Let $(\Omega,\Sigma,\mathbb{P})$ be a probability space and $X:\Omega\rightarrow\mathbb{R}$ a random variable. Denote by $\phi_X(u)$ the characteristic function of $X$. It is well known that if $X$ has moments up to $k$-th order, then $\phi_X(u)$ is $k$-times continuously differentiable and we have $\mathbb{E}_{\mathbb{P}}[X^k]=(-i)^k\cdot\frac{d^k}{du^k}\phi_X(0).$ Similarly, for a vector valued random variable $Y:\Omega\rightarrow\mathbb{R}^d$ we have $\mathbb{E}_{\mathbb{P}}[Y^\alpha]=(-i)^{\vert\alpha\vert}\cdot\frac{\partial^{\vert\alpha\vert}}{\partial u_{\alpha_1}...\partial u_{\alpha_d}}\phi_Y(0),$ where $Y^\alpha=Y_1^{\alpha_1}\cdots Y_d^{\alpha_d}$ and $\alpha$ is a multi index in $\mathbb{N}^d_0$, $\vert\alpha\vert=\alpha_1+\cdots+\alpha_d$. Is there a natural analog of this formula for a matrix valued random variable $Z:\Omega\rightarrow\mathcal{S}_n$, where $\mathcal{S}_n$ is the set of symmetric $n\times n$ matrices? Can somebody guess it or give a good reference?
Characteristic function of matrix valued random variable
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matrices
probability-theory
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0That is actually also part of the question. But I think that the quantity to use here is something like $\mathbb{E}[1/n\cdot tr(Y)^k]$... – 2012-07-03