Talk:Kolmogorov's zero–one law
There is a minor sloppyness in language. The term is the limit of the series .
- That's not a series; that's a sequence! Michael Hardy 00:36, 1 Nov 2004 (UTC)
The later converges or not, the prior exists or not.
- What does it mean for a sum of random variables not to exist? I think you've misunderstood the point.
Don't know how to correct this without an overhead of explanation.
Btw., what are the exact requirements on for the series to converge at all? Certainly is needed, but not sufficient. I even believe the convergences of the series would be equivalent to .
- No on both accounts. See eg. the central limit theorem.
- ouch, now I see where I was wrong: in many theorems the random variables have to be identically distributed, not so here. 18.104.22.168 12:16, 31 Oct 2004 (UTC)
How about a rigorous definition for a tail event or tail sigma algebra? —Preceding unsigned comment added by 22.214.171.124 (talk) 07:48, 23 April 2009 (UTC)
Are and the same or do I miss something? Pr.elvis (talk) 14:15, 10 August 2009 (UTC)
A couple of issues I have with this article
First, I think you need to say more about the definition of a tail event, because consider:
Let be i.i.d. Bernoulli random variables on satisfying . Define as the set of all such that . Using the axiom of choice, choose a family of subsets of such that for each there exists unique such that the symmetric difference of and is finite. Now let be a subset of of cardinality whose complement is also of cardinality . Let be the set . Let be the σ-algebra on generated by the and let be the σ-algebra generated by . An arbitrary element of can be expressed in the form where . Define the probability of such an event to be . Then the event is independent of all finite subsets of the and is uniquely determined by the 'tail' for any , but has probability 1/2.
The reason why this doesn't contradict the theorem is because the theorem should only be stated for events belonging to .
Another issue is that you define a tail event as being one that is independent of all finite subsets of the , but this itself follows from the hypothesis that the are independent. In fact, under the article's definition of 'tail event', surely we don't even need to assume that the are independent for the result to be true (provided it's only stated for events in )? —Preceding unsigned comment added by 126.96.36.199 (talk) 10:38, 6 August 2010 (UTC)