Summing Differently-Distributed, Correlated Random Variables I have four mutually correlated time series. Each contains 40 observations but is fundamentally and decisively differently distributed from its peers. The first series follows a Laplace distribution, whereas the second series follows a log-normal distribution, while the third series follows a Levy distribution, and the fourth follows Johnson's Bounded distribution. I also have another set of data that follow an ARIMA process. My task is to find the 99% and 99.9% bounds (which means I can't just re-sample from the original series) of the product of a) the sum of the simulated four data series and b) simulations from the ARIMA process.
My question is this: How can I sum across such differently distributed, yet-correlated processes and multiply them by yet another differently-distributed process? Is this even feasible?
Please note that I must use Python (with which I am more than comfortable) for this project. While I remain comfortable with stochastic processes, my linear algebra is a bit rusty, as it's been 10 years since I've dealt with eigenvectors / eigenvalues.
Thank you for your time and kind assistance!
 A: For sums of positively correlated variables:
Suppose you have multiple distributions, and $m$ is the vector of their medians, and $q$ is the vector of their 99th percentiles. Suppose $R$ is the matrix of correlations between the four distributions. If all the variables are positively correlated, then $$\sqrt{(q-m)^\top R\,(q-m)} + \sum_i m_i$$ is a reasonable estimate for the 99th percentile of their sum, which would be exact if they were correlated normal distributions.
For products of positively correlated variables:
Suppose that you have multiple positive distributions, with vectors $m$ and $q$ as above, and now $L$ is the correlation matrix of the logs of those distributions. If all the logs are positively correlated, then $$\exp\left(\sqrt{(\ln q- \ln m)^\top L\,(\ln q - \ln m)} + \sum_i \ln m_i \right)$$
is a reasonable estimate for the 99th percentile of their product, which would be exact if they were correlated lognormal distributions.
For the case described in the comments: The sum is of electric outputs from different dams, which are then multiplied by an electric price. The assumption of positive correlations among the dam outputs is probably reasonable, since if they're all getting the same price, they're probably all in the same region, and thus all getting large inflows of water at similar times.
On the other hand, there is probably a negative correlation between dam outputs and electric price, since high electric supply from the dams could cause the electric price to go down. So the 99th percentile of the product could reasonably come the 99th percentile of quantity and the 1st percentile of price, or from the 1st percentile of quantity and the 99th percentile of price. This means that the formulas above won't work, and the formula should probably look at both sides of the distributions.
For products of variables that may be negatively correlated: Let $m$ and $L$ be as before, and let $\sigma$ be the vector of the standard deviations of the logs of the variables. Let $z$ be the z-score for the quantile in question, $\pm2.33$ for the 99th percentile or $\pm3.09$ for the 99.9th percentile, with a sign depending on which side of the distribution you want. Then
$$\exp\left(z\sqrt{\sigma^\top L\,\sigma} + \sum_i \ln m_i \right)$$
is a place to start for estimating the 99th percentile of the product, which would be exact if the variables were correlated lognormal disributions. On the other hand, causal interplay of the variables could make this estimate inappropriate, and a more sensitive model of their interaction would be useful.
Update: Here is an answer I like better, which treats positive and negative correlations in parallel, continuously varying between them and with the right limiting behavior. Let $m$, $q$, $M$ and $L$ be as before. Let $s$ be the vector of the 1st percentiles of the variables.
If $\rho_i$ is the correlation of the $i^{th}$ variable with the sum, and
$$b_i = \frac{1+\rho_i}{2}(q_i-m_i) + \frac{1-\rho_i}{2}(m_i-s_i)$$
Then
$\sqrt{b^\top M\,b} + \sum_i m_i$
is a reasonable estimate for the 99th percentile of the sum, which would be exact if the variables were correlated normals.
Similarly if $\rho_i$ is the correlation of the $i^{th}$ variable with the product, and
$$a_i = \frac{1+\rho_i}{2}\ln\left(\frac{q_i}{m_i}\right) + \frac{1-\rho_i}{2}\ln\left(\frac{m_i}{s_i}\right)$$
then $\exp\left(\sqrt{a^\top L\, a} + \sum_i \ln m_i \right)$
is a reasonable estimate for the 99th percentile of the product, which would be exact if the variables were correlated lognormals.
