Let $X_i$ real random variables for $i \in 1, \dots n$ such that $$ \mathbb{P}\left(\|X_i\|\geq \frac{1}{n}\log\left(\frac{1}{\delta}\right) \right)\leq\delta $$
Is there a chance to find a constant $c>0$ independent of $n$ and $\delta$ such that
$$ \mathbb{P}\left(\sum_{i=1}^n\|X_i\|\geq \log\left(\frac{c}{\delta}\right) \right)\leq\delta? $$ Or does anyone know similar bound for this problem?
What i tried so far was using that $$ \mathbb{P}\left(\|X_i\|\geq \frac{1}{n}\log\left(\frac{1}{\delta}\right) \right)\leq\delta $$ implies a bound for moments (using the layer cake inequality) so we have $$ \mathbb{E}[\|X_i\|^q] \leq \frac{C^q}{n^q}. $$ From this i wanted to use markov ineq. : $$ \mathbb{P}\left(\sum_{i=1}^n\|X_i\|\geq \log\left(\frac{c}{\delta}\right) \right)\leq\frac{c}{\delta}\mathbb{E}[\exp(\sum_{i=1}^n\|X_i\|)]. $$ and then use the expansion of the exponential function... However i can not assume that the variables are independent so i don't know how to pull the sum out of the expectation. Any suggestions are verry appreciated.