# Relation between Borel–Cantelli lemmas and Kolmogorov's zero-one law

I was wondering what is the relation between the first and second Borel–Cantelli lemmas and Kolmogorov's zero-one law?

The former is about limsup of a sequence of events, while the latter is about tail event of a sequence of independent sub sigma algebras or of independent random variables. Both have results for limsup/tail event to have either probability 0 or 1. I guess there are relations between but cannot identify them.

Can the former be viewed as a special case of the latter? How about in reverse direction?

Thanks and regards!

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## 1 Answer

The first Borel-Cantelli is simply the fact that the probability of a union is at most the sum of the probabilities; it has nothing in common with Kolmogorov zero-one law.

What Kolmogorov zero-one law tells you in the setting of the second Borel-Cantelli lemma is that the probability of the limsup is $0$ or $1$, because (1) the limsup is always in the tail $\sigma$-algebra, (2) you are considering independent events, hence their tail $\sigma$-algebra is trivial. Then the second Borel-Cantelli lemma itself tells you that this probability is in fact $1$ under the non summability condition which you know.

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Thanks! Could you elaborate (1) why "considering independent events, hence their tail σ-algebra is trivial"? What is their tail σ-algebra? (2) why "this probability is in fact 1 under the (non) summability condition"? Doesn't the second Borel-Cantelli lemma only consider nonsummarbility condition which implies probability 1? Does it consider nonsummability condition? – Tim Feb 25 '11 at 15:10
@Tim: (1) The definition of a tail event is on the page you link to. (2) Doesn't the second Borel-Cantelli lemma only consider nonsummability condition which implies probability $1$? Yes, this is the statement of second BC lemma. Does it consider nonsummability condition? Same question, same answer. Sorry but I fail to see what you want me to add to what I wrote in my post. – Did Feb 25 '11 at 16:47
Thanks! For (2), I made a typo, in my comment, but after your update, I don't have doubts any more. For (1), what is the tail σ-algebra here (not asked for definition)? I want to see how trivial it is. Thanks! – Tim Feb 25 '11 at 16:57
@Tim: As the WP page you linked to explains, the tail $\sigma$-algebra of a sequence of $\sigma$-algebras $(\mathcal{F}_n)_n$ is the $\sigma$-algebra $\bigcap_{n=1}^\infty G_n$, where each $G_n$ is the $\sigma$-algebra generated by $\bigcup_{k=n}^\infty \mathcal{F}_k$. – Did Feb 25 '11 at 17:04