# Finding the sum of realizations of normal random variables equalling zero

Is there some method which will allow me to find some set of (random) numbers $z_1,\dots,z_n$ such that

$z_1 c_1 + z_2 c_2 + ... + z_n c_n = 0$

where for $k=1,\dots,n$, the $c_k$ are fixed coefficients and $z_k$ are realizations of a standard normal random variable? Many thanks in advance!

EDIT: The considered scenario is a multivariate normal distribution with known covariance matrix $\mathbf{\Sigma}$ and Cholesky decomposition $\mathbf{C}\mathbf{C'}=\mathbf{\Sigma}$. I am using a Monte-Carlo approach, where I am interested in all realizations of the vector $\mathbf{z} = (z_1, z_2,..., z_n)'$ where, using $\mathbf{x} = \mathbf{C}\mathbf{z}$, the value of $x_n = z_1 c_1 + z_2 c_2 + ... + z_n c_n$ is equal to zero. I hope this is a bit clearer.

• Do you want the sum to be a random variable with mean zero, or do you want the sum to be equal to zero for every realization? In the second case I think you have no chance other than letting z_n be the deterministic solution to the linear equation. – Thomas Jun 13 '14 at 8:15
• @Thomas the latter: the sum of the realizations should be zero. Could you please clarify the last part of you comment? How would I solve this? – ws6079 Jun 13 '14 at 8:18
• It seems slightly unclear what you are asking. Do you need a joint distribution fo random variables $Z_1,\ldots,Z_n$ such that the marginal distribution of $Z_k$ is standard normal (for all $k$) and the linear combination is guaranteed to be 0? Or something else? It might help if you add some information about what you are trying to do. – Juho Kokkala Jun 13 '14 at 8:45