Neighbourhoods of Poisson Randomness, Independence, and Uniformity

As we have mentioned before, colloquially in applications, it is very common to encounter the usage of ‘random’ to mean the specific case of a Poisson process §1.1.3 whereas formally in statistics, the term random has a more general meaning: probabilistic, that is dependent on random variables. When we speak of neighbourhoods of randomness we shall mean neighbourhoods of a Poisson process and then the neighbourhoods contain perturbations of the Poisson process. Similarly, we consider processes that are perturbations of a process controlled by a uniform distribution on a finite interval, yielding neighbourhoods of uniformity. The third situation of interest is when we have a bivariate process controlled by independent exponential, gamma or Gaussian distributions; then perturbations are contained in neighbourhoods of independence. These neighbourhoods all have well-defined metric structures determined by information theoretic maximum likelihood methods. This allows trajectories in the space of processes, commonly arising in practice by altering input conditions, to be studied unambiguously with geometric tools and to present a background on which to describe the output features of interest of processes and products during changes.