Pointwise convergence

This lecture discusses pointwise convergence, first for sequences of random variables and then for sequences of random vectors.

Table of contents

Pointwise convergence of a sequence of random variables
Pointwise convergence of a sequence of random vectors
Solved exercises

Pointwise convergence of a sequence of random variables

Let be a sequence of random variables defined on a sample space .

Let us consider a single sample point and a generic random variable belonging to the sequence.

is a function . However, once we fix , the realization associated to the sample point is just a real number. By the same token, once we fix , the sequence is just a sequence of real numbers.

Therefore, for a fixed , it is very easy to assess whether the sequence is convergent; this is done employing the usual definition of convergence of sequences of real numbers.

If, for a fixed , the sequence is convergent, we denote its limit by , to underline that the limit depends on the specific we have fixed.

A sequence of random variables is said to be pointwise convergent if and only if the sequence is convergent for any choice of .

Definition Let be a sequence of random variables defined on a sample space . We say that is pointwise convergent to a random variable defined on if and only if converges to for all . is called the pointwise limit of the sequence and convergence is indicated by

Roughly speaking, using pointwise convergence we somehow circumvent the problem of defining the concept of distance between random variables: by fixing , we reduce ourselves to the familiar problem of measuring distance between two real numbers, so that we can employ the usual notion of convergence of sequences of real numbers.

Example Let be a sample space with two sample points ( $omega _{1}$ and $omega _{2}$ ). Let be a sequence of random variables such that a generic term of the sequence satisfies [eq16] We need to check the convergence of the sequences for all , i.e. for and for : (1) the sequence , whose generic term is , is a sequence of real numbers converging to ; (2) the sequence , whose generic term is , is a sequence of real numbers converging to . Therefore, the sequence of random variables converges pointwise to the random variable , where is defined as follows: [eq25]

Pointwise convergence of a sequence of random vectors

The above notion of convergence generalizes to sequences of random vectors in a straightforward manner.

Let be a sequence of random vectors defined on a sample space , where each random vector has dimension .

If we fix a single sample point , the sequence is a sequence of real vectors.

By the standard criterion for convergence, the sequence of real vectors is convergent to a vector if where is the distance between a generic term of the sequence and the limit .

The distance between and is defined to be equal to the Euclidean norm of their difference: [eq36] where the second subscript is used to indicate the individual components of the vectors and .

Thus, for a fixed , the sequence of real vectors is convergent to a vector if

A sequence of random vectors is said to be pointwise convergent if and only if the sequence is convergent for any choice of .

Definition Let be a sequence of random vectors defined on a sample space . We say that is pointwise convergent to a random vector defined on if and only if converges to for all (i.e. ). is called the pointwise limit of the sequence and convergence is indicated by

Now, denote by the sequence of the -th components of the vectors . It can be proved that the sequence of random vectors is pointwise convergent if and only if all the sequences of random variables are pointwise convergent:

Proposition Let be a sequence of random vectors defined on a sample space . Denote by the sequence of random variables obtained by taking the -th component of each random vector . The sequence converges pointwise to the random vector if and only if converges pointwise to the random variable $X_{ullet ,i}$ (the -th component of ) for each .

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Let the sample space be:

i.e. the sample space is the set of all real numbers between and . Define a sequence of random variables as follows:

Find the pointwise limit of the sequence .

Solution

For a fixed sample point , the sequence of real numbers has limit

Therefore, the sequence of random variables converges pointwise to the random variable defined as follows:

Exercise 2

Suppose the sample space is as in the previous exercise:

Define a sequence of random variables as follows:

Find the pointwise limit of the sequence .

Solution

For a given sample point , the sequence of real numbers has limit

(note that this limit is encountered very frequently and you can find a proof of it in most calculus textbooks). Thus, the sequence of random variables converges pointwise to the random variable defined as follows:

Exercise 3

Suppose the sample space is as in the previous exercises:

Define a sequence of random variables as follows:

Define a random variable as follows:Does the sequence converge pointwise to the random variable ?

Solution

For , the sequence of real numbers has limitHowever, for , the sequence of real numbers has limitThus, the sequence of random variables does not converge pointwise to the random variable , but it converges pointwise to the random variable defined as follows: [eq84]

How to cite

Please cite as:

Taboga, Marco (2021). "Pointwise convergence", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/asymptotic-theory/pointwise-convergence.