Suppose that $f \colon [0, 1] \to \mathbb{R}$ is a $1$-Lipschitz function. Define the uniform norm $\|f\|_\infty = \sup_{x} |f(x)|$.

Given $\{U_j\}_{j=1}^\infty$ independent and identically distributed uniform random variables on the unit interval $[0, 1]$, I am curious to know about the behavior of the random variable
$$ \Delta_n(f) := \Big|\max_{j \leq n} |f(U_j)| - \|f\|_\infty\Big| $$ I would roughly expect that $\Delta_n \sim \frac{1}{n}$, with high probability simply because on average one of the $n$ samples $U_1, \dots, U_n$ should lie within $1/n$ of the maximum, and therefore, the error should be of this order. I am also interested to what degree this can be made uniform over Lipschitz functions $f$, i.e., by studying the maximal quantity $\sup_{f \in \mathcal{F}} \Delta_n(f)$ where $\mathcal{F}$ is a suitable family of Lipschitz functions (say with $f(0) = 0$). Is this question studied? If so, are there any references available?