Let $\Theta$ be a subset of a metric space. Suppose $(X\_\theta)\_{\theta \in \Theta}$ is a random process on $\Theta$ which is $L$-Lipschitz and with the property that there exists constants $A, B>0$ such that for every $\epsilon>0$ and $\theta \in \Theta$, it holds that $P(X\_\theta \ge \epsilon) \le A\exp(-B\epsilon^2)$.


Question
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* What upper bounds can be obtained on $P(\sup\_{\theta \in \Theta} X\_\theta \ge \epsilon)$ in terms of the Lipschitz constant $L$, and the covering number of $\Theta$ ?
* In case the conditions are not sufficient, what can be added in order to obtain any interesting answers ?



Use covering number to get uniform concentration from pointwise concentration
If for all $\theta \in\Theta$ we have $P(X\_\theta\ge\epsilon)\le
A\exp(-B\epsilon^2)$ and $\Theta$ has $\epsilon$-packing number $M(\epsilon)$ and additionally $X\_\theta$ is $L$-Lipschitz in $\theta$, then
$$P(\sup\_{\theta\in\Theta} X\_\theta>\epsilon(L+1))
\le AM(\epsilon)\exp(-B\epsilon^2)
.$$


