# Order statistics on the spacings between order statistics for the uniform distribution

For any natural $$n$$, let $$U_1,\dots,U_n$$ be independent identically distributed random variables each uniformly distributed on the interval $$[0,1]$$. As usual, let $$U_{n:1}\le\cdots\le U_{n:n}$$ denote the corresponding order statistics. Consider
$$$$\label{eq:G_i} G_i:=U_{n:i}-U_{n:i-1} \quad \text{for}\quad i=1,\dots,n+1,$$$$ where $$U_{n:0}:=0$$ and $$U_{n+1:n+1}:=1$$.

One may refer to the $$G_i$$'s as the gaps or, as it is usually done in the literature, the spacings between the consecutive order statistics. Let now $$G_{n+1:1}\le\cdots\le G_{n+1:n+1}$$ denote the ordered gaps $$G_1,\dots,G_{n+1}$$, so that the random sets $$\{G_{n+1:1},\dots, G_{n+1:n+1}\}$$ and $$\{G_1,\dots,G_{n+1}\}$$ are the same.

Fisher obtained the distribution of the largest gap, $$G_{n+1:n+1}$$. Fisher's result was generalized in this note, where the distribution of the $$k$$th smallest gap, $$G_{n+1:k}$$, was obtained, for each $$k=1,\dots,n+1$$.

Does anyone know other references to these or other related results in the literature?

I have just found out that this result is due to Irwin.

• Nov 26, 2019 at 9:09
• @BrendanMcKay : Thank you for your comment. Yes, that paper by Barton and David is where I found the reference to Irwin. Nov 26, 2019 at 14:42
• Note also that exercise 667 in Whitworth's DDC Exercises precedes Fisher by roughly 30 years.
– esg
Nov 27, 2019 at 18:07