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In this formulation, it still isn't true. Take $\{X_i\}_{i=1}^n$i.i.d standard Gaussian. Take $\{Y_i\}_{i=1}^{n-1}$ i.i.d. standard Gaussian. Take $Y_n=\sum_{i=1}^n X_i/\sqrt{n}$. Then $|EX_i Y_j|\leq …

Another way to measure the concentration is through the rate function, which is larger for $Y$ than for $X$. See Section 7.2 of Dembo-Zeitouni "Large Deviations Techniques and Applications".

There is a large literature, too long to review here. Deheuvels 1989 and Chistyakov 1996 come to mind. But also try: Google search and consider the relevant hits. Hope this helps.

I think the following answers your question (and more): http://www-personal.umich.edu/~romanv/papers/product-random-deterministic.pdf

A general bound on the variance is given by the Borell (Tsirelson-Ibragimov-Sudakov) inequality, see http://webee.technion.ac.il/people/adler/borell.pdf Without more structure on A I don't think i …

For $p>1$, the random variables you discuss do not possess exponential moments; You are in the regime of large deviations with stretched exponential tails. See for example the following recent paper b …

Why not use second moment? For fixed $r$, the correlation between the events that $B_{G_n}(v,r)$ is a tree and the same event for $v'\neq v$ is small, of order $f(c)/n$. Try $r=2$ to see what I mean. …

This is worked out in some detail in the paper of Fan, Grama and Liu, J. Math Anal. Appl. 448 (2017), 538-566 (see in particular Theorem 2.1 there, and the references). Unfortunately I do not have an …