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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.

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    I suggest giving up immediately on the two-dimensional case, as it seems to be extremely messy. There is very little hope to get any closed form expressions; analyzing asymptotic behavior might be possible through Poisson approximation. Overall this sounds quite interesting, however, it is very desirable that you motivate why do you want to study this $S$. – 2017-02-20
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    I thought it could provide a cool way to test whether a pseudorandom number generator is sufficiently random. If you have some idea about the asymptotic distribution of S, then you could use it to determine whether a large set of numbers "looks" uniform via hypothesis testing. For example, we could rule out a perfectly evenly spaced set of numbers because S = 0 is very unusual. – 2017-02-20
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    Interesting. Could you add this motivation part to your question? I will try to write my thoughts when I have time. – 2017-02-20

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