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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.

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  • $\begingroup$ 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. $\endgroup$ – Thomas Jun 13 '14 at 8:15
  • $\begingroup$ @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? $\endgroup$ – ws6079 Jun 13 '14 at 8:18
  • $\begingroup$ 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. $\endgroup$ – Juho Kokkala Jun 13 '14 at 8:45
  • $\begingroup$ For the record, you should not refer to the collection of coefficients as "the $c_n$" because $c_n$ is the last coefficient. Better use "the $c_k$$ or some other free index. I submitted an edit to this end, but it needs to be reviewed. $\endgroup$ – Thomas Jun 13 '14 at 9:26
  • $\begingroup$ Based on the edit and the comments, I assume that you are asking how to produce samples from the conditional distribution of $z$ given $x_n=0$. The current wording is in my opinion still a bit strange - finding numbers that 'are realizations' is not clear. $\endgroup$ – Juho Kokkala Jun 13 '14 at 12:27
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$z_1,\ldots,z_n,x_n$ together form a $n+1$-dimensional multivariate normal distribution. Proof: as $z$s are independent normal, linear combinations of them are normal. Furthermore, as $x_n$ is defined as a linear combination of the $z$s, any linear combination of the $z$s and $x_n$ is a linear combination of the $z$s and thus normal.

Mean of the multinormal distribution in question is $\mathbf{0}$. The covariance is such that the covariance of any $z_i,z_j$ is 0 if $i\neq j$ and 1 if $i=j$. Covariance of $z_i,x_n$ is $c_k$ and the variance of $x_n$ is $\sum_{k=1}^n c_n^2$. Now, to obtain the conditional distribution of $(z_1,\ldots,z_n)$ conditional on $x_n = 0$, we may directly apply the known equations for conditional distributions of multivariate normal distribution, with $\mathbf{x}_1 = (z_1,\ldots,z_n),~\mathbf{x}_2=x_n,\mathbf{a}=0$.

As the observed value 0 equals the mean of $x_n$, the conditional mean equals the unconditional mean, i.e., \begin{equation} E((z_1,\ldots,z_n) \mid x_n = 0 ) = (0,\ldots,0). \end{equation} To obtain the covariance, we need to evaluate the Schur complement, \begin{equation} \bar{\Sigma} = \Sigma_{11} - \Sigma_{12}\Sigma^{-1}_{22}\Sigma_{21}. \end{equation} Plugging in the values of the distribution in question, \begin{equation} = I_n - \begin{pmatrix} c_1 & \ldots & c_n \end{pmatrix}^T\left(\sum_{k=1}^n c_k^2\right)^{-1}\begin{pmatrix}c_1 & \ldots & c_n \end{pmatrix}. \end{equation} Evaluating this element by element, we get \begin{array}{lll} Cov(z_i,z_i) & = & 1 - \frac{c_i^2}{\sum_{k=1}^n c_k^2},& i \in \{1,\ldots,n\} \\ Cov(z_i,z_j) & = & - \frac{c_ic_j}{\sum_{k=1}^n c_k^2}, & i\neq j;~ i,j \in \{1,\ldots,n\}. \end{array}

Since the conditional distribution is multivariate normal, you can sample from it using standard methods, e.g., the Cholesky decomposition approach.

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Let the $c_i$ be fixed and let $z_1,\dots,z_{n-1}$ be normal. Set $z_n := -\frac{1}{c_n} \sum_{i=1}^{n-1} c_i z_i$. These random variables satisfy the linear constraint by construction and since $z_n$ is a linear combination of standard normals it is normal itself.

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  • $\begingroup$ A linear combination of standard normals is not guaranteed to be standard normal. $\endgroup$ – Juho Kokkala Jun 13 '14 at 8:39
  • $\begingroup$ Yes, right. I removed "standard". $\endgroup$ – Thomas Jun 13 '14 at 8:40
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    $\begingroup$ Without additional assumptions about the joint distribution of $z_1,\ldots,z_{n-1}$ (namely, that they are multivariate normal), the linear combination may be non-normal. $\endgroup$ – Juho Kokkala Jun 13 '14 at 8:42
  • $\begingroup$ Right, my problem is that I can not get the distribution of $z_n$ to meet my constraint (zero mean and unit-variance) $\endgroup$ – ws6079 Jun 13 '14 at 8:43
  • $\begingroup$ @JuhoKokkala for my scenario I know that the $z_n$ are uncorrelated and independent (for the unconditional case) $\endgroup$ – ws6079 Jun 13 '14 at 8:45

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