Let $U_1, \dots, U_n$ denote independent and identically distributed samples on the unit interval $[0, 1]$.
Let $U_{(i)}$ denote the order statistics, so that $U_{(1)} \leq U_{(2)} \leq \cdots \leq U_{(n)}$. We take $U_{(0)} = 0$.
Now define the differences $$ D_{j} := U_{(j)} - U_{(j-1)}, \quad \mbox{for}~j = 1, 2, \ldots, n. $$ Question: What is the distribution of the maximum $\max_j D_j$? In particular, I am interested in its density and the expectation $\mathbb{E}[\max_{j \leq n} D_j]$.
My thoughts: Marginally, I know that $D_j \sim \mathrm{Beta}(1, n)$. Moreover, except for the boundary constraints, $D_j$ should "roughly" behave independently. Additionally for $n$ large, independent Beta(1, $n$) random variables should concentrate very strongly around the mean, and so my conjecture is that $$ \mathbb{E}[\max_j D_j] = (1 + o(1)) \frac{1}{n+1}, \quad \mbox{as}~n \to \infty. $$ I tried to check this by using the formula for the joint density of the order statistics, but had difficulty solving the integral. Any ideas or suggestions? I was thinking there could potentially be an argument based on coupling the dependent random variables $D_j$ to some sequence of independent Beta samples; I tried this but didn't succeed to find a good coupling.