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PostPosted: Sat, 5 May 2012 16:00:33 UTC 
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Let be X_1, X_2, \dots X_n iid. Standard Cauchy random variables. How to find the characteristic function of Y_n:=n^{-1}(X_1+X_2+ \dots X_n)?
I know that the characteristic function of standard Cauchy is \varpfi=e^{-|t|} 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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PostPosted: Sat, 5 May 2012 16:04:00 UTC 
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mrgrieco wrote:
Let be X_1, X_2, \dots X_n iid. Standard Cauchy random variables. How to find the characteristic function of Y_n:=n^{-1}(X_1+X_2+ \dots X_n)?
I know that the characteristic function of standard Cauchy is \varphi=e^{-|t|} 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 \varphi_{aX}(t)=\varphi_X(at), for all constant a\in\mathbb{R}.

_________________
\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


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PostPosted: Sun, 6 May 2012 11:35:19 UTC 
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outermeasure wrote:
mrgrieco wrote:
Let be X_1, X_2, \dots X_n iid. Standard Cauchy random variables. How to find the characteristic function of Y_n:=n^{-1}(X_1+X_2+ \dots X_n)?
I know that the characteristic function of standard Cauchy is \varphi=e^{-|t|} 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 \varphi_{aX}(t)=\varphi_X(at), for all constant a\in\mathbb{R}.


Thank you very much, so it is also \varphi_{Y_n}=e^{-|t|}

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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PostPosted: Sun, 6 May 2012 18:20:19 UTC 
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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 \{0,\pm 1\}-valued.

_________________
\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


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