure convergence.
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
Again, similar to the proof for convergence in probability, but this time joint convergence is already in the assumptions.
The following sections contain more details about the Continuous Mapping theorem.
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).
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 ).
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.
The Continuous Mapping theorem applies also to random matrices because random matrices are just random vectors whose elements have been arranged into columns.
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 (provided the matrices are invertible). This is a consequence of the fact that matrix inversion is a continuous transformation.
Below you can find some exercises with explained solutions.
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 .
The sequenceconverges in distribution to 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.
Shao, J. (2007) Mathematical statistics, Springer.
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