Let $X_1, \ldots X_N \sim \operatorname{Unif}[0,1]$ and consider the intervals between successive order statistics: $[0, X_{(1)}], [X_{(1)}, X_{(2)}], \ldots, [X_{(N)}, 1]$.

What is the distribution of this vector of interval lengths?

I hypothesize that it is Dirichlet$(1, \ldots, 1)$, but I cannot prove it and my literature search is taking me to obscure 1800s papers. Any thoughts?

Let $X_1, \ldots, X_N \sim \operatorname{Unif}[0,1]$ and consider the intervals between successive order statistics: $[0, X_{(1)}], [X_{(1)}, X_{(2)}], \ldots, [X_{(N)}, 1]$.

What is the distribution of this vector of interval lengths?

I hypothesize that it is Dirichlet$(1, \ldots, 1)$, but I cannot prove it and my literature search is taking me to obscure 1800s papers. Any thoughts?

Let $X_1, ... X_N \sim Unif[0,1]$ and consider the intervals between successive order statistics: $[0, X_{(1)}], [X_{(1)}, X_{(2)}], ..., [X_{(N)}, 1]$.

What is the distribution of this vector of interval lengths?

I hypothesize that it is Dirichlet$(1, ..., 1)$, but I cannot prove it and my literature search is taking me to obscure 1800s papers. Any thoughts?

Let $X_1, \ldots X_N \sim \operatorname{Unif}[0,1]$ and consider the intervals between successive order statistics: $[0, X_{(1)}], [X_{(1)}, X_{(2)}], \ldots, [X_{(N)}, 1]$.

What is the distribution of this vector of interval lengths?

I hypothesize that it is Dirichlet$(1, \ldots, 1)$, but I cannot prove it and my literature search is taking me to obscure 1800s papers. Any thoughts?