Multivariate logistic distribution The normal distribution can be generalized into the multivariate normal distribution.
Can the logistic distribution also be generalized into a similar multivariate distribution? 
Is there a multivariate generalization of the logistic distribution which depends on the covariance matrix $\Sigma$, similar to the multivariate normal distribution? The multivariate distribution should be such that its marginals are univariate logistic distributions.
 A: Using copulas you can create a multivariate distribution generalized from any univariate distribution, so yes it is possible to find a multivariate distribution with all the marginal distributions equal to logistic distributions, however it will probably not be a simple function of a covariance matrix, that relationship is pretty unique to the normal distribution.
A: Yes. In fact, the multivariate normal and logistic distributions are members of the more general family of elliptically-contoured distributions, which can be derived from their univariate counterparts.
Both the univariate and multivariate normal distributions share the same probability density generator, which is proportional to
$$
g(u) = exp(-u/2)
$$
That is, if $u =\big(\dfrac{x-\mu}{\sigma}\big)^2$ we have the normal distribution and if $u = (x-\mu)'\Sigma^{-1}(x-\mu)$ we have the multivariate normal distribution.
The same story goes for the univariate and multivariate logistic distributions, which share the same probability density generator, which is proportional to
$$
g(u) = \dfrac{exp(-u)}{(1+exp(-u))^2}.
$$
If $u =\big(\dfrac{x-\mu}{\sigma}\big)^2$ we have the logistic distribution and if $u = (x-\mu)'\Sigma^{-1}(x-\mu)$ we have the multivariate logistic distribution.
Furthermore, it is well known that the parent and marginal distributions  of any elliptically contoured distributions share the same type of distribution.
See the wikipedia page on elliptical distributions for more details
https://en.wikipedia.org/wiki/Elliptical_distribution
A: I don't think any such distribution is known to literature. 
Books on continuous multivariate distributions (such as Kotz '00) and books on the logistic distribution (such as N. Balakrishnan '92) don't mention any such generalization.
Most multivariate distributions discussed there contains at most two parameters which in some cases govern the covariance between the variables (besides the parameters $\mu_i$ and $\sigma_i$ for the mean and standard deviation in each variable $i$). No single distribution is given which uses (as much parameters as) the covariance matrix $\Sigma$. 
However, that does not guarantee no such distribution is possible.
