Intuition behind various modes of convergence of random variables

Question

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.

Answer 1

I feel like this question has been largely addressed on the web. I will limit myself to post three videos on which the intuition behind the convergence of random variables is presented. Of course, someone braver, more energetic, and with more imagination than me may be willing to spend some time on writing yet another explanation:

Convergence in Random Variables (part 1/3)

Convergence in Random Variables (part 2/3)

Convergence of random variables (part 3/3)