Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-28T05:51:41.168Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic behavior of projections of supercritical multi-type continuous-state and continuous-time branching processes with immigration

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  22 November 2021

Mátyás Barczy Open the ORCID record for Mátyás Barczy [Opens in a new window] ,
Sandra Palau  and
Gyula Pap
Mátyás Barczy*
Affiliation:
University of Szeged
Sandra Palau*
Affiliation:
Universidad Nacional Autónoma de México
Gyula Pap*
Affiliation:
University of Szeged
*
*Postal address: MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H–6720 Szeged, Hungary.
**Postal address: Department of Statistics and Probability, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, México.
*Postal address: MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H–6720 Szeged, Hungary.
Get access Rights & Permissions [Opens in a new window]

Abstract

Under a fourth-order moment condition on the branching and a second-order moment condition on the immigration mechanisms, we show that an appropriately scaled projection of a supercritical and irreducible continuous-state and continuous-time branching process with immigration on certain left non-Perron eigenvectors of the branching mean matrix is asymptotically mixed normal. With an appropriate random scaling, under some conditional probability measure, we prove asymptotic normality as well. In the case of a non-trivial process, under a first-order moment condition on the immigration mechanism, we also prove the convergence of the relative frequencies of distinct types of individuals on a suitable event; for instance, if the immigration mechanism does not vanish, then this convergence holds almost surely.

Keywords

Multi-type continuous-state and continuous-time branching processes with immigrationmixed normal distribution

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F15: Strong theorems
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 4 , December 2021 , pp. 1023 - 1060
Check for updates
Copyright
© The Author(s) 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Athreya, K. B. (1968). Some results on multitype continuous time Markov branching processes. Ann. Math. Statist. 39, 347–357.10.1214/aoms/1177698395CrossRefGoogle Scholar
Athreya, K. B. (1969). Limit theorems for multitype continuous time Markov branching processes. I. The case of an eigenvector linear functional. Z. Wahrscheinlichkeitsth. 12, 320–332.10.1007/BF00538753CrossRefGoogle Scholar
Athreya, K. B. (1969). Limit theorems for multitype continuous time Markov branching processes. II. The case of an arbitrary linear functional. Z. Wahrscheinlichkeitsth. 13, 204–214.10.1007/BF00539201CrossRefGoogle Scholar
Badalbaev, I. S. and Mukhitdinov, A. (1990). Limit distributions of some functionals in multitype branching processes. Theory Prob. Appl. 35, 625–638.10.1137/1135095CrossRefGoogle Scholar
Barczy, M., Li, Z. and Pap, G. (2015). Yamada–Watanabe results for stochastic differential equations with jumps. Internat. J. Stoch. Anal. 2015, 460472, 23 pp.Google Scholar
Barczy, M., Li, Z. and Pap, G. (2015). Stochastic differential equation with jumps for multi-type continuous state and continuous time branching processes with immigration. ALEA Latin Amer. J. Prob. Math. Statist. 12, 129–169.Google Scholar
Barczy, M., Li, Z. and Pap, G. (2016). Moment formulas for multi-type continuous state and continuous time branching processes with immigration. J. Theoret. Prob. 29, 958–995.10.1007/s10959-015-0605-0CrossRefGoogle Scholar
Barczy, M., Palau, S. and Pap, G. (2020). Almost sure, $L_1$ - and $L_2$ -growth behavior of supercritical multi-type continuous state and continuous time branching processes with immigration. Sci. China Math. 63, 2089–2116.10.1007/s11425-019-1552-1CrossRefGoogle Scholar
Barczy, M., Palau, S. and Pap, G. (2018). Asymptotic behavior of projections of supercritical multi-type continuous state and continuous time branching processes with immigration. Preprint. Available at https://arxiv.org/abs/1806.10559.Google Scholar
Barczy, M. and Pap, G. (2016). Asymptotic behavior of critical, irreducible multi-type continuous state and continuous time branching processes with immigration. Stoch. Dynamics 16, 1650008, 30 pp.Google Scholar
Buraczewski, D., Damek, E. and Mikosch, T. (2016). Stochastic Models with Power-Law Tails. Springer, Cham.10.1007/978-3-319-29679-1CrossRefGoogle Scholar
Crimaldi, I. and Pratelli, L. (2005). Convergence results for multivariate martingales. Stoch. Process. Appl. 115, 571–577.10.1016/j.spa.2004.10.004CrossRefGoogle Scholar
Duffie, D., Filipović, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Ann. Appl. Prob. 13, 984–1053.10.1214/aoap/1060202833CrossRefGoogle Scholar
He, X. and Li, Z. (2016). Distributions of jumps in a continuous-state branching process with immigration. J. Appl. Prob. 53, 1166–1177.10.1017/jpr.2016.72CrossRefGoogle Scholar
Horn, R. A. and Johnson, C. R. (2013). Matrix Analysis, 2nd edn. Cambridge University Press.Google Scholar
Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland/Kodansha, Amsterdam/Tokyo.Google Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin.10.1007/978-3-662-05265-5CrossRefGoogle Scholar
Jagers, P. (1969). The proportions of individuals of different kinds in two-type populations. A branching process problem arising in biology. J. Appl. Prob. 6, 249–260.10.2307/3211996CrossRefGoogle Scholar
Janson, S. (2004). Functional limit theorems for multitype branching processes and generalized PÓlya urns. Stoch. Process. Appl. 110, 177–245.10.1016/j.spa.2003.12.002CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York.Google Scholar
Kesten, H. and Stigum, B. P. (1966). Additional limit theorems for indecomposable multidimensional Galton–Watson processes. Ann. Math. Statist. 37, 1463–1481.10.1214/aoms/1177699139CrossRefGoogle Scholar
Kyprianou, A. E., Palau, S. and Ren, Y.-X. (2018). Almost sure growth of supercritical multi-type continuous-state branching process. ALEA Latin Amer. J. Prob. Math. Statist. 15, 409–428.CrossRefGoogle Scholar
Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.CrossRefGoogle Scholar
Van der Vaart, A. W. (1998) Asymptotic Statistics. Cambridge University Press.10.1017/CBO9780511802256CrossRefGoogle Scholar
Yakovlev, A. Y. and Yanev, N. M. (2010). Limiting distributions for multitype branching processes. Stoch. Anal. Appl. 28, 1040–1060.10.1080/07362994.2010.515486CrossRefGoogle ScholarPubMed