What is the expectation of the Cholesky factor of a Wishart distributed random matrix? Let a $d-\text{dimensional}$ Wishart random variable with $\nu$ degrees of freedom $\Sigma$ be distributed according to $\mathcal{W}(\Sigma|\Sigma_0, \nu) \propto |\Sigma|^\frac{\nu-d-1}2\exp{(-\frac12\text{tr}\ \Sigma_0^{-1}\Sigma)}$.
What is the expected value of $\text{chol} \ \Sigma$?
 A: In 1933 Bartlett described the Wishart distribution in terms of the distribution of the factors after Cholesky decomposition. I can not read the original source (On the theory of statistical regression) but this 'Bartlett decomposition' has been described elsewhere many times. See for instance the Wikipedia article on the Wishart distribution, which describes it as:
$$W \sim LAA^TL$$
with $L$ the cholesky factor of the $\Sigma_0$ matrix. And $A$ a triangular matrix that is standard normal distributed for the non-diagonal elements and the root of a chi-square distributed variable for the diagonal elements (which is just a chi distributed variable).
The elements of the cholesky factor $LA$ is then a weighted sum of normal distributed variables and a chi distributed variable. To compute the mean of this sum you take the sum of the means of the individual variables. For the normal distributed variables the mean is obvious. For the chi distributed variable you can compute it using a ratio of gamma functions.
