Generating Singular Multi-Normal Random Vector by Using Square Root Matrix Approach

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The joint distribution of a random vector is called singular multi-normal distribution, when the corresponding variance covariance matrix or the correlation matrix is only positive semi-definite. To generate the random vector with this property, traditional Cholesky decomposition method is failed because it needs the variance covariance matrix to be positive definite. In this paper, I propose a square matrix approach to overcome this problem. Specifying a particular positive semi-definite variance covariance matrix, the square root matrix approach can still decompose this matrix and perform the Monte Carlo simulation of singular multi-normal distribution. Several numerical example and graphical representation are illustrated as well as some probability evaluation. The techniques to present the singularity are developed and applied to the numerical examples.