Let $\zeta_n$ be a sequence of centered jointly generalized chi-square random variables, i.e. $\zeta_n = \sum_{k=1}^{m_n} a_{k,n}(\xi_{k,n}^2 - E[\xi_{k,n}^2])$, and $\xi_{k,n}$ are centered jointly Gaussian.

I'm looking at the results of the form $$ \zeta_n = O\big((E[\zeta_n^2])^{1/2}\log n\big), n\to\infty, $$ almost surely.

It is easy to deduce from hypercontractivity that $\zeta_n = o((E[\zeta_n^2])^{1/2}n^a)$ for all $a>0$, but thanks to the exponential tails of chi-square I conjecture something like logarithmic behavior.

Even a simple case where $\xi_{k,n}$ are independent $N(0,\sigma_n^2)$, and $a_{n,k}=1$ would be interesting.