Hanson-Wright's inequality such as this provides a bound on the second-order chaos of the form
$$P(x^\top A x -\mathbb{E}[x^\top A x] > t) $$
For example, $x\sim N(0, I_n)$ and $A$ is some $n\times n$ matrix.
Generally, it bounds the MGF of $x^\top A x$ in terms of $x^\top A x'$ where $x'$ is an independent copy of $x$ via decoupling inequality for zero mean random vectors with independent entries.$$\mathbb{E}[\exp(x^\top A x)] \leq \mathbb{E}[\exp(4x^\top A x')]$$
The second form is easier to analyze than the first.
My question is, is there any result on bounding the relation between $P(x^\top A x - \mathbb{E}[x^\top A x] > t)$ and $P(x^\top A x' -\mathbb{E}[x^\top A x'] > t)$
Perhaps something like$$P(x^\top A x - \mathbb{E}[x^\top A x] > t) < C \cdot P(c(\cdot x^\top A x' -\mathbb{E}[x^\top A x']) > t)$$for some constants $C$ and $c$. Any help is appreciated. Thanks!
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