Let $X\in\mathbb{R}^{d}$ have independent, mean zero subgaussian entries, and $A_{1},\ldots,A_{k}$ be fixed $d\times d$ matrices that have zeros on the diagonal. I would like to upper bound the quantity \begin{equation} \mathbb{E}\bigg[\bigg(\sum_{i=1}^{k}(X^{\top}A_{i}X)^{2}\bigg)^{q}\bigg], \end{equation} for $q\in\mathbb{N}$.

Without the square on the quadratic form, this computation is easy as one can pull the summation inside and use results for the moments of subexponential random variables ($X^{\top}BX$ is subexponential.) With the square, however, it seems difficult. My idea is to use a decoupling trick to replace $X^{\top}A_{i}X$ with $X^{\top}A_{i}X'$, condition on $X'$, and then pull the summation in ($X'$ is an independent copy of $X$).

Vershynin's textbook on High Dimensional Probability (Theorem 6.1.1) gives \begin{equation} \mathbb{E}[f(X^{\top}AX)]\le \mathbb{E}[f(4X^{\top}AX')]. \end{equation} for $f:\mathbb{R}\rightarrow\mathbb{R}$ convex and $A$ diagonal-free.

A multivariate version of this result might be helpful. Any hints?


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