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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 \begin{equation} \forall\, \omega \in \tilde{\Omega}\,,\quad \lim_{n\to\infty}X_n(\omega)=X(\omega) \end{equation}

  1. Convergence in probability

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

  1. Convergence in $L^p$

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

  1. Convergence in Law

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

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

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

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