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Limit theorems for pure death processes coming down from infinity

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  15 September 2017

Serik Sagitov  and
Thibaut France
Serik Sagitov*
Affiliation:
Chalmers University of Technology and University of Gothenburg
Thibaut France*
Affiliation:
École Polytechnique
*
* Postal address: Mathematical Sciences, Chalmers University, Gothenburg, 412 96, Sweden. Email address: serik@chalmers.se
** Postal address: École Polytechnique, route de Saclay, 91128 Palaiseau Cedex, France.
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Abstract

In this paper we treat a pure death process coming down from infinity as a natural generalization of the death process associated with the Kingman coalescent. We establish a number of limit theorems including a strong law of large numbers and a large deviation theorem.

Keywords

Almost sure convergencelarge deviationKingman's coalescent

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60F10: Large deviations 60F15: Strong theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 3 , September 2017 , pp. 720 - 731
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Bansaye, V., Méléard, S. and Richard, M. (2016). Speed of coming down from infinity for birth-and-death processes. Adv. Appl. Prob. 48, 1183–1210. Google Scholar
[2] Berestycki, J., Berestycki, N. and Limic, V. (2010). The Λ-coalescent speed of coming down from infinity. Ann. Prob. 38, 207–233. Google Scholar
[3] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation (Encyclopedia Math. Appl. 27). Cambridge University Press. Google Scholar
[4] Depperschmidt, A., Pfaffelhuber, P. and Scheuringer, A. (2015). Some large deviations in Kingman's coalescent. Electron. Commun. Prob. 20, 7. Google Scholar
[5] Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248. Google Scholar
[6] Klesov, O. I. (1983). The rate of convergence of series of random variables. Ukrainian Math. J. 35, 309–314. Google Scholar
[7] Pakes, A. G. (1992). Divergence rates for explosive birth processes. Stoch. Process. Appl. 41, 91–99. Google Scholar
[8] Sagitov, S. M. (1996). On an explosive branching process. Theory Prob. Appl. 40, 575–577. Google Scholar
[9] Waugh, W. A. O'N. (1974). Modes of growth of counting processes with increasing arrival rates. J. Appl. Prob. 11, 237–247. Google Scholar