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How a sequence of random variables $X_n$ can converge in probability to a random variable $X$, but not converge pointwise for any point of the domain?

A sequence of random variables $X_n$ converges in probability to $X$ if for all $\epsilon > 0$ $$\lim_{n\rightarrow \infty}\mathbb{P}(|X_n-X|\leq \epsilon)=1$$

Intuitively, $X_n$ is considered close to $X$ when $|X_n-X|\leq \epsilon$; therefore, $\mathbb{P}(|X_n-X|\leq \epsilon)$ is the probability that $X_n$ is close to $X$. But I can't see how to get any information for pointwise convergence from that.

Besides, if the random variables viewed as functions which are not converging pointwise then what does the convergence refer to?

It will be great help if anyone provide a solution which will be self-contained, as I am learning this subject (measure-theoretic probability) by my own.

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  • $\begingroup$ your question has been studied before here at MSE. One nice posting that is the following one $\endgroup$ Jul 8 at 13:30
  • $\begingroup$ Nice thread @OliverDiaz. Could you help me with that " if the random variables viewed as functions which are not converging pointwise then what does the convergence refer to?" $\endgroup$
    – falamiw
    Jul 8 at 13:58
  • $\begingroup$ The stance of real valued random variables (measurable functions) on a probability space $(\Omega,\mathscr{F},\mathbb{P})$, denoted by $L_0(\mathbb{P}$), admits a pseudo metric $d$ given by $d(X,Y)=\mathbb{E}[\min(1,|X-Y|)$. Under this pseudo metric, convergence in probability is equivalent to convergence in $d$. $\endgroup$ Jul 8 at 14:13
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    $\begingroup$ Convergence in probability means that for every epsilon, the measure of the sets where $X_n$ is far away from its limit tends to zero. These measurable sets need not be related (pointwise), they just need to small. This is how the counterexamples are constructed anyway. $\endgroup$
    – Will M.
    Jul 8 at 14:30
  • $\begingroup$ nice intuitive explanation @WillM. Could you explain convergence almost surely like that? $\endgroup$
    – falamiw
    Jul 8 at 14:35
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Here is a simple counterexample. Let $\{X_n\}_n$ be a sequence of independent rv's taking only the values 0 and 1 with probability $\left\{1-\frac{1}{n};\frac{1}{n}\right\}$ respectively.

It is evident that $X_n\xrightarrow{\mathcal{P}}0$ but, using B.C. II you get

$$\sum_{n=1}^{\infty}\mathbb{P}[|X_n|>\epsilon]=\sum_{n=1}^{\infty}\frac{1}{n}=\infty$$

Thus the sequence does not converge a.s. and thus it does not even converge pointwise

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  • $\begingroup$ Aha, I forgot Borel–Cantelli lemma, thanks, @tommik. Could you help me with this "if the random variables viewed as functions which are not converging pointwise then what does the convergence refer to?" I was really struggling the convergence of probability theory analog with function convergence. Like what the analog of almost surely convergence in function? $\endgroup$
    – falamiw
    Jul 8 at 13:17
  • $\begingroup$ @falamiw Convergence in probability is convergence in measure. Convergence almost surely is convergence almost everywhere. See this page of Folland's book that has a table of translations: google.com/… $\endgroup$
    – Mason
    Jul 8 at 22:08
  • $\begingroup$ Thanks, @Mason. It clear some of my confusion. Actually I am studying this subject by my own which seems pretty hard. Because I haven’t taken any measure theory course, All I did is real analysis 🙂. It will be great help if you can suggest me a book where theories are explained in details. $\endgroup$
    – falamiw
    Jul 9 at 6:49
  • $\begingroup$ @falamiw "Measure Theory and Integration" by Michael Taylor and "Real Analysis: Modern Techniques and Their Applications" By Gerald Folland are good for me so far. I am studying by myself. I use Taylor's book as the main text, and read Folland to clear my doubts and get another perspective. $\endgroup$
    – Mason
    Jul 9 at 18:58
  • $\begingroup$ Thanks, @Mason. I really like "Real Analysis: Modern Techniques and Their Applications" By Gerald Folland. As it go through all the topics which I need to learn. $\endgroup$
    – falamiw
    Jul 10 at 16:23

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