I'm interested in methods for proving one-sided bounds of the form $$ \mathbb{P}[\frac{1}{n}\sum_{i=1}^n X^4_i \geq 3+t]\leq Ce^{-nt} $$ where $X_i$ are standard normal random variables. I've run a few basic random experiments, it seems that this should be possible for t relatively large (but independent of $n$). Note that I don't care about concentration, just a loose upper bound.
Is this possible?