Estimates on the discrepancy of random sequences

Question

The discrepancy of a $[0,1]$-valued sequence $n \mapsto \alpha_n$ is the quantity $$D(N; \alpha) \stackrel{\text{def}}{=} \sup_{(a,b) \subset [0,1]} \left|\frac{\#\{1 \leq n \leq N : \alpha_n \in (a,b)\}}{N} - (b-a)\right|$$

This is of course a very well-studied concept. We typically care about the case where $\alpha$ is a so-called equidistributed sequence i.e., when $D(N; \alpha) \to 0$. Then the natural questions pertain to the rate of convergence. There are various estimates for $D(N;\alpha)$. The tightest lower bound (I believe due to W.M. Schmidt) is that $D(N; \alpha) \gtrsim \frac{\log N}{N}$, in the sense that there exist absolute constants $c,C>0$ such that (i) for every sequence $\alpha$, $\limsup_{N} \frac{D(N; \alpha)}{\frac{\log N}{N}} \geq c$ and (ii) there exists a sequence $\alpha$ with $\limsup_{N} \frac{D(N; \alpha)}{\frac{\log N}{N}} \leq C$ (an example of the latter is the van der Corput sequence).

The setting where this is typically studied is deterministic i.e., when $(\alpha_n)$ is a fixed (non-random) sequence. I would like to study this in a probabilistic setting.

Let $X=(X_i)_{i \in \mathbb{N}}$ be a sequence of independent $\text{Uniform}([0,1])$ random variables. Then it is not hard to show that $D(N; X) \to 0$ almost surely i.e., that $X$ is equidistributed almost surely. A natural question to ask (among others) is about estimating $\mathbb{E}[D(N; X)]$.

Is there any literature on this, or, more broadly, studying the discrepancy of random sequences?

Answer 1

$\mathbb{E}[D(N; X)]$ is of order $N^{-1/2}$. This, and more precise information, follows from the analysis of the Kolmogorov-Smirnov test [1].
See, in particular, [2] and the references described on page 98 of the book [3]. Interesting extensions to higher dimensions are in [4].

[1] https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test

[2] Niederreiter, H. "Metric theorems on the distribution of sequences." In Proc. Sympos. Pure Math, vol. 24, pp. 195-212. 1973.

[3] Kuipers, L. and Niederreiter, H., 2012. Uniform distribution of sequences. Courier Corporation.

[4] Stute, Winfried. "Convergence rates for the isotrope discrepancy." The Annals of Probability (1977): 707-723.

Answer 2

Doing a quick search I just found this:

Probabilistic discrepancy bound for Monte Carlo point sets https://arxiv.org/pdf/1211.1058.pdf

Where they conclude on an upper bound as $D < c(q) N^{-1/2}$ with probability $q$. Are there better estimates?