a fast uniform order statistic generator Can someone provide me with the mathematical expression for this code/function as a fast way to generate $n$ sorted  $U[0,1]$  random numbers:
rsunif <- function(n) { n1 <- n+1
   cE <- cumsum(rexp(n1)); cE[seq_len(n)]/cE[n1] }

 A: The R code means returning
$$(E_1,E_1+E_2,\ldots,E_1+\cdots+E_n)\Big/\sum_{i=1}^{n+1} E_i$$ and the result follows from checking that the differences between the cumulated sums of exponentials renormalised by the overall sum has the same distribution as the differences between order statistics for a uniform sample, $S_i=U_{(i)}-U_{(i-1)}$. This is described and established in the bible of simulation, Devroye's Non-Uniform Random Variate Generation (1986, pp. 207-219):



and also in Biau & Devroye Lectures on the Nearest Neighbor Method (pp.5-7)
Running a test to compare this spacing method with a direct ordering of uniform variates shows a clear advantage for this approach (for n=100 and 10⁷ replications, using R benchmark tool).
      test replications elapsed relative user.self sys.self user.child
2   direct          1e7 355.213    4.722   355.112    0.024          0
1 spacings          1e7  75.221    1.000    75.208    0.000          0

although increasing $n$ to $n=10^3$ reduces the gain:
      test replications elapsed relative user.self sys.self user.child
2   direct          1e6  96.225    1.886     96.20        0          0
1 spacings          1e6  51.029    1.000     51.02        0          0

