Overview

In this section, we study the different ways a sequence of random variables can converge to a limit. This is central to probability theory, as it allows us to approximate complex random systems with simpler ones (like the General Limit Theorems).

Key Topics

1. Modes of Convergence: Almost sure, in probability, in $L^p$, and in distribution.
2. Convergence Hierarchy: How the different modes of convergence relate to each other.
3. Weak Convergence: A deep dive into convergence in distribution, the weakest but arguably most useful mode.
4. Skorohod’s Theorem: Relates weak convergence to almost sure convergence via a change of probability space.
5. Portmanteau Theorem: Equivalent definitions of weak convergence involving open/closed sets and bounded continuous functions.
6. Helly’s Selection Theorem: A compactness result for sequences of distribution functions.
7. Tightness Condition: Crucial condition for the existence of convergent subsequences.
8. Continuity Theorem: The link between weak convergence of measures and pointwise convergence of characteristic functions (Lévy’s Continuity Theorem).