What Is Convergence in Probability with Examples?

TL;DR

Convergence in probability occurs when the sequence of random variables approaches a limit — typically zero. For example, a sequence of discrete random variables can converge to zero while its expectation diverges to infinity, highlighting that convergence in probability does not guarantee convergence of expectations. Another example shows that the minimum of identically distributed random variables also converges to zero.

Transcript

We will now go through two examples of convergence in probability. Our first example is quite trivial. We're dealing with a sequence of random variables Yn that are discrete. Most of the probability is concentrated at 0. But there is also a small probability of a large value. Because the bulk of the probability mass is concentrated at 0, it is a go... Read More

Key Insights

• 🍸 Convergence in probability focuses on the bulk of the distribution, while the expectation is sensitive to the tail of the distribution.
• ❓ The first example illustrates that convergence in probability does not imply convergence of expectations.
• 🛀 The second example shows the convergence in probability of the minimum values of identically distributed random variables to 0.
• ⛔ When proving convergence in probability, the first step is to make a conjecture about the limit and then calculate the probability of being epsilon away from that conjectured limit.

Questions & Answers

Q: How is convergence in probability different from convergence of expectations?

Convergence in probability focuses on the bulk of the distribution and only requires that the tail of the distribution has a small probability. It does not guarantee convergence of expectations. The expectation is sensitive to the tail of the distribution and can be significantly different from the limit of the random variable.

Q: Can you explain the first example of convergence in probability?

In the first example, a sequence of discrete random variables, Yn, is considered. The majority of the probability is concentrated at 0, with a small probability of a large value. Checking the definition of convergence in probability, it is determined that the sequence converges to 0 because the probability of Yn being epsilon or more away from 0 goes to 0 as n goes to infinity.

Q: Why is the expectation of the first example different from the limit of the random variable?

While the sequence of random variables, Yn, converges to 0 in probability, the expectation does not converge to 0. This is because even though the bulk of the distribution is concentrated at 0, there is a small probability assigned to a very large value (n squared) which significantly impacts the expectation and causes it to go to infinity.

Q: How is convergence in probability verified in the second example?

In the second example, the random variables are independent and identically distributed, with a common distribution uniform on the unit interval. The random variable Yn, defined as the minimum of the first n X's, is conjectured to converge to 0. To verify this, the probability that Yn is larger than or equal to epsilon is calculated. It is shown that this probability goes to 0 for any positive epsilon, confirming the convergence in probability of Yn to 0.

Summary & Key Takeaways

• The first example deals with a sequence of discrete random variables, where the bulk of the probability mass is concentrated at 0. The sequence is shown to converge in probability to 0, but the expectation does not converge to 0 and goes to infinity.

• The second example explores the minimum values of identically distributed random variables. The random variable Yn is defined as the minimum value of the first n X's. It is conjectured that Yn converges to 0, and this is verified by calculating the probability that Yn is larger than or equal to epsilon and showing that it converges to 0.