Let $X_1, ..., X_n$ be i.i.d. sub-Gaussian random variables with mean $0$ and variance $1$. That is, we have $Pr[|X_i| > t] \leq \exp(1-t^2/K^2)$ for all $t>0$ and a parameter $K$.

Then we can use Bernsein-type inequalities to prove deviation bounds for the summation $S := \sum_{i=1}^n X_i^2$ (after appropriate shifting). Namely, we can prove, for an absolute constant $c > 0$,

$Pr[S > n+t] \leq 2 \exp(-c\min(t^2/(K^4 n), t/K^2))$.

Now suppose that, instead, we wish to derive deviation bounds for the summation $S_2 := \sum_{i=1}^n X_i^4$. Obviously,

$Pr[S_2 > t] \leq Pr[S > \sqrt{t}]$ since $S_2 \leq S^2$, and one can use the above Bernstein-type bound for $S$. But is it possible to do better? Are there tighter concentration bounds for summation of powers of independent sub-exponential random variables?