Proof of a simple time-step propagation scheme for Monte Carlo simulation

An explicit proof of a simple time-step propagation scheme is given in the framework of basic probability theory. It can be used in Monte Carlo simulations solving the Boltzmann transport equation. If the stochastically selected first scattering event occurs before a given time t1, the particle is propagated as usual until the end of the corresponding free-flight time; otherwise, however, the propagation can be stopped in this scheme at t1 and a new random number can be generated to decide whether the first scattering event occurs before the next specified time t2.