Arbitrary Event Initial Conditions for Branching Poisson Processes
ANTHONY J. LAWRANCE · Journal of the Royal Statistical Society Series B (Statistical Methodology) · 1972
Summary Arbitrary event initial conditions for the branching Poisson process (the Bartlett–Lewis cluster process) are obtained. These correspond to the process starting at an arbitrary event, where an arbitrary event is defined in the Khintchine fashion. The conditions give the joint distribution of the number and structure of the initial subsidiary processes; in particular, the marginal distribution of the number of initial subsidiary processes is shown to be the mixture of a Poisson variable and a Poisson variable plus one. It is shown that these conditions marginally repeat themselves at each subsequent event, and thus that the sequence of intervals between events forms a stationary sequence of random variables.