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I have proved that a sequence of random variables $(M_n)_{n\in\mathbb N}$ diverges to $+\infty$ almost surely. I.e I have proved that $$\bigcap_{c\in\mathbb Q^+}\bigcup_{N=1}^{\infty}\bigcap_{n=N}^\infty\{M_n>c\}$$ has probability 1 thanks to Borel-Cantelli lemma.

Is that enough to imply that those variables are positive almost surely ?

Thank you!

EDIT : My question actually is, "Is that enough to imply that those variables are positive almost surely for sufficiently large $n$ ?"

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  • $\begingroup$ No, the sequence can have first few values negative. $\endgroup$ – Kavi Rama Murthy Sep 14 '18 at 9:32
  • $\begingroup$ If we take a sequence of sets $(A_n)_{n=1}^\infty$ with $A_1\supset A_2\supset \ldots$, $\mathbb{P}(A_n)>0$, and $\mathbb{P}(A_n)\to 0$, then $M_n:= -1_{A_n}+ n 1_{A_n^c}$ tends to $+\infty$ a.s. but is negative on $A_n$. $\endgroup$ – bangs Sep 14 '18 at 9:33
  • $\begingroup$ I just edited my question : does it hold that for sufficiently large $n$ the $M_n$ are positive almost surely ? $\endgroup$ – A Ht Sep 14 '18 at 9:49
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    $\begingroup$ Not necessarily for $n$ large enough not random, but there exists some almost surely finite random variable $N$ such that $M_n>0$ for every $n>N$, almost surely. $\endgroup$ – Did Sep 14 '18 at 10:09

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