Im looking for a good concentration inequality that bounds the following probability:
$$ P\left(\max_{M\in\{1,... ,n\}}\left| \sum_{i=1}^M(Z_i^2-1)\phi_{M,i}\right|>\delta\right), $$ where $Z_i$ are iid standard normal variables and $\phi_{M,i}$ is a non stochastic Term depending on $M$ and $i$. Unfortunately, just bounding the max with the sum over all n is not enough... so i would be verry glad for any suggestion.