Various best-choice problems related to the planar homogeneous Poisson process in finite or semi-infinite rectangle are studied. The analysis is largely based on the properties of the one-dimensional box-area process associated with the sequence of records. We prove a series of distributional identities involving exponential and uniform random variables and resolve the Petruccelli-Porosinski-Samuels paradox on coincidence of asymptatic values in certain discrete time optimal stopping problems.
The paper is available on arXiv.org