The main body of this paper has two rather distinct parts, the first part (Section 2) containing the derivation of an explicit representation for the joint distribution of accident times in quite a general situation, and the remainder of the paper dealing with a very special case which leads to a discussion of testing the hypothesis that a distribution is uniform over (0, 1) against alternatives which are exponential truncated to (0, 1). The second part may be read profitably with little reference to the probabilistic arguments of Section 2.
Grace E. Bates. "Joint Distributions of Time Intervals for the Occurrence of Successive Accidents in a Generalized Polya Scheme." Ann. Math. Statist. 26 (4) 705 - 720, December, 1955. https://doi.org/10.1214/aoms/1177728429