# Questions tagged [borel-cantelli-lemmas]

For questions involving the Borel-Cantelli lemma or the second Borel-Cantelli lemma. Use this tag along with (probability-theory), (real-analysis) or (measure-theory).

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### Borel Cantelli Lemma Lim-sup Question

I am stuck on this one step in the proof of the first Borel-Cantelli Lemma. We have infinite $A_1, A_2, \ldots$ where the sum of their probabilities is finite. (convergence) Let $B$ be the event that ...
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### Expected value of a series

While learning about the Borel-Cantelli Lemma, I came across this article and I am having trouble following the Introduction. I cite the relevant text here: Consider an infinite sequence of games, ...
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### Understanding this passage in Borel Cantelli Lemma N.2

I'm trying to understand a passage in the proof of Borel Cantelli Lemma 2. Be $(\Omega, \mathcal{F}, P)$ a probability space and $(A_n)$ a sequence in $\mathcal{F}$ such that $(A_n)$ are pair ...
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### Almost surely positive variables converging in law to an almost surely positive variable

I've been having some trouble with the following question: Let $X_1,X_2,...$ be almost surely positive random variables converging in distribution. Can you always find reals $(c_n)_{n\geq 1}$ such ...
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### Borel Cantelli and convergence almost surely

I was wondering if it's possible to use the Borel Cantelli Theorem in order to ensure that the almost sure convergence DOESN'T exist. We know that: Let $A_n$ be a sequence of events in a probability ...
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### On the Borel-Cantelli lemma - Is this argumentation valid?

In order to show that a $X_n$ converges to $X$ almost surely, one can often use the Borel-Cantelli lemma. I suppose it is a sufficient but not necessary condition for almost sure convergence. (?) Let'...
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