Abstract
The aim of this paper is to introduce a multitype branching process with random migration following the research initiated with the Galton–Watson process with migration introduced in [N. M. Yanev and K. V. Mitov, Controlled branching processes: The case of random migration, C. R. Acad. Bulgare Sci. 33 1980, 4, 473–475]. We focus our attention in what we call the critical case. Sufficient conditions are provided for the process to have unlimited growth or not. Furthermore, using suitable normalizing sequences, we study the asymptotic distribution of the process. Finally, we obtain a Feller-type diffusion approximation.
Funding source: Agencia Estatal de Investigación
Award Identifier / Grant number: PID2019-108211GB-I00
Funding statement: The authors are supported by grant PID2019-108211GB-I00 funded by MICIU/AEI/10.13039/501100011033. Pedro Martín-Chávez is also grateful to the Ministerio de Ciencias, Innovación y Universidades for support from a predoctoral fellowship Grant No. FPU20/06588.
Acknowledgements
We would like to thank the referee for her/his comments that helped us improve the paper.
References
[1] M. Barczy, M. González, P. Martín-Chávez, and I. del Puerto, Diffusion approximation of critical controlled multi-type branching processes, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 118 (2024), Article No. 101. 10.1007/s13398-024-01593-0Search in Google Scholar
[2] Y. S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Martingales, Springer Texts Statist., Springer, New York, 1997. 10.1007/978-1-4612-1950-7Search in Google Scholar
[3] E. E. Dyakonova, Branching processes that are close to critical with migration (in Russian), Teor. Veroyatnost. i Primenen. 41 (1996), no. 1, 186–192; translation in Theory Probab. Appl. 41 (1996), 186–192. Search in Google Scholar
[4] M. González, R. Martínez and M. Mota, On the unlimited growth of a class of homogeneous multitype Markov chains, Bernoulli 11 (2005), no. 3, 559–570. 10.3150/bj/1120591189Search in Google Scholar
[5] M. González, R. Martínez and M. Mota, Rates of growth in a class of homogeneous multidimensional Markov chains, J. Appl. Probab. 43 (2006), no. 1, 159–174. 10.1239/jap/1143936250Search in Google Scholar
[6] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University, Cambridge, 2013. Search in Google Scholar
[7] L. V. Khan, Limit theorems for a Galton–Watson branching process with immigration (in Russian), Sibirsk. Mat. Zh. 21 (1980), no. 2, 183–194; translation in Sib. Math. J. 21 (1980), 283–292. Search in Google Scholar
[8] A. Pakes, Immigration–emigration processes, Encyclopedia of Statistical Sciences, John Wiley and Sons, New York (2006), 1–5. 10.1002/0471667196.ess1212.pub2Search in Google Scholar
[9] I. Rahimov, Random Sums and Branching Stochastic Processes, Lect. Notes Stat 96, Springer, New York, 1995. 10.1007/978-1-4612-4216-1Search in Google Scholar
[10] V. A. Vatutin and A. M. Zubkov, Branching processes. II, J. Soviet. Mazh. 67 (1993), 3407–3485. 10.1007/BF01096272Search in Google Scholar
[11] G. P. Yanev and N. M. Yanev, Branching processes with two types emigration and state-dependent immigration, Athens Conference on Applied Probability and Time Series Analysis, Vol. I (1995), Lect. Notes Stat. 114, Springer, New York (1996), 216–228. 10.1007/978-1-4612-0749-8_15Search in Google Scholar
[12] G. P. Yanev and N. M. Yanev, Limit theorems for branching processes with random migration components, Pliska Stud. Math. Bulgar. 13 (2000), 199–205. Search in Google Scholar
[13] N. M. Yanev and K. V. Mitov, Controlled branching processes: The case of random migration, C. R. Acad. Bulgare Sci. 33 (1980), no. 4, 473–475. Search in Google Scholar
[14] N. M. Yanev and K. V. Mitov, Life periods of critical branching processes with random migration, Theory Probab. Appl. 28 (1983), no. 3, 458–467. 10.1137/1128045Search in Google Scholar
[15] N. M. Yanev and K. V. Mitov, Subcritical branching migration processes (in Russian), Pliska Stud. Math. Bulgar. 7 (1984), 75–82. Search in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- 80th Jubilee of Professor Nikolay Yanev
- My 55 Years in Stochastics
- On Subcritical Markov Branching Processes with a Specified Limiting Conditional Law
- Galton–Watson Theta-Processes in a Varying Environment
- Population Dependent Two-Sex Branching Process with Random Mating and Overlapping Generations
- Critical Multitype Branching Processes with Random Migration
- Branching Processes Under Nonstandard Conditions
Articles in the same Issue
- Frontmatter
- 80th Jubilee of Professor Nikolay Yanev
- My 55 Years in Stochastics
- On Subcritical Markov Branching Processes with a Specified Limiting Conditional Law
- Galton–Watson Theta-Processes in a Varying Environment
- Population Dependent Two-Sex Branching Process with Random Mating and Overlapping Generations
- Critical Multitype Branching Processes with Random Migration
- Branching Processes Under Nonstandard Conditions