Abstract
Random measures and point processes; Cox processes, randomization, and thinning; mixed Poisson and binomial processes; independence and symmetry criteria; Markov transition and rate kernels; embedded Markov chains and explosion; compound and pseudo-Poisson processes; ergodic behavior of irreducible chains
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The Poisson distribution was introduced by De Moivre (1711–12) and Poisson (1837) as an approximation to the binomial distribution. The associated process arose much later from miscellaneous applications. Thus, it was considered by Lundberg (1903) to model streams of insurance claims, by Rutherford and Geiger (1908) to describe the process of radioactive decay, and by Erlang (1909) to model the incoming traffic to a telephone exchange. Poisson random measures in higher dimensions appear implicitly in the work of Lévy (1934–35), whose treatment was later formalized by Ito (1942b).
The independent-increment characterization of Poisson processes goes back to Erlang (1909) and Levy (1934–35). Cox processes, originally introduced by Cox (1955) under the name of doubly stochastic Poisson processes, were thoroughly explored by Kingman (1964), Krickeberg (1972), and Grandell (1976). Thinnings were first considered by Rényi (1956). The binomial construction of general Poisson processes was noted independently by Kingman (1967) and Mecke (1967). One-dimensional uniqueness criteria were obtained, first in the Poisson case by Rényi (1967), and then in general by Mönch (1971), Kallenberg (1973a, 1986), and Grandell (1976). The mixed Poisson and binomial processes were studied extensively by Matthes et al. (1978) and Kallenberg (1986).
Markov chains in continuous time have been studied by many authors, beginning with Kolmogorov (1931a). The transition functions of general pure jump-type Markov processes were explored by Pospisil (1935–36) and Feller (1936, 1940), and the corresponding sample path properties were examined by Doeblin (1939b) and Doob (1942b). The first continuous-time version of the strong Markov property was obtained by Doob (1945).
Kingman (1993) gives an elementary introduction to Poisson processes with numerous applications. More detailed accounts, set in the context of general random measures and point processes, appear in Matthes et al. (1978), Kallenberg (1986), and Daley and Vere-Jones (1988). Introductions to continuous-time Markov chains are provided by many authors, beginning with Feller (1968). For a more comprehensive account, see Chung (1960). The underlying regenerative structure was examined by Kingman (1972).
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Kallenberg, O. (2002). Poisson and Pure Jump-Type Markov Processes. In: Foundations of Modern Probability. Probability and Its Applications. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4015-8_12
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DOI: https://doi.org/10.1007/978-1-4757-4015-8_12
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