# Intuition behind various modes of convergence of random variables

There are 5 widely-invoked modes of convergence for a random variable, listed below from strongest to weakest:

### 1. Complete convergence

$\sum_{n=1}^{\infty} P \left( | X_n - X | > \epsilon \right) < \infty$

### 2. Almost sure convergence

$P \left( \lim_{n \to \infty} X_n = X \right) = 1$

### 3. Convergence in $r^{th}$ mean

$\lim_{n \to \infty} E \big[ \;| X_n - X |^r \; \big] = 0$

### 4. Convergence in probability

$\lim_{n \to \infty} P \left( | X_n - c | > \epsilon \right) = 0$

### 5. Convergence in distribution

$\lim_{n \to \infty} F_{X_n} = F_{X}$ at all continuity points

I'm seeking (to whatever extent possible) intuitive motivations for, and connections between, these modes of convergence. I know how to prove their interrelationships formally, but I find these concepts rather abstract and would like to develop a more well-rounded understanding. I often find visual and/or simulation-based intuitions especially helpful, but use your creative pedagogy.