Miscellaneous

Estimating Mean and Variance

Cholesky Factorization

The Cholesky factorization is a factorization of a square matrix where is a lower-triangular matrix.

lemma: If is positive definite symmetric then a lower triangular matrix exists such that and is unique modulo sign.

Simulating Random Vectors with Matched Variance

It is frequently the case that sequences of correlated sequences of random numbers need to be generated in simulations. Let $X$ be a row vector of zero-mean random variables with variance $\Sigma =EX{X}^T$. Assuming the variance is positive definite, by the Cholesky factorization there is a lower triangular matrix $L$ such that $\Sigma =L{L}^T$. Let $Y$be a row vector of iid zero-mean unit-variance random variables. Define $Z=LY$ . Then $E\left[Z{Z}^T\right]=E\left[LY(LY{)}^T\right]=LE\left[Y{Y}^T\right]L^T=L{L}^T=\Sigma$ and so $Z$ and $Y$ have identical mean and variance.