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 Post subject: On Borell's Theorem (Gaussian processes)Posted: Sat, 4 Jun 2011 20:04:06 UTC
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Joined: Mon, 28 Dec 2009 16:23:32 UTC
Posts: 10
Let be a Gaussian process with mean and bounded (with probability 1) sample paths. Borell's Theorem states then that for all we have
,
where is the median of , is the supremum of and is the tail of a standard normal distribution.
I need to show that under the assumptions of this theorem (we can use the theorem as well) has finite all moments, i.e. exists and is finite fot all .

Thank you for any help.

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 Post subject: Re: On Borell's Theorem (Gaussian processes)Posted: Sun, 5 Jun 2011 07:01:43 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6008
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Nobody1111 wrote:
Let be a Gaussian process with mean and bounded (with probability 1) sample paths. Borell's Theorem states then that for all we have
,
where is the median of , is the supremum of and is the tail of a standard normal distribution.
I need to show that under the assumptions of this theorem (we can use the theorem as well) has finite all moments, i.e. exists and is finite fot all .

Thank you for any help.

You forgot a finiteness assumption of , e.g. Brownian motion on .

Once you have finiteness, it is just is rapidly decreasing.

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 Post subject: Posted: Tue, 7 Jun 2011 07:50:21 UTC
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Joined: Mon, 28 Dec 2009 16:23:32 UTC
Posts: 10
Assuming that and are finite, I have still some problems with proving that is finite.
My first problem is how toexploit the knowledge about the tail distribution of supremum to calculate the -th moment. I know the formula
if is the cdf of X. But is there any similar formula valid for higher moments?
The second problem is that the bound in Borell's theorem holds for and we don't know what happens if .
I have been trying to solve the problem without Borell's theorem but I gave it up.

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 Post subject: Posted: Tue, 7 Jun 2011 11:43:30 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6008
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Nobody1111 wrote:
Assuming that and are finite, I have still some problems with proving that is finite.
My first problem is how toexploit the knowledge about the tail distribution of supremum to calculate the -th moment. I know the formula
if is the cdf of X. But is there any similar formula valid for higher moments?
The second problem is that the bound in Borell's theorem holds for and we don't know what happens if .
I have been trying to solve the problem without Borell's theorem but I gave it up.

Answer to your first question: follows from a simple change of variable in Cavalieri's formula (or Tonelli's theorem).

Answer to your second question: Since the process has mean 0 (and assumed to have continuous paths?), you don't really have to worry about u<0 --- remember and reflect the process will give you an immediate bound.

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 Post subject: Posted: Thu, 9 Jun 2011 14:27:58 UTC
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Joined: Mon, 28 Dec 2009 16:23:32 UTC
Posts: 10
Thank you very much. I have been thinking about your post and now everything is clear for me.
Cheers

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