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 Post subject: Charasteristic function of sum of Cauchy distributionPosted: Sat, 5 May 2012 16:00:33 UTC
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Joined: Sat, 26 Mar 2011 16:45:36 UTC
Posts: 21
Let be iid. Standard Cauchy random variables. How to find the characteristic function of ?
I know that the characteristic function of standard Cauchy is and I know that if two random variables are independent then the charasteristic function of their sum is the product of the charasteristic functions. But I don't know how to find the char. function of Y_N.

Any help would be appreciated.

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 Post subject: Re: Charasteristic function of sum of Cauchy distributionPosted: Sat, 5 May 2012 16:04:00 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
mrgrieco wrote:
Let be iid. Standard Cauchy random variables. How to find the characteristic function of ?
I know that the characteristic function of standard Cauchy is and I know that if two random variables are independent then the charasteristic function of their sum is the product of the charasteristic functions. But I don't know how to find the char. function of Y_N.

Any help would be appreciated.

Recall also , for all constant .

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 Post subject: Re: Charasteristic function of sum of Cauchy distributionPosted: Sun, 6 May 2012 11:35:19 UTC
 Member

Joined: Sat, 26 Mar 2011 16:45:36 UTC
Posts: 21
outermeasure wrote:
mrgrieco wrote:
Let be iid. Standard Cauchy random variables. How to find the characteristic function of ?
I know that the characteristic function of standard Cauchy is and I know that if two random variables are independent then the charasteristic function of their sum is the product of the charasteristic functions. But I don't know how to find the char. function of Y_N.

Any help would be appreciated.

Recall also , for all constant .

Thank you very much, so it is also

I have another question, I already mentioned that if two random variables are independent then the charasteristic function of their sum is the product of the charasteristic functions. My question is that is is a sufficient and necessary condition, I think it is not, but I can't find a counterexample.

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 Post subject: Re: Charasteristic function of sum of Cauchy distributionPosted: Sun, 6 May 2012 18:20:19 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6009
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
mrgrieco wrote:
I have another question, I already mentioned that if two random variables are independent then the charasteristic function of their sum is the product of the charasteristic functions. My question is that is is a sufficient and necessary condition, I think it is not, but I can't find a counterexample.

There exists a counterexample when X and Y are -valued.

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