Title: Matrix product moments in normal variables,
Abstract: Let ${\cal X }=XX^{\prime}$ be a random matrix associated with a centered
$r$-column centered Gaussian vector $X$ with a covariance matrix $P$. In this
article we compute expectations of matrix-products of the form $\prod_{1\leq
i\leq n}({\cal X } P^{v_i})$ for any $n\geq 1$ and any multi-index parameters
$v_i\in\mathbb{N}$. We derive closed form formulae and a simple sequential
algorithm to compute these matrices w.r.t. the parameter $n$. The second part
of the article is dedicated to a non commutative binomial formula for the
central matrix-moments $\mathbb{E}\left(\left[{\cal X }-P\right]^n\right)$. The
matrix product moments discussed in this study are expressed in terms of
polynomial formulae w.r.t. the powers of the covariance matrix, with
coefficients depending on the trace of these matrices. We also derive a series
of estimates w.r.t. the Loewner order on quadratic forms. For instance we shall
prove the rather crude estimate $\mathbb{E}\left(\left[{\cal X
}-P\right]^n\right)\leq \mathbb{E}\left({\cal X }^n-P^n\right)$, for any $n\geq
1$