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Math Help - Existence of series of random variables

  1. #1
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    Existence of series of random variables

    Is it possible that following series of random variables exists?
    P({\omega \in \Omega : lim_{n -> \infty} X_{n}=0})=0.5

    Thx for any help
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  2. #2
    Moo
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    Hello,

    No, have a look at Kolmogorov's 0-1 law
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    Please explain me why,
    in Kolmogorov's 0-1 law we have A \in F instead of \omega  \in \Omega
    It contradicts ituition, it seems to me that P(X_{n}=0)=1/2 and P(X_{n}=n)=1/2 satisfy that.
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    I think considering \omega or the measurability of the rv's doesn't change much thing... I'll have to give a deeper look into the definition of a tail event but I'm quite tired.

    Anyway, I think your example doesn't fit what you want...
    Let's assume the rv's are independent.

    We want to find \displaystyle P(\lim_{n\to\infty} X_n=0).


    We have \forall \epsilon>0,\displaystyle \sum_n P(X_n>\epsilon)=\sum_n P(X_n=n)=\infty.
    This condition, added to the independence, lets us use Borel-Cantelli's lemma (part II) and we get that \displaystyle \forall \epsilon>0,P(\limsup_n \{X_n>\epsilon\})=1 \Leftrightarrow \forall \epsilon>0,P(\liminf_n \{X_n<\epsilon\})=0 \quad (\star)

    Now, (for the first equality, we can take \epsilon \in\mathbb Q^+ because it's dense in \mathbb R^+)

    \displaystyle \begin{aligned} P(\lim_n X_n=0)&=P(\forall \epsilon \in \mathbb Q^+,\exists N\in\mathbb N,\forall n>N,X_n<\epsilon) \\<br />
&=P(\bigcap_{\epsilon\in\mathbb Q^+}\bigcup_{N\in\mathbb N}\bigcap_{n>N}\{X_n<\epsilon\}) \\<br />
&=P(\bigcap_{\epsilon\in\mathbb Q^+} \liminf_n \{X_n<\epsilon\}) \\<br />
&<P(\liminf_n \{X_n<\epsilon_0\}),\text{ for a given } \epsilon_0\in\mathbb Q^+ \end{aligned}

    By (\star), this final probability is 0. Hence \displaystyle P(\lim_n X_n=0)=0.

    I agree though that this is kind of counterintuitive. So if there's a mistake somewhere just tell me!
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