Let $X_1, \ldots, X_n$ be $n$ points chosen uniformly on [0, 1]. Let $Y_i = \displaystyle \min_{j \neq i} \{ |X_i - X_j| \}$ be the shortest distance between $X_i$ and any of the other points. Can we say anything interesting about the sample variance of $Y_i$? In other words, if $\overline{Y} = \frac{1}{n}\sum Y_i$, define
$$ S = \frac{1}{n - 1} \displaystyle \sum_{i=1}^n (Y_i - \overline{Y})^2 $$
Do we know anything about S, e.g., its expectation?
What if the points $X_i$ are chosen uniformly from the unit square instead of the unit interval?
This question is motivated by pseudorandom numbers. Given a large set of numbers between 0 and 1, if we knew enough about the asymptotic distribution of $S$, then we could perform a hypothesis test to see whether the numbers "look" uniform. For example, we could rule out a perfectly evenly spaced set of numbers because $S = 0$ is very unusual.