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Convergence in probability - Exercise set 1

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

Exercise 1.1

Let $U$ be a random variable having a uniform distribution on the interval $left[ 0,1ight] $. In other words, $U$ is an absolutely continuous random variable with support:[eq1]and probability density function:[eq2]Now, define a sequence of random variables [eq3] as follows:[eq4]where [eq5] is the indicator function of the event [eq6].

Find the probability limit (if it exists) of the sequence [eq7].

nav_button Solution

A generic term X_n of the sequence, being an indicator function, can take only two values:

By the previous inequality, $m$ goes to infinity as n goes to infinity and:[eq12]Therefore, the probability that X_n is equal to zero converges to 1 as n goes to infinity. So, obviously, [eq3] converges in probability to the constant random variable[eq14]because for any $ arepsilon >0$[eq15]

Exercise 1.2

Does the sequence in the previous exercise also converge almost surely?

nav_button Solution

We can identify the sample space Omega with the support of $U$:[eq16]and the sample points omega in Omega with the realizations of $U$: i.e. when the realization is $U=u$, then $omega =u$. Almost sure convergence requires that[eq17]where E is a zero-probability event and the superscript $c$ denotes the complement of a set. In other words, the set of sample points omega for which the sequence [eq18] does not converge to [eq19] must be included in a zero-probability event E. In our case, it is easy to see that, for any fixed sample point [eq20], the sequence [eq21] does not converge to [eq22], because infinitely many terms in the sequence are equal to 1. Therefore:[eq23]and, trivially, there does not exist a zero-probability event including the set [eq24]Thus, the sequence does not converge almost surely to X.

Exercise 1.3

Let [eq3] be an IID sequence of continuous random variables having a uniform distribution with support:[eq26]and probability density function:[eq27]

Find the probability limit (if it exists) of the sequence [eq7].

nav_button Solution

As n tends to infinity, the probability density tends to become concentrated around the point $x=0$. Therefore, it seems reasonable to conjecture that the sequence [eq3] converges in probability to the constant random variable[eq14]To rigorously verify this claim we need to use the formal definition of convergence in probability. For any $ arepsilon >0$:[eq31]

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