Let $X_1,\dots,X_n\in {\bf R}^d$ be $d$-dimensional iid zero mean random vectors with covariance matrix $\Sigma$. I am interested in tail bounds for the Euclidean norm $$N_n\equiv \frac{1}{\sqrt{n}}\|X_1 + X_2 + \dots X_n\|.$$ The main assumption should be that the one-dimensional marginals of $X_1$ have finite $L^p$ norm for some $p>4$, ie that there exists $c>0$ such that: $${\bf (A)}_p:\forall v\in {\bf R}^d, {\bf Ex}\langle X_1,v\rangle^p\leq c \langle v,Cv\rangle^{p/2}$$ where $\langle,\rangle$ is the usual inner product. More specifically, I am interested in Central-Limit-Theorem like bounds of the form: $$(\star)\Pr(N_n>\sqrt{a_1 tr(\Sigma) + a_2 \|\Sigma\|_{{\rm op}}t})\leq e^{-t} + C n^{-\beta}$$ where $a_1,a_2$ are universal and $\beta,C$ depend only on $c$ and $p$ (and not on the dimension $d$); $tr(\Sigma)$ is the trace; and $\|\Sigma\|_{{\rm op}}$ is the operator norm.

If the $X_i$ are Gaussian, this result follows from an explicit calculation without the $n^{-\beta}$ term. A similar result holds if the $X_i$ are subgaussian [1].

Much less seems to be known for more general distributions. All results I found (see eg. [2]) assume less about the $X_i$'s but also give poorer bounds with respect to dependence on d. On the other hand, these results also go for an actual CLT whereas I just want the looser inequality $(\star)$.

Is there a (perhaps obvious) obstruction to this? Do you know any references that prove/disprove $(\star)$ or something close?

[1] A tail inequality for quadratic forms of subgaussian random vectors. Daniel Hsu, Sham M. Kakade, and Tong Zhang. Electronic Communications in Probability, 17(52):1-6, 2012.

[2] On the rate of convergence of the multivariate CLT. F. Götze Annals of Probability, 19 (2): 724-739, 1991.