# An upper bound on $\mathbb{E}\bigg[\bigg(\sum_{i=1}^{k}(X^{\top}A_{i}X)^{2}\bigg)^{q}\bigg]$

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 $$$$\mathbb{E}\bigg[\bigg(\sum_{i=1}^{k}(X^{\top}A_{i}X)^{2}\bigg)^{q}\bigg],$$$$ for $$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 $$$$\mathbb{E}[f(X^{\top}AX)]\le \mathbb{E}[f(4X^{\top}AX')].$$$$ for $$f:\mathbb{R}\rightarrow\mathbb{R}$$ convex and $$A$$ diagonal-free.

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