Pointwise convergence - Exercise set 1

This exercise set contains some solved exercises on pointwise convergence. The theory needed to solve these exercises is introduced in the lecture entitled Pointwise convergence.

Exercise 1.1

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 1.2

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 1.3

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 limit:However, for , the sequence of real numbers has limit: Thus, the sequence of random variables does not converge pointwise to the random variable , but it converges pointwise to the random variable defined as follows: [eq28]

by Marco Taboga

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