I'm tryig to prove the next proposition but I'm lost:
Let $X=(X^{1},\ldots,X^{d})$ be a $d-$dimensional Lévy process with Gaussian coefficient $Q$ and Lévy measure $\Pi.$ Then the Levy processes $X^1,\ldots,X^{d}$ are independent if and only if $Q$ is diagonal matrix and the support of $\Pi$ is contained in the coordinate axes.
I've been working with the definition of characterístic exponent: $$E(\exp(i\langle\lambda,X\rangle))=\exp\{-t\Psi(\lambda)\},\space\lambda\in\mathbb{R}^d.$$
Here $$\Psi(\lambda)=i\langle a,\lambda\rangle+\frac{1}{2}Q(\lambda)+\int_{\mathbb{R}^{d}}(1-e^{ix\lambda}-i\langle\lambda x\rangle1_{\{|x|<1\}})\Pi(dx),\space a\in\mathbb{R}^{d}.$$
Utilizing the above, and independence of Lévy processes we get $$\displaystyle\sum_{j=1}^{d}\Psi(\lambda_{j})=\Psi(\lambda)$$ and
$$\displaystyle\sum_{j=1}^{d}(\frac{1}{2}q_{j}+\int_{\mathbb{R}}(1-e^{i\lambda_{j}x_{j}}+i\lambda_{j}x_{j}1_{\{|x_{j}|<1\}})\Pi(dx_{j}))=\frac{1}{2}Q(\lambda)+\int_{\mathbb{R}^{d}}(1-e^{i\langle\lambda,x\rangle}+i\langle\lambda,x\rangle1_{\{|x|<1\}})\Pi(dx)$$
So we have that that $Q$ is matrix diagonable and the support of $\Pi$ it's on the coordinate axes if the equality between integrals on the behind equality holds, but I don't get the last. Because of independence and Fubini theorem we will have $$\displaystyle\prod_{j=1}^{d}\int_{\mathbb{R}}e^{i\lambda_{j}x_{j}}\Pi(dx_{j})=\displaystyle\sum_{j=1}^{d}\int_{\mathbb{R}}e^{i\lambda_{j}x_{j}}\Pi(dx_{j})$$ isn't it? I'm not sure of the behind. I have doubts of this because of the integration domain (the indicator functions.) but I don't know if this is eough to have such equality.
For the other implication I guess the same steps as above works or this purpose, but I'm not sure.
Any kind of help is thanked in advanced.
Edit:I've proved the first implication; for the second I have troubles.
I'd like to prove that
$$\int_{\mathbb{R}^{d}}(1-e^{ix\lambda}-i\langle\lambda x\rangle1_{\{|x|<1\}})\Pi(dx)=\displaystyle\sum_{j=1}^{d}\int_{\mathbb{R}}(1-e^{i\lambda_{j}x_{j}}+i\lambda_{j}x_{j}1_{\{|x_{j}|<1\}})\Pi(dx_{j}))$$ utilizing the fact that $\Pi$ has support contained in coordinate axes. I've been worked with the first term of the above equality, but I don't get a sum of integrals as I'd like.
Is there another way to prove this?
Again, thanks for the help in advanced.