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Let $ p_1, p_2,p_3 \sim U([0, 1]^d) $ be three points in the $d$-dimensional unit hypercube which are uniform randomly independently sampled.

This post shows that distance $ D(p_1, p_2) = \sqrt{\sum_{i=1}^d { \left (p_1^{(i)} - p_2^{(i)} \right )}^2} $ can be approximated by a normal distribution with the mean $ \sqrt{\frac{d}{6} - \frac{7}{120}} $ and with the variance $ \frac{7}{120}$.

I wonder how does $D(p_1,p_2) + D(p_2, p_3)$ distribute?

I found by simulation that it looks like it could be approximated by a normal distribution as well but I could not proove it.

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    But $D(p_1,p_2)$ and $D(p_2,p_3)$ are not independent. – 2017-02-06
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    I do not see why they are weakly correlated (and simulation does not shows a low correlation as well). The result variance is something like: $ 1.5 \times \frac{7}{120} $. – 2017-02-06

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