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Regarding the second line of the first proof by @JDL in Altenate definitions of almost sure convergence

Are the following equivalent?

$P(\omega:\exists n\in\mathbf{N}, \forall m\in\mathbf{N}, \exists i>m \,\,\, \text{s.t.} \,\, |X_i(\omega) - X(\omega)| < 1/n)$

AND

$P(\omega:\forall n\in\mathbf{N}, \exists m\in\mathbf{N},\ \text{s.t.}\ \forall i>m \,\, |X_i(\omega) - X(\omega)| \ge 1/n)$

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    $\begingroup$ FYI, the answer you refer to is certainly not the most direct nor the clearest one that one can imagine, and it confuses several times the union of events $\{Y_n\geqslant y\}$ with the event $\{\sup Y_n\geqslant y\}$. Even if one denotes unions as supremums, it is not true that $$\sup_n\{Y_n\geqslant y\}=\{\sup_nY_n\geqslant y\}$$ $\endgroup$ – Did Oct 11 '16 at 6:43
  • $\begingroup$ Re the current page, I mention that there is no question in your post since the highlighted formulas are not statements. $\endgroup$ – Did Oct 11 '16 at 6:45
  • $\begingroup$ Finally, if what you are interested in is the result on the other page, not the suboptimal approaches proposed there to prove it, you should say so and I might go as far as posting a two-lines complete, purely real-analysis proof (since there is not an ounce of probability involved in the result, actually). $\endgroup$ – Did Oct 11 '16 at 6:51
  • $\begingroup$ @Did That would be the most helpful. Should I post a new question? I am actually concerned about the equivalence of the two definitions. $\endgroup$ – Qwerty Oct 11 '16 at 9:00
  • $\begingroup$ @Did Can you please help? $\endgroup$ – Qwerty Oct 11 '16 at 12:30
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The original question asks to prove the equivalence: $$P[\omega:\lim_{n\to\infty}X_n(\omega) = X(\omega)] = 1\iff\lim_{n\to\infty}P[\omega:\sup_{k>n}|X_k(\omega) - X(\omega)|>\epsilon] = 0$$

Adding the missing "for every positive $\epsilon$" on the RHS and correcting the notations on both sides, one should show the equivalence of the properties:

  • $P(A)=1$ where $A=[\lim\limits_{n\to\infty}X_n=X]$
  • $\lim\limits_{n\to\infty}P(A_n^\epsilon)=1$ for every positive $\epsilon$, where $A_n^\epsilon=[Y_n\leqslant\epsilon]$ and $Y_n=\sup\limits_{k>n}|X_k-X|$

To show this equivalence, first note that the sequence of random variables $(Y_n)$ is nonincreasing, hence $$A=\bigcap_{\epsilon>0}[\lim\limits_{n\to\infty}Y_n\leqslant\epsilon]=\bigcap_{\epsilon>0}[\exists n,Y_n\leqslant\epsilon]=\bigcap_{\epsilon>0}\bigcup_nA_n^\epsilon$$ Now, for every positive $\epsilon$, the sequence of events $(A_n^\epsilon)$ is nondecreasing, hence $$P(A)=\inf_{\epsilon>0}P\left(\bigcup_nA_n^\epsilon\right)=\inf_{\epsilon>0}\left(\lim_{n\to\infty}P(A_n^\epsilon)\right)$$ which proves the desired equivalence. Note that the last identity above nowhere uses the hypothesis that $P(A)=1$, thus, for every $p$ in $[0,1]$, $$P(A)\geqslant p\iff\forall\epsilon>0,\lim\limits_{n\to\infty}P(A_n^\epsilon)\geqslant p$$


Nota bene: The step $$\bigcap_{\epsilon>0}[\lim\limits_{n\to\infty}Y_n\leqslant\epsilon]=\bigcap_{\epsilon>0}[\exists n,Y_n\leqslant\epsilon]$$ is trickier than may appear at first sight, since the identity $$[\lim\limits_{n\to\infty}Y_n\leqslant\epsilon]=[\exists n,Y_n\leqslant\epsilon]$$ is false for some fixed $\epsilon$ (consider $Y_n=\epsilon+\frac1n$). But the following inclusions save the day: $$[\lim\limits_{n\to\infty}Y_n\leqslant\epsilon]\subseteq[\exists n,Y_n\leqslant2\epsilon]\subseteq[\lim\limits_{n\to\infty}Y_n\leqslant2\epsilon]$$ since indeed they imply that $$\bigcap_{\epsilon>0}[\lim\limits_{n\to\infty}Y_n\leqslant\epsilon]\subseteq\bigcap_{\epsilon>0}[\exists n,Y_n\leqslant2\epsilon]\subseteq\bigcap_{\epsilon>0}[\lim\limits_{n\to\infty}Y_n\leqslant2\epsilon]=\bigcap_{\epsilon>0}[\lim\limits_{n\to\infty}Y_n\leqslant\epsilon]$$ Another option is to use strict inequalities, since $$[\lim\limits_{n\to\infty}Y_n<\epsilon]=[\exists n,Y_n<\epsilon]$$

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  • $\begingroup$ Can you help me understand the followoing things better? 1>$P(A)=\inf_{\epsilon>0}P\left(\bigcup_nA_n^\epsilon\right)$ 2>$[\lim\limits_{n\to\infty}Y_n\leqslant\epsilon]\subseteq[\exists n,Y_n\leqslant2\epsilon]$. I am not much experienced in handling limits (sups and infs) with inclusions/exclusions. $\endgroup$ – Qwerty Oct 12 '16 at 4:32
  • $\begingroup$ Not sure what you are asking really, partly because of your vague formulations. $\endgroup$ – Did Oct 12 '16 at 5:41
  • $\begingroup$ I was asking, for e.g. how to get that $[\lim_{n\to \infty} Y_n\le \epsilon]$ a subset of $[\exists n, Y_n\le 2\epsilon]$. I am not getting the intuition. $\endgroup$ – Qwerty Oct 12 '16 at 6:00
  • $\begingroup$ Well, if $Y_n>2\epsilon$ for every $n$, do you think that $\lim Y_n\leqslant\epsilon$ can happen? Nothing probabilistic there, just plain real analysis. $\endgroup$ – Did Oct 12 '16 at 7:45
  • $\begingroup$ I need more help. Can you please show the steps you jumped in going from $[\lim\limits_{n\to\infty}X_n=X]$ to $\bigcap\limits_{\epsilon>0}[\lim\limits_{n\to\infty}\sup_{k>n}|X_k-X|\le\epsilon]$ ? $\endgroup$ – Qwerty Oct 12 '16 at 10:07

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