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Let $X=\lVert M^\top p\rVert_2$, where $M$ is an $n\times n$ non-random matrix and $p\sim N(0,I_{n\times n})$ is an $n\times 1$ vector, and$\lVert \cdot\rVert_2$ is the Euclidean norm.

Using some linear algebra, we can determine that \begin{equation} \mathbb{E}[X^2]=tr(MM^\top), \end{equation} where $tr(.)$ is the trace operator.

Now suppose instead we are interested in the 4th moment - i.e., \begin{equation} \mathbb{E}[X^4], \end{equation} How can this be obtained? For instance, is there an extension of Isserlis' Theorem which is appropriate in this setting?

Arya McCarthy
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Carl
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1 Answers1

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\begin{align} E[X^4] &= E[X^2X^2] \\ &= \text{Cov}(X^2,X^2) + E[X^2]^2 \\ &= \text{Var}(X^2) + E[X^2]^2 \end{align} Note that $X^2 = \| M^T p \|_2^2 = p^T M M^T p$, which follows a generalized chi-squared distribution when $p\sim N(0,I_{n\times n})$. Then, \begin{align} E[X^4] &= \text{Var}(X^2) + E[X^2]^2 \\ &= 2\text{Tr}\left[MM^TMM^T\right] + \text{Tr}\left[MM^T\right]^2 \end{align}

mhdadk
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