What are the interpretations of these different types of stochastic convergences?

Thanks for passing by! I am studying an introduction to stocastic processes and I found 4 types of stocastic convergences with their definitions as follows:

Let $$(X_n)_{n\geq0}$$ be a sequence of $$r.v.$$ on $$(\Omega,\mathcal{F},\mathcal{P})$$ we say..

1. Almost sure convergence

$$(X_n)_{n\geq0}$$ converges almost surely to the r.v. $$X$$ iff there exists an event $$\tilde{\Omega} \in \mathcal{F}$$ of probability one such that $$$$\forall\, \omega \in \tilde{\Omega}\,,\quad \lim_{n\to\infty}X_n(\omega)=X(\omega)$$$$

1. Convergence in probability

$$(X_n)_{n\geq0}$$ converges in probability to the r.v. $$X$$ iff $$\forall\,\epsilon \gt0$$ : $$$$\lim_{n\to\infty} P(|X_n - X|\gt\epsilon)=0$$$$

1. Convergence in $$L^p$$

$$(X_n)_{n\geq0}$$ converges in $$L^p$$ to the r.v. $$X$$ iff: $$$$\lim_{n\to\infty} ||X_n - X||_p=0$$$$

1. Convergence in Law

$$(X_n)_{n\geq0}$$ converges in law to the r.v. $$X$$ iff: $$$$\lim_{n\to\infty} E[f(X_n)] = E[f(X)]$$$$

This last one having some other formulations in terms of $$F_x$$ and also the Characteristic Function of the distribution.

I have seen youtube videos with demonstrations and read some interesting sources on the internet, but I still can't really get it. I think I understand this theoretical formulations, but I do not arrive to imagine its applications in concrete cases.

Why do they always use "closed" formulas to define the values in the sequence of $$X_n$$? For me it's quite disturbing since I understood that $$X_n$$ is more a sequence of observations/realizations of a r.v. rather that something that is defined by a mathematical formula.

So any type of clarification on this would be extremely appreciated, Thanks a lot in advance

• I found this post stats.stackexchange.com/questions/134701/… by @Silverfish extremely useful in clarifying Convergence in Probability and in Law/Distribution. – nico_so Jul 19 at 21:39
• For convergence in law, the correct definition is "...for all continuous bounded $f: \mathbb{R}\rightarrow \mathbb{R}$". – Michael Jul 20 at 10:46