Consider a system of noninteracting particles moving at constant speeds in a Euclidean space. It is shown that, if at two or more epochs the velocities are independent of the positions and are independent and identically distributed, then the positions must be given by a Cox process of a very special structure.
"On the Independence of Velocities in a System of Noninteracting Particles." Ann. Probab. 6 (5) 885 - 890, October, 1978. https://doi.org/10.1214/aop/1176995435