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 Post subject: Almost surely convergencePosted: Wed, 18 Apr 2012 19:05:21 UTC
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Joined: Sat, 26 Mar 2011 16:45:36 UTC
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How can the following statement be proved?

For any sequence of random variables, there exists a deteministic sequence of numbers such that almost surely.

I would be grateful if you could help me.

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 Post subject: Re: Almost surely convergencePosted: Wed, 18 Apr 2012 21:11:03 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
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mrgrieco wrote:
How can the following statement be proved?

For any sequence of random variables, there exists a deteministic sequence of numbers such that almost surely.

I would be grateful if you could help me.

I think this is just Borel-Cantelli, but it's been a while since I did this exercise.

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 Post subject: Re: Almost surely convergencePosted: Thu, 19 Apr 2012 19:36:35 UTC
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Joined: Sat, 26 Mar 2011 16:45:36 UTC
Posts: 21
mrgrieco wrote:
How can the following statement be proved?

For any sequence of random variables, there exists a deteministic sequence of numbers such that almost surely.

I would be grateful if you could help me.

I think this is just Borel-Cantelli, but it's been a while since I did this exercise.

You might be right, but to tell the truth I don't really see how the lemma could be used in this case. Which form of the lemma do you mean and how does it give the result?

Thank you very much in advance!

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 Post subject: Re: Almost surely convergencePosted: Thu, 19 Apr 2012 22:14:49 UTC
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Joined: Fri, 1 Jul 2011 01:17:26 UTC
Posts: 467
mrgrieco wrote:
mrgrieco wrote:
How can the following statement be proved?

For any sequence of random variables, there exists a deteministic sequence of numbers such that almost surely.

I would be grateful if you could help me.

I think this is just Borel-Cantelli, but it's been a while since I did this exercise.

You might be right, but to tell the truth I don't really see how the lemma could be used in this case. Which form of the lemma do you mean and how does it give the result?

Thank you very much in advance!

You haven't put any condition on c_n. To use Borel-Cantelli, define c_n so that ∑P(X_n/c_n) converges.

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 Post subject: Re: Almost surely convergencePosted: Fri, 20 Apr 2012 06:38:53 UTC
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mathematic wrote:
You haven't put any condition on c_n. To use Borel-Cantelli, define c_n so that ∑P(X_n/c_n) converges.

X_n/c_n is a random variable and only takes event as input. I think you mean something like converges.

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 Post subject: Re: Almost surely convergencePosted: Sat, 21 Apr 2012 01:23:05 UTC
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Joined: Fri, 1 Jul 2011 01:17:26 UTC
Posts: 467
outermeasure wrote:
mathematic wrote:
You haven't put any condition on c_n. To use Borel-Cantelli, define c_n so that ∑P(X_n/c_n) converges.

X_n/c_n is a random variable and only takes event as input. I think you mean something like converges.

You are right.

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