Given a homogeneous Poisson process on R with intensity λ, we prove that it is possible to partition the points into two sets, as a deterministic function of the process, and in an isometry-equivariant way, so that each set of points forms a homogeneous Poisson process, with any given pair of intensities summing to λ. In particular, this answers a question of Ball [3], who proved that in d = 1,...

We finished discussing Discrete-Time Markov Chains in the previous lecture, and are now heading towards Continuous-Time Markov Chains. Discrete-time Markov Chains are totally synchronized, whereas CTMCs are not. In preparation for CTMCs, we need to discuss the Exponential distribution and the Poisson arrival process. We say that a random variable X is distributed exponentially with rate λ, X ∼ ...

The fractional Poisson process (FPP) is a counting process with independent and identically distributed inter-event times following the Mittag-Leffler distribution. This process is very useful in several fields of applied and theoretical physics including models for anomalous diffusion. Contrary to the well-known Poisson process, the fractional Poisson process does not have stationary and indep...