an elegant result on exponential spacings

A question on X validated I spotted in the train back from Lyon got me desperately seeking a reference in Devroye’s Generation Bible despite the abyssal wireless and a group of screeching urchins a few seats away from me… The question is about why

\sum_{i=1}^{n}(Y_i - Y_{(1)}) \sim \text{Gamma}(n-1, 1)

when the Y’s are standard exponentials. Since this reminded me immediately of exponential spacings, thanks to our Devroye fan-club reading group in Warwick,  I tried to download Devroye’s Chapter V and managed after a few aborts (and a significant increase in decibels from the family corner). The result by Sukhatme (1937) is in plain sight as Theorem 2.3 and is quite elegant as it relies on the fact that

\sum_{i=1}^n y_i=\sum_{j=1}^n (n-j+1)(y_{(j)}-y_{(j-1)})=\sum_{j=2}^n (y_{(j)}-y_{(1)})

hence sums up as a mere linear change of variables! (Pandurang Vasudeo Sukhatme (1911–1997) was an Indian statistician who worked on human nutrition and got the Guy Medal of the RSS in 1963.)

4 Responses to “an elegant result on exponential spacings”

  1. Georges Henry Says:

    And by the way, I do not understand your identity, which does not seem quite correct.

  2. Georges Henry Says:

    Without calculation, just recall that

    (Y_{(1)},\ldots,Y_{(n)})\sim (\frac{Y_n}{n},\  \frac{Y_n}{n}+ \frac{Y_{n-1}}{n-1},\cdots,  \frac{Y_n}{n}+ \frac{Y_{n-1}}{n-1}+\cdots+Y_1) which makes trivial the fact that
    \sum_{k=1}^n(Y_k-Y_{(1)})\sim \sum_{k=1}^{n-1}Y_k.

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