Let $d\in\mathbb N$, $X$ be a $\mathbb R^d$-valued random variable on a probability space $(\Omega,\mathcal A,\operatorname P)$ with density $$p(x):=\prod_{i=1}^df(x_i)\;\;\;\text{for }x\in\mathbb R^d$$ for some positive $f\in C^1(\mathbb R)$ with $$\int f(x)\:{\rm d}x=1$$ with respect to the Lebesgue measure on $\mathcal B(\mathbb R^d)$ and $Z$ be a $\mathbb R^d$-valued random variable on $(\Omega,\mathcal A,\operatorname P)$ independent of $X$ and distributed according to $\mathcal N_d(0,\sigma^2I_d)$, where $\sigma^2=d^{\varepsilon-1}$ for some $\varepsilon\in(0,1)$. Let $g:=\ln f$ and assume $g'$ is Lipschitz continuous and $(X_1,\xi_1),\ldots,(X_d,\xi_d)$ are mutually independent..
How can we show that $$\operatorname E\left[\left|d^{-\frac12}\sum_{i=1}^dg'(X_i)\xi_i\right|^{2q}\right]$$ is bounded by a constant only depending on $q$ (and not on $d$)?
I've read here on page 25 that the claim can be proved by considering the polynomial expansion. However, it's not clear to me how this helps.