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Branching processes in generalized autoregressive conditional environments

Part of: Inference from stochastic processes Limit theorems Markov processes Stochastic processes

Published online by Cambridge University Press:  11 January 2017

Irene Hueter
Irene Hueter*
Affiliation:
Columbia University
*
* Postal address: Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, NY 10027, USA. Email address: [email protected]
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Abstract

Branching processes in random environments have been widely studied and applied to population growth systems to model the spread of epidemics, infectious diseases, cancerous tumor growth, and social network traffic. However, Ebola virus, tuberculosis infections, and avian flu grow or change at rates that vary with time—at peak rates during pandemic time periods, while at low rates when near extinction. The branching processes in generalized autoregressive conditional environments we propose provide a novel approach to branching processes that allows for such time-varying random environments and instances of peak growth and near extinction-type rates. Offspring distributions we consider to illustrate the model include the generalized Poisson, binomial, and negative binomial integer-valued GARCH models. We establish conditions on the environmental process that guarantee stationarity and ergodicity of the mean offspring number and environmental processes and provide equations from which their variances, autocorrelation, and cross-correlation functions can be deduced. Furthermore, we present results on fundamental questions of importance to these processes—the survival-extinction dichotomy, growth behavior, necessary and sufficient conditions for noncertain extinction, characterization of the phase transition between the subcritical and supercritical regimes, and survival behavior in each phase and at criticality.

Keywords

Branching processes in random environmentGARCHGalton–Watson processextinctionphase transitionlimit theorems

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 62M10: Time series, auto-correlation, regression, etc. 60G10: Stationary processes 60F05: Central limit and other weak theorems
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 4 , December 2016 , pp. 1211 - 1234
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Athreya, K. B. and Karlin, S. (1971). On branching processes with random environments. I. Extinction probabilities. Ann. Math. Statist. 42, 1499–1520. CrossRefGoogle Scholar
[2] Athreya, K. B. and Karlin, S. (1971). Branching processes with random environments. II. Limit theorems. Ann. Math. Statist. 42, 1843–1858. Google Scholar
[3] Athreya, K. B. and Ney, P. E. (2004). Branching Processes, Dover, Mineola, NY. Google Scholar
[4] Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31, 307–327.Google Scholar
[5] Coffey, J. and Tanny, D. (1984). A necessary and sufficient condition for noncertain extinction of a branching process in a random environment (BPRE). Stoch. Proc. Appl. 16, 189–197.Google Scholar
[6] Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1007.CrossRefGoogle Scholar
[7] Ferland, R. Latour, A. and Oraichi, D. (2006). Integer-valued GARCH process. J. Time Series Anal. 27, 923–942.Google Scholar
[8] Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.Google Scholar
[9] Heinen, A. (2001). Modeling time series count data: the autoregressive conditional poisson model. Doctoral Thesis, Department of Economics, University of California, San Diego.Google Scholar
[10] Heyde, C. C. (1970). Extension of a result of Seneta for the super-critical Galton–Watson process. Ann. Math. Statist. 41, 739–742.Google Scholar
[11] Hueter, I. (2014). Interventions in GARCE branching processes. Preprint.Google Scholar
[12] Kesten, H. and Stigum, B. P. (1966). A limit theorem for multidimensional Galton–Watson processes. Ann. Math. Statist. 37, 1211–1223.Google Scholar
[13] Levinson, N. (1959). Limiting theorems for Galton–Watson branching process. Illinois. J. Math. 3, 554–565.Google Scholar
[14] Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin.Google Scholar
[15] Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of \italic{L} log \italic{L} criteria for mean behavior of branching processes. Ann. Prob. 23, 1125–1138.CrossRefGoogle Scholar
[16] Neumann, M. H. (2011). Absolute regularity and ergodicity of Poisson count processes. Bernoulli 17, 1268–1284.CrossRefGoogle Scholar
[17] Rechavi, O. et al. (2014). Starvation-induced transgenerational inheritance of small RNAs in c. elegans. . Cell 158, 277–287.Google Scholar
[18] Seneta, E. (1968). On recent theorems concerning the supercritical Galton–Watson process. Ann. Math. Statist. 39, 2098–2102.CrossRefGoogle Scholar
[19] Seneta, E. (1975). Normed-convergence theory for supercritical branching processes. Stoch. Proc. Appl. 3, 35–43.Google Scholar
[20] Skinner, M. K. (2014). A new kind of inheritance. Scientific American 311, 44–51.Google Scholar
[21] Smith, W. L. (1968). Necessary conditions for almost sure extinction of a branching process with random environment. Ann. Math. Statist. 39, 2136–2140.Google Scholar
[22] Smith, W. L. and Wilkinson, W. E. (1969). On branching processes in random environments. Ann. Math. Statist. 40, 814–827.Google Scholar
[23] Smith, W. L. and Wilkinson, W. E. (1971). Branching processes in Markovian environments. Duke Math. J. 38, 749–763.CrossRefGoogle Scholar
[24] Tanny, D. (1977). Limit theorems for branching processes in a random environment. Ann. Prob. 5, 100–116.CrossRefGoogle Scholar
[25] Tanny, D. (1978). Normalizing constants for branching processes in random environments (B.P.R.E.). Stoch. Proc. Appl. 6, 201–211.Google Scholar
[26] Tanny, D. (1988). A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means. Stoch. Proc. Appl. 28, 123–139.Google Scholar
[27] Weiss, C. H. (2009). Modelling time series of counts with overdispersion. Statist. Meth. Appl. 18, 507–519.Google Scholar
[28] Wilkinson, E. E. (1969). On calculating extinction probabilities for branching processes in random environments. J. Appl. Prob. 6, 478–492.CrossRefGoogle Scholar
[29] Zhu, F. (2011). A negative binomial integer-valued GARCH model. J. Time Series Anal. 32, 54–67.CrossRefGoogle Scholar
[30] Zhu, F. (2012). Modeling overdispersed or underdispersed count data with generalized Poisson integer-valued GARCH models. J. Math. Anal. Appl. 389, 58–71.Google Scholar
[31] Zhu, F. (2012). Zero-inflated Poisson and negative binomial integer-valued GARCH models. J. Statist. Planning Infer. 142, 826–839.Google Scholar