# Joint Probability Distribution of Partitioned Sums

Let $X_{1,1},\dots,X_{M,N}$ be $M \times N$ random variables with Log-Rayleigh distribution, i.e. pdf $$f(z) = \frac{(e^z)^2}{\beta^2} \exp \left( - \frac{(e^z)^2}{2 \beta^2} \right)$$ and with covariance matrix $C = [c_{ij}]$ with $1 \leq i,j \leq MN$ between all $MN$ random variables and Gaussian copula. Further, we have $N$ different sum variables $Y_1, \dots, Y_N$ with $$Y_i = \sum_{j=1}^M X_{j,i}$$

I would like to know the joint probability density of $Y_1, \dots, Y_N$ dependent on $C$. How can I approach this?

• Just a warning to readers - this uses "log-Rayleigh" in a different sense to the log in "lognormal" or "log-logistic" (e.g. lognormal means "the distribution of a variate whose log is normally distributed). Instead, from the pdf, this looks like it's the density of the log of a Rayleigh-distributed random variable. [This potential ambiguity is not the OP's fault; it is found all over.] – Glen_b Dec 28 '15 at 2:28
• To which direction of the matrix $(X_{i,j})$ does the covariance apply? I mean, do you have $M$ or $N$ independent replicas of Rayleigh vectors? – Xi'an Dec 28 '15 at 8:30
• @Xi'an I clarified the size of the covariance matrix. If I understand correctly there are no replica vectors. If it helps, the covariance matrix $C$ is a distance matrix from an array $a_i \in \mathbb{R}$ with $1 \leq i \leq MN$, i.e. $C_{i,j} = 1 / (1 + |a_i - a_j|)$. – Sebastian Schlecht Dec 28 '15 at 10:35
• @Xi'an: Is there any information missing to fully specify the problem? – Sebastian Schlecht Jan 12 '16 at 9:28