Multivariate Distribution I am dealing with a multi-dimensional problem for which i have marginal information (chi-squared distribution) together with an idea of dependence.
When is this question well–posed?
 A: If you have a multivariate vector $x = \{ x_1, x_2, ..., x_p \} $ where the entries are independent and marginal distributions that are $\chi^2_{k_1}, ..., \chi^2_{k_p}$ respectively, then the multivariate density is
$$f(x_1, ..., x_n) = \prod_{i=1}^{n} \dfrac{x_i^{(k_i/2-1)} e^{-x_i/2}}{2^{k_i/2} \Gamma\left(\frac{k_i}{2} \right)}$$
Edit: In response to the OP clarification, the joint distribution of $X'X$ when $X$ is a $p$-dimensional multivariate normal is a Wishart distribution, which as density 
$${\displaystyle \frac{1}{2^{np/2} \left|{\mathbf V}\right|^{n/2} \Gamma_p\left|\frac {n} {2}\right | }{\left|\mathbf{X}\right|}^{(n-p-1)/2} e^{-(1/2)\operatorname{tr}({\mathbf V}^{-1}\mathbf{X})}}$$
The diagonal of $X'X$ could fit the description you gave because the individual entries would have a $\chi^2$ distribution and there would be dependencies reflected in ${\bf V}$. Not sure how to derive the distribution of ${\rm diag}(X'X)$ but I'd guess it's possible. If you figure it out, post an answer. 
