Thinning of point processes-covariance analyses

Abstract

This dissertation addresses a class of problems in point process theory called 'thinning'. By thinning we mean an operation whereby a point process is split into two point processes by some rule. We obtain the covariance structure between the thinned processes under various thinning rules. We first obtain this structure for independent Bernoulli thinning of an arbitrary point process. We show that if the point process is a renewal (stationary or ordinary) process, the thinned processes will be uncorrelated if and only if the renewal process is Poisson in which case the thinned processes are independent. Thus, we have a situation where zero correlation implies independence. We also show that while the intervals between events in the thinned processes may be uncorrelated, the counts need not be.

Next, we obtain the covariance structure between the thinned processes resulting from a mark dependent thinning of a Markov renewal process with a Polish mark space. These results are used to study the overflow queue where we show that in equilibrium the input and overflow processes are uncorrelated as are the output and overflow processes. We thus provide an example where two uncorrelated but dependent renewal processes, neither of which is Poisson but which produce a Poisson process when superposed.

Next, we study Bernoulli thinning of an alternating Markov process and show that the thinned process are uncorrelated if and only if the process is Poisson in which case the thinned processes are independent. Finally, we obtain the covariance structure obtained when a renewal process is thinned by an independent Markov chain. We show that if the renewal process is Poisson and the chain is stationary, the thinned processes will be uncorrelated if and only if the Markov chain is a Bernoulli process. We also show how to design the chain to obtain a pre-specified covariance function.

We then close the dissertation by summarizing the work presented and indicating areas of further research including a short discussion of the use of Laplace functionals in the determination of joint distributions of thinned processes.