Assume that $f_1, f_2, f_3,\ldots$ are i.i.d. random functions $[0,1]\mapsto \mathbb{R}$ such that

(1) random variables $M_k=\sup_{x\in[0,1]}f_k(x)$ have exponential tails,

(2) $f$'s are a.s. Lipschitz, and random variables $L_k=\inf\{h>0: f_k \text{ is Lipschitz with constant $h$}\}$ have exponential tails,

(3) $\mathbb{E} f_k(x) = 0$ for all $x\in[0,1]$.

My question is: what can be said about the law of r.v. $X_n=\sup_{x\in[0,1]}\sum_{k=1}^n f_k(x)$? Can one show that $\mathbb{P}[X_n>an] < e^{-bn}$ (where $a>0$ and $b$ depends on $a$)? And/or for any $\varepsilon>0$ there exists large enough $v=v(\varepsilon)$ such that $\mathbb{P}[X_n>v\sqrt{n}]<\varepsilon$?

Assume that $f_1, f_2, f_3,\ldots$ are i.i.d. random functions $[0,1]\mapsto \mathbb{R}$ such that

(1) random variables $M_k=\sup_{x\in[0,1]}f_k(x)$ have exponential tails,

(2) $f$'s are a.s. Lipschitz, and random variables $L_k=\inf\{h>0: f_k \text{ is Lipschitz with constant $h$}\}$ have exponential tails,

(3) $\mathbb{E} f_k(x) = 0$ for all $x\in[0,1]$.

My question is: what can be said about the law of r.v. $X_n=\sup_{x\in[0,1]}\sum_{k=1}^n f_k(x)$? Can one show that $\mathbb{P}[X_n>an] < e^{-bn}$ (where $a>0$ is arbitrary and $b$ depends on $a$)? And/or for any $\varepsilon>0$ there exists large enough $v=v(\varepsilon)$ such that $\mathbb{P}[X_n>v\sqrt{n}]<\varepsilon$?