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The characteristic function of a random vector $X=(X_1, X_2)$ is $$ \phi(u_1, u_2)= \alpha(-iu_1)^{\alpha} \Gamma (- \alpha, - iu_1) e^{iu_2^2}$$ Are $X_1$ and $X_2$ independent?

$X_1,X_2$ are independent $\iff \phi_X (u_1, u_2)= \phi_{X_1} (u_1) \phi_{X_2}(u_2)$

Question1: I calculate $ \phi_{X_2} (u_2)= \phi_X (0, u_2)=0$, but than $ \phi_{X_2}$ is not a characteristic function because a characteristic function must be 1 in 0. So is there a mistake in the task or did I have made a mistake?

Question2: $\phi(u_1, u_2)$ can be written as the product of a function g in $u_1$ and a function h in $u_2$. Why it follows that these are the characteristic functions of $X_1, X_2$? I already know that a finite measure is uniquely determined by its characteristic function. So i guess you need a proof that the function in $u_1$ and the function in $u_2$ are characteristic functions? If $g(0)=1, h(0)=1$ it is clear.

Lauren Veganer
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    Long story short: since $\phi(u_1,u_2)$ can be written as the product of a function in $u_1$ and a function in $u_2$, they are independent. – angryavian Sep 28 '16 at 21:39
  • What is $\gamma(\cdot,\cdot)$? – saz Sep 29 '16 at 05:25
  • Gamma function: https://en.wikipedia.org/wiki/Gamma_function – Lauren Veganer Sep 29 '16 at 06:17
  • I use: Be $X=(X_1,.,,X_n)$ a random vector so $ \phi_{X_j} (u_j)= \phi_X (0,0,..0,u_j,0,..,0)$ are the characteristic function of $X_j$. – Lauren Veganer Sep 29 '16 at 06:19
  • Fully solved there. – Did Sep 29 '16 at 06:22
  • I don´t understand all in the contribution, so i editet two questions, that i still have on this topic. – Lauren Veganer Sep 29 '16 at 06:52
  • "So i guess you need a proof that the function in u1 and the function in u2 are characteristic functions?" This is direct: if $\phi_X(u_1,u_2)=g(u_1)h(u_2)$ for every $(u_1,u_2)$ then $\phi_X(u,0)=g(u)h(0)=\phi_{X_1}(u)$ and $\phi_X(0,u)=g(0)h(u)=\phi_{X_2}(u)$ for every $u$ hence $g=c\phi_{X_1}$ and $h=\phi_{X_2}/c$ for some nonzero $c$. – Did Sep 29 '16 at 12:58
  • Thanks for your answer! Two question:1) why you know that the c is the same for g and h, from your calculation it only follows: $\exists c,k: cg= \phi_{X_1}, kh= \phi_{X_2}$, 2) But for the example above is the problem that $ g(0)=0$. – Lauren Veganer Sep 29 '16 at 13:27
  • @LaurenVeganer Usually, the Gamma function is a function of one variable and not two... so how is $\Gamma(x,y)$ defined? – saz Oct 01 '16 at 08:21

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