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A simple proof of Feller's characterization of the compound Poisson distributions
Affiliation:1. Department of Conservation Biology, J. F. Blumenbach Institute of Zoology and Anthropology, University of Goettingen, Bürgerstr. 50, 37073 Goettingen, Germany;2. School of Biological Sciences, University of Aberdeen, Zoology Building, Tillydrone Avenue, Aberdeen AB24 2TZ, UK;3. Sahel Square, Parsia Complex, Sarv Building, PO Box 14938-89881, Tehran, Iran;4. Department of Environmental Sciences, Faculty of Natural Resources, University of Tehran, P.O. Box 4111, Karaj, Iran;5. Department of Environmental Sciences, Faculty of Natural Resources and Environment, Islamic Azad University, Science and Research Branch, Simon Bulivar Blvd, P.O. Box 14515/775, Tehran, Iran;6. Human Dimensions of Natural Resources Department & Center for Human-Carnivore Coexistence, Colorado State University, Fort Collins, CO 80525, USA;7. Kooch Foundation for Communities and Biodiversity Conservation, Banafsheh Street, P.O. Box 1477977456, Tehran, Iran;8. Department of Computational Landscape Ecology, UFZ—Helmholtz Centre for Environmental Research, Permoserstr. 15, 04318 Leipzig, Germany;9. Department of Natural Environment and Biodiversity, College of Environment and Research Group of Biodiversity and Biosafety, Research Center for Environment and Sustainable Development, Department of Environment, Tehran, Iran.;10. Department of Environment, Faculty of Natural Resources, University of Kurdistan, Iran;11. CIBIO, Research Centre in Biodiversity and Genetic Resources, Campus de Vairão, University of Porto, 4485-661 Vairão, Portugal;12. Faculty of Biological Sciences and Technology, Shahid Beheshti University, G.C, Daneshjoo St, PO Box 1983969411, Tehran, Iran.;13. Wildlife Sciences, University of Goettingen, Buesgenweg 3, 37077 Goettingen, Germany.;14. U.S. Geological Survey, Eastern Ecological Science Center, MD, USA.;15. CSIRO Land and Water, PMB 44, Winnellie, Darwin 0822, Northern Territory, Australia.;1. School of Economics and Management, Nanjing University of Science and Technology, China;2. School of Finance, Nanjing Audit University, China;3. School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, China;4. Antai College of Economics and Management, Shanghai Jiao Tong University, China
Abstract:A classical result of W. Feller is that every distribution that is infinitely divisible and concentrated on the non-negative integers is compound Poisson. We give a simple proof that uses some of the recursive formulas that have recently become popular among actuaries.
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