Tensor moments of mixture of gaussians

Question

Suppose we have the following Mixture of Gaussians model: $$ p(\mathbf{x}) = \sum_{i=1}^k w_i \frac{1}{{2\pi\sigma^2}^{D/2}}\exp\left(-\frac{\left\lVert\mathbf{x}-\mathbf{a_i}\right\rVert^2}{2\sigma^2}\right)$$ where $\mathbf{x},\mathbf{a_i}\in\mathbb{R}^D$, $\sum_i w_i = 1$ and the covariance matrices are all equal to $\sigma^2I_{D\times D}$. The task is to compute $\mathbb{E}[\mathbf{x}],\mathbb{E}[\mathbf{x}\mathbf{x}^T],\mathbb{E}[\mathbf{x}\otimes\mathbf{x}\otimes\mathbf{x}].$ I am able to compute it but, especially for the tensor product, the computation gets very messy and I feel like I am lacking intuition on how to write it (and reconduct it to lower dimensional objects). Do you have a simple way of computing $\mathbb{E}[\mathbf{x}\otimes\mathbf{x}\otimes\mathbf{x}]$? Hints on how to understand better tensor moments?

Answer 1

Can't you just use $$ \mathbb{E}[Y(\mathbf{x})]=\sum_i w_i\mathbb{E}_{\mathbf{y}\sim N(\mathbf{a}_i,\sigma^2I)}[Y(\mathbf{y})] $$ for all $Y$ which the expectation makes sense? So it reduces to essentially tensor moment of radially symmetric Gaussians.