Talk:Kolmogorov's zero–one law

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There is a minor sloppyness in language. The term $\sum _{k=1}^{\infty }X_{k}$ is the limit of the series ${\big (}\sum _{k=1}^{N}X_{k}{\big )}_{N=1}^{\infty }$.

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 $X_{k}^{}$ for the series to converge at all? Certainly $\operatorname {E} (X_{k})=0$ is needed, but not sufficient. I even believe the convergences of the series would be equivalent to $\operatorname {P} \{X_{k}=0\}=1$.

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. 217.230.28.82 12:16, 31 Oct 2004 (UTC)

How about a rigorous definition for a tail event or tail sigma algebra? —Preceding unsigned comment added by 71.207.219.120 (talk) 07:48, 23 April 2009 (UTC)

Are $F_{k}$ and $B_{k}$ 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 $X_{i}(\omega )\ :\ i\in \mathbb {N}$ be i.i.d. Bernoulli random variables on $\Omega$ satisfying $\operatorname {P} (X_{i}=1)=1/2$. Define $I_{X}\subset \mathbb {N}$ as the set of all $i\in \mathbb {N}$ such that $X_{i}=1$. Using the axiom of choice, choose a family ${\mathcal {A}}$ of subsets of $\mathbb {N}$ such that for each $B\subset \mathbb {N}$ there exists unique $A\in {\mathcal {A}}$ such that the symmetric difference of $A$ and $B$ is finite. Now let ${\mathcal {A}}'$ be a subset of ${\mathcal {A}}$ of cardinality $2^{\aleph _{0}}$ whose complement is also of cardinality $2^{\aleph _{0}}$. Let $Z$ be the set $\{\omega \in \Omega :\exists A\in {\mathcal {A}}'{\textrm {s.t.}}\,I_{X}\bigtriangleup A\,{\textrm {is}}\,{\textrm {finite}}\}$. Let ${\mathcal {F}}$ be the σ-algebra on $\Omega$ generated by the $X_{i}$ and let ${\mathcal {G}}$ be the σ-algebra generated by ${\mathcal {F}}\cup \{Z\}$. An arbitrary element of ${\mathcal {G}}$ can be expressed in the form $(A\cap Z)\cup (B\cap Z^{c})$ where $A,B\in {\mathcal {F}}$. Define the probability of such an event to be $(\mathop {P} (A)+\mathop {P} (B))/2$. Then the event $Z$ is independent of all finite subsets of the $X_{i}$ and is uniquely determined by the 'tail' $(X_{n},X_{n+1},X_{n+2},\ldots )$ for any $n$, 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 ${\mathcal {F}}$.

Another issue is that you define a tail event as being one that is independent of all finite subsets of the $X_{i}$, but this itself follows from the hypothesis that the $X_{i}$ are independent. In fact, under the article's definition of 'tail event', surely we don't even need to assume that the $X_{i}$ are independent for the result to be true (provided it's only stated for events in ${\mathcal {F}}$)? —Preceding unsigned comment added by 92.15.133.253 (talk) 10:38, 6 August 2010 (UTC)