Let $U_1, U_2,...,U_n$ be iid uniform random variables on $[0,1]$.
$U_{1,n}\leq U_{2,n}\leq...\leq U_{n,n}$ be the order statistics.
$U_{0,n}=0$ and $U_{n+1,n}=1$
The spacings are $Q_{i,n}=U_{i+1,n}-U_{i,n}$ with $0\leq i\leq n$
Show that for all $0 \leq i \neq j \leq n$ the vector
$(nQ_{i,n},nQ_{j,n})$ converges in distribution and find the limit
For $n\to\infty$, the process locally appears as a Poisson process, as the correlations caused by the fixed number of events become negligible. Thus for $(X,Y)=(nQ_{i,n},nQ_{j,n})$ the distribution converges to one where $X$ and $Y$ are independent and each follows an exponential distribution with rate $1$.