# Almost sure convergence

This lecture introduces the concept of almost sure convergence. In order to understand this lecture, you should first understand the concepts of almost sure property and almost sure event, explained in the lecture entitled Zero-probability events, and the concept of pointwise convergence of a sequence of random variables, explained in the lecture entitled Pointwise convergence.

## Almost sure convergence of a sequence of random variables

Let be a sequence of random variables defined on a sample space . The concept of almost sure convergence (or a.s. convergence) is a slight variation of the concept of pointwise convergence. As we have seen, a sequence of random variables is pointwise convergent if and only if the sequence of real numbers is convergent for all . Achieving convergence for all is a very stringent requirement. Therefore, this requirement is usually weakened, by requiring the convergence of for a large enough subset of , and not necessarily for all . In particular, is usually required to be a convergent sequence almost surely: if is the set of all sample points for which the sequence is convergent, its complement must be included in a zero-probability event:In other words, almost sure convergence requires that the sequences converge for all sample points , except, possibly, for a very small set of sample points ( must be included in a zero-probability event). This is summarized by the following definition.

Definition Let be a sequence of random variables defined on a sample space . We say that is almost surely convergent (a.s. convergent) to a random variable defined on if and only if the sequence of real numbers converges to almost surely, i.e., if and only if there exists a zero-probability event such that is called the almost sure limit of the sequence and convergence is indicated by

The following is an example of a sequence that converges almost surely.

Example Suppose the sample space isIt is possible to build a probability measure on , such that assigns to each sub-interval of a probability equal to its length:(see the lecture entitled Zero-probability events). Remember that in this probability model all the sample points are assigned zero probability (each sample point, when considered as an event, is a zero-probability event):Now, consider a sequence of random variables defined as follows:When , the sequence of real numbers converges to becauseHowever, when , the sequence of real numbers is not convergent to becauseDefine a constant random variable as follows: We have thatBut because which means that the eventis a zero-probability event. Therefore, the sequence converges to almost surely. Note, however, that does not converge pointwise to because does not converge to for all .

## Almost sure 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 . Also in the case of random vectors, the concept of almost sure convergence is obtained from the concept of pointwise convergence by relaxing the assumption that the sequence converges for all . Remember that the sequence of real vectors converges to a real vector if and only if Instead, it is required that the sequence converges for almost all (i.e., almost surely).

Definition Let be a sequence of random vectors defined on a sample space . We say that is almost surely convergent to a random vector defined on if and only if the sequence of real vectors converges to the real vector almost surely, i.e., if and only if there exists a zero-probability event such that is called the almost sure 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 almost surely convergent if and only if all the sequences of random variables are almost surely 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 almost surely to the random vector if and only if converges almost surely to the random variable (the -th component of ) for each .

## Solved exercises

Below you can find some exercises with explained solutions.

### Exercise 1

Let the sample space bei.e. the sample space is the set of all real numbers between 0 and 1. Sub-intervals of are assigned a probability equal to their length:

Define a sequence of random variables as follows:

Define a random variable as follows:

Does the sequence converge almost surely to ?

Solution

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

For , the sequence of real numbers has limit

Therefore, the sequence of random variables does not converge pointwise to becausefor . However, the set of sample points such that does not converge to is a zero-probability event: Therefore, the sequence converges almost surely to .

### Exercise 2

Let and be two sequences of random variables defined on a sample space . Let and be two random variables defined on such that

Prove that

Solution

Denote by the set of sample points for which converges to :The fact that converges almost surely to implies thatwhere .

Denote by the set of sample points for which converges to :The fact that converges almost surely to implies thatwhere .

Now, denote by the set of sample points for which converges to :

Observe that if then converges to , because the sum of two sequences of real numbers is convergent if the two sequences are convergent. Therefore,Taking the complement of both sides, we obtainBut and as a consequence . Thus, the set of sample points such that does not converge to is included in the zero-probability event , which means that

### Exercise 3

Let the sample space bethat is, the sample space is the set of all real numbers between 0 and 1. Sub-intervals of are assigned a probability equal to their length:

Define a sequence of random variables as follows:

Find an almost sure limit of the sequence.

Solution

If or , then the sequence of real numbers is not convergent:

For , the sequence of real numbers has limitbecause for any we can find such that for any (as a consequence for any ).

Thus, the sequence of random variables converges almost surely to the random variable defined asbecause the set of sample points such that does not converge to is a zero-probability event:

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