Skip to main content Accessibility help
×
Home
Hostname: page-component-747cfc64b6-nvdzj Total loading time: 0.248 Render date: 2021-06-15T14:01:08.200Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

Speed of coming down from infinity for birth-and-death processes

Published online by Cambridge University Press:  11 January 2017

Vincent Bansaye
Affiliation:
École Polytechnique
Sylvie Méléard
Affiliation:
École Polytechnique
Mathieu Richard
Affiliation:
École Polytechnique

Abstract

We describe in detail the speed of `coming down from infinity' for birth-and-death processes which eventually become extinct. Under general assumptions on the birth-and-death rates, we firstly determine the behavior of the successive hitting times of large integers. We identify two different regimes depending on whether the mean time for the process to go from n+1 to n is negligible or not compared to the mean time to reach n from ∞. In the first regime, the coming down from infinity is very fast and the convergence is weak. In the second regime, the coming down from infinity is gradual and a law of large numbers and a central limit theorem for the hitting times sequence hold. By an inversion procedure, we deduce that the process is almost surely equivalent to a nonincreasing function when the time goes to 0. Our results are illustrated by several examples including applications to population dynamics and population genetics. The particular case where the death rate varies regularly is studied in detail.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below.

References

[1] Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5, 348.CrossRefGoogle Scholar
[2] Allen, L. J. S. (2011). An Introduction to Stochastic Processes with Applications to Biology, 2nd edn. CRC Press, Boca Raton, FL.Google Scholar
[3] Anderson, W. J. (1991). Continuous-Time Markov Chains: An Applications-Oriented Approach. Springer, New York.CrossRefGoogle Scholar
[4] Berestycki, J., Berestycki, N. and Limic, V. (2010). The λ-coalescent speed of coming down from infinity. Ann. Prob. 38, 207233.CrossRefGoogle Scholar
[5] Billingsley, P. (1986). Probability and Measure, 2nd edn. John Wiley, New York.Google Scholar
[6] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation (Encyclopedia Math. Appl. 27). Cambridge University Press.Google Scholar
[7] Cattiaux, P. et al. (2009). Quasi-stationary distributions and diffusion models in population dynamics. Ann. Prob. 37, 19261969. CrossRefGoogle Scholar
[8] Donnelly, P. (1991). Weak convergence to a Markov chain with an entrance boundary: ancestral processes in population genetics. Ann. Prob. 19, 11021117.CrossRefGoogle Scholar
[9] Karlin, S. and McGregor, J. L. (1957). The differential equations of birth-and-death processes, and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85, 489546.CrossRefGoogle Scholar
[10] Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.Google Scholar
[11] Klesov, O. I. (1983). The rate of convergence of series of random variables. Ukrain. Mat. Zh. 35, 309314.Google Scholar
[12] Kot, M. (2001). Elements of Mathematical Ecology. Cambridge University Press.CrossRefGoogle Scholar
[13] Lambert, A. (2005). The branching process with logistic growth. Ann. Appl. Prob. 15, 15061535.CrossRefGoogle Scholar
[14] Limic, V. and Talarczyk, A. (2010). Diffusion limits for mixed with Kingman coalescents at small times. Preprint. Available at http://arxiv.org/abs/1409.6200v1.Google Scholar
[15] Slade, P. F. and Wakeley, J. (2005). The structured ancestral selection graph and the many-demes limit. Genetics 169, 11171131.CrossRefGoogle ScholarPubMed
[16] Sibly, R. M. et al. (2005). On the regulation of populations of mammals, birds, fish, and insects. Science 309, 607610.CrossRefGoogle ScholarPubMed
[17] Taylor, H. M. and Karlin, S. (1998). An Introduction to Stochastic Modeling, 3rd edn. Academic Press, San Diego, CA.Google Scholar
[18] Van Doorn, E. A. (1991). Quasi-stationary distributions and convergence to quasi-stationarity of birth–death processes. Adv. Appl. Prob. 23, 683700.CrossRefGoogle Scholar
10
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Speed of coming down from infinity for birth-and-death processes
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Speed of coming down from infinity for birth-and-death processes
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Speed of coming down from infinity for birth-and-death processes
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *