Is it possible that following series of random variables exists?

Thx for any help

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- Jan 8th 2011, 12:48 PMslavertExistence of series of random variables
Is it possible that following series of random variables exists?

Thx for any help - Jan 8th 2011, 01:34 PMMoo
Hello,

No, have a look at Kolmogorov's 0-1 law :) - Jan 9th 2011, 03:03 AMslavert
Please explain me why,

in Kolmogorov's 0-1 law we have instead of

It contradicts ituition, it seems to me that satisfy that. - Jan 9th 2011, 10:02 AMMoo
I think considering 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 .

We have .

This condition, added to the independence, lets us use Borel-Cantelli's lemma (part II) and we get that

Now, (for the first equality, we can take because it's dense in )

By , this final probability is 0. Hence .

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