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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

Let [eq1] be a sequence of random variables defined on a sample space Omega. Let us consider a single sample point omega in Omega and a generic random variable X_n belonging to the sequence. X_n is a function [eq2]. However, once we fix omega, the realization [eq3] associated to the sample point omega is just a real number. By the same token, once we fix omega, the sequence [eq4] is just a sequence of real numbers. Therefore, for a fixed omega, it is very easy to assess whether the sequence [eq5] is convergent; this is done employing the usual definition of convergence of sequences of real numbers. If, for a fixed omega, the sequence [eq6] is convergent, we denote its limit by [eq7], to underline that the limit depends on the specific omega we have fixed. A sequence of random variables is said to be pointwise convergent if and only if the sequence [eq8] is convergent for any choice of omega.

Definition Let [eq1] be a sequence of random variables defined on a sample space Omega. We say that [eq1] is pointwise convergent to a random variable X defined on Omega if and only if [eq11] converges to [eq12] for all omega in Omega. X is called the pointwise limit of the sequence and convergence is indicated by[eq13]

Roughly speaking, using pointwise convergence we somehow circumvent the problem of defining the concept of distance between random variables: by fixing omega, 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 [eq14] be a sample space with two sample points ($omega _{1}$ and $omega _{2}$). Let [eq15] be a sequence of random variables such that a generic term X_n of the sequence satisfies[eq16]We need to check the convergence of the sequences [eq17] for all omega in Omega, i.e. for [eq18] and for [eq19]: (1) the sequence [eq20], whose generic term is [eq21] is a sequence of real numbers converging to 0; (2) the sequence [eq22], whose generic term is [eq23] is a sequence of real numbers converging to 1. Therefore, the sequence of random variables [eq1] converges pointwise to the random variable X, where X 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 [eq1] be a sequence of random vectors defined on a sample space Omega, where each random vector X_n has dimension Kx1. If we fix a single sample point omega in Omega, the sequence [eq27] is a sequence of real Kx1 vectors. By the standard criterion for convergence, the sequence of real vectors [eq28] is convergent to a vector [eq29] if [eq30]where [eq31] is the distance between a generic term of the sequence [eq32] and the limit [eq29]. The distance between [eq34] and [eq29] 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 [eq37] and [eq29]. Thus, for a fixed omega, the sequence of real vectors [eq39] is convergent to a vector [eq40] if [eq41]

A sequence of random vectors [eq1] is said to be pointwise convergent if and only if the sequence [eq43] is convergent for any choice of omega.

Definition Let [eq1] be a sequence of random vectors defined on a sample space Omega. We say that [eq1] is pointwise convergent to a random vector X defined on Omega if and only if [eq11] converges to [eq12] for all omega in Omega (i.e. $orall omega ,$ [eq48]). X is called the pointwise limit of the sequence and convergence is indicated by[eq49]

Now, denote by [eq50] the sequence of the i-th components of the vectors X_n. It can be proved that the sequence of random vectors [eq1] is pointwise convergent if and only if all the K sequences of random variables [eq50] are pointwise convergent:

Proposition Let [eq1] be a sequence of random vectors defined on a sample space Omega. Denote by [eq50] the sequence of random variables obtained by taking the i-th component of each random vector X_n. The sequence [eq1] converges pointwise to the random vector X if and only if [eq50] converges pointwise to the random variable $X_{ullet ,i}$ (the i-th component of $X $) for each $i=1,ldots ,K$.

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Let the sample space Omega be:[eq57]

i.e. the sample space Omega is the set of all real numbers between 0 and 1. Define a sequence of random variables [eq1] as follows:[eq59]

Find the pointwise limit of the sequence [eq1].

Solution

For a fixed sample point omega, the sequence of real numbers [eq61] has limit[eq62]

Therefore, the sequence of random variables [eq1] converges pointwise to the random variable X defined as follows:[eq64]

Exercise 2

Suppose the sample space Omega is as in the previous exercise:[eq65]

Define a sequence of random variables [eq1] as follows:[eq67]

Find the pointwise limit of the sequence [eq1].

Solution

For a given sample point omega, the sequence of real numbers [eq61] has limit[eq70]

(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 [eq1] converges pointwise to the random variable X defined as follows:[eq72]

Exercise 3

Suppose the sample space Omega is as in the previous exercises:[eq73]

Define a sequence of random variables [eq1] as follows:[eq75]

Define a random variable X as follows:[eq76]Does the sequence [eq1] converge pointwise to the random variable X?

Solution

For [eq78], the sequence of real numbers [eq79] has limit[eq80]However, for $omega =1$, the sequence of real numbers [eq17] has limit[eq82]Thus, the sequence of random variables [eq1] does not converge pointwise to the random variable X, but it converges pointwise to the random variable Y defined as follows:[eq84]

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