An $n$-dimensional random vector $X=(x_1, \ldots, x_n)$ follows an elliptical distribution with mean $\mu \in \mathbb{R}^n$ and positive definite covariance matrix $\Sigma \in \mathbb{R}^{n \times n}$ if $$Z \stackrel{d}{=}\mu + \xi AU,$$ where $A \in \mathbb{R}^{n \times n}$ satisfies $A A^\top= \Sigma$, $U \in \mathbb{R}^n$ uniformly distributed on $\mathbb{S}_{n-1}$ and $\xi$ an unspecified random variable, independent of $U$ and $\mathbf{E}(\xi^2)=n$.
Now according to http://www.leg.ufpr.br/lib/exe/fetch.php/wiki:internas:biblioteca:applied_multivariate_statistics.pdf, page 170 we have:
Any linear combination of elliptically distributed variables are elliptical.
Marginal distributions of elliptically distributed variables are elliptical.
The characteristic function $\phi(t)=\mathbf{E}(e^{it^\top X})$ of $X$ is of the form: $$\phi(t)=e^{it^\top \mu}\psi(t^\top \Sigma t)$$
Now first the understanding:
In number 1 is written, "elliptical distributed variable", not vector, so does this mean, that for x,y being elliptical distributed, $\lambda x+\mu y$ for constants $\lambda$, $\mu \in \mathbb{R}$ is only elliptical distributed, if $x$ and $y$ are both 1-dimenisonal distributed but it is not true for the general case, that $x$ and $y$ are $n$-dimensional distributed? Is there any regulations about that $x$ and $y$ must be independent or identically distributed?(identically meaning elliptically distributed with the same parameters)
states that for $X=(x_1, \ldots, x_n)$ $x_i$ is 1-dimensional elliptical distributed for all $i \in \{1, \ldots, n\}$, right?
Also I am looking for ideas how to prove the three statements... Is there anyone able to give me an idea on how to start a proof?