I have a sequence of normally distributed random variables. Let's call it $S_1$. I want to generate 4 more series, each of which has its own correlation with $S_1$ and its own variance. Let's call them $S_2$, $S_3$, $S_4$ and $S_5$.
If I use a covariance matrix, I have to designate the covariances between $S_1$ and each of the other series, which is fine. But I also have to designate covariances between each of the other series (e.g. between $S_2$ and $S_3$). I will therefore end up with a correlation between $S_2$ and $S_3$, and there is no reason why there should be one.
How can I do generate $S_2$ to $S_5$ without forcing a correlation between them?