Define the multivariate Gaussian PDF being proportional to $ e^{-\pi ||x||^2}.$
In a paper I'm reading is written that
This distribution can be expressed as the sum of $n$ orthogonal 1-dimensional Gaussian distributions...
But this statement is not clear to me. First of all, what do they mean with the sum of distributions? I understand that if we add up univariate Gaussians we end up again with a univariate Gaussian, but I don't understand how to get a multivariate distribution from the "sum" of univariate ones.
Secondly, I don't understand what they mean with orthogonal (in general, I do not know what does it mean for a set of distributions to be orthogonal, maybe orthogonal with respect to certain inner product?).
I already spent enough time surfing the web in order to get information about this, without any concrete result I'm afraid. Any help will be highly appreciated!
Context and Background
The paper I'm reading is worst-case to average-case reductions based on gaussian measures, and the statement is on top of page 8. About my background, I don't have any previous knowledge on normal distributions, and I'm having a tough fight in order to learn all these topics by my own.