Advanced Probability Theory Seminar: Convergence and Large Deviations

On October 24, 2025, I had the pleasure of delivering a seminar on Advanced Probability Theory to an engaged group of students at the school of Mathematical Sciences, Zhejiang University.

Seminar Focus

The session centered around key concepts from Probability Theory and Examples (PTE) by Durrett. I delved into two critical areas:

1. Section 2.5: Convergence of Random Series
2. Section 2.7: Large Deviations

Key Takeaways

This is my first time leading a discussion class. Although it was exhausting, I gained a lot. I have deeper insights into the mathematical underpinnings of probability theory, especially in the context of random series and large deviations. And I hope to inspire others to explore these ideas further.

I first introduced the concept of the tail sigma-algebra, explained Kolmogorov’s 0-1 law and the Hewitt-Savage 0-1 law, and then we discussed the convergence of series of independent and identically distributed random variables as well as issues of infinite expectation.

Then, we began to focus on the theory of large deviations, discussing the probability convergence of the sum of random variables being greater than or equal to n times the expectation and a series of related properties.

Lecture Notes

You can find my Lecture Notes here: Advanced Probability Theory Seminar: Convergence and Large Deviations.