# Supremum of centered jointly generalized chi-square random variables

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.

Here is how it works in the latter simple case, where I also take $m_n = n$. Now $E[\zeta_n^2] = 2 n \sigma_n^4$.
Put $r_n = a\log n$, $\lambda = \sigma_n^{-2} n^{-1/2}$. Then
\begin{gathered} P\left(\zeta_n> (E[\zeta_n^2])^{1/2} r_n \right) = P\left(\zeta_n> \sqrt{2n}\sigma_n^2 r_n \right)\le \frac{E[e^{\lambda\zeta_n}]}{e^{\lambda \sqrt{2n}\sigma_n^2 r_n }} \\ = (1-2\lambda \sigma_n^2)^{-n/2} \exp\{ -\lambda n\sigma_n^2 - r_n\sqrt{2}\} = \exp\Big\{ -n^{1/2} - \frac n2 \log \big(1-2n^{-1/2}\big)- r_n\sqrt2 \Big\} \\ \le \exp\Big\{ K - r_n\sqrt{2}\Big\} \end{gathered} with some constant $K$. This means that taking $a>1/\sqrt{2}$ and using the Borel-Cantelli lemma we arrive at the desired result.