Abstract
Poisson point processes can be used as a cornerstone in the construction of very different stochastic objects such as, for example, infinitely divisible distributions, Markov processes with complex dynamics, objects of stochastic geometry and so forth.
In this chapter, we briefly develop the general framework of random measures and construct the Poisson point process and characterize it in terms of its Laplace transform. As an application we construct a certain subordinator and show that the Poisson point process is the invariant measure of systems of independent random walks. Via the connection with subordinators, in the third section, we construct two distributions that play prominent roles in population genetics: the Poisson-Dirichlet distribution and the GEM distribution.
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Klenke, A. (2014). The Poisson Point Process. In: Probability Theory. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-5361-0_24
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DOI: https://doi.org/10.1007/978-1-4471-5361-0_24
Publisher Name: Springer, London
Print ISBN: 978-1-4471-5360-3
Online ISBN: 978-1-4471-5361-0
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