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.


1 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)

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    $\begingroup$ Have a look at our guide on how to write a good answer. This is really rather short by our standards and mostly consists of links to external sites. If these go dead (for instance, if the creator removes these videos from Youtube) then your answer will suffer from "linkrot". For that reason we prefer it if you could expand on this a little to include at least a brief summary of what the links are saying? $\endgroup$
    – Silverfish
    Aug 25, 2016 at 22:20
  • $\begingroup$ I like these videos very much, but they only cover what I would consider the three most intuitive modes (in probability, in distribution, and almost sure). Can anything be said to tie the other two into the framework? $\endgroup$
    – half-pass
    Sep 3, 2016 at 12:38

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