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Homogeneous Branching Processes with Non-Homogeneous Immigration

  • Ibrahim Rahimov EMAIL logo

Abstract

The stationary immigration has a limited effect over the asymptotic behavior of the underlying branching process. It affects mostly the limiting distribution and the life-period of the process. In contrast, if the immigration rate changes over time, then the asymptotic behavior of the process is significantly different and a variety of new phenomena are observed. In this review we discuss branching processes with time non-homogeneous immigration. Our goal is to help researchers interested in the topic to familiarize themselves with the current state of research.

MSC 2010: 60J80; 62F12; 60G99

Acknowledgements

My sincere thanks to A. M. Zubkov and A. G. Pakes for useful discussions and valuable comments.

References

[1] M. K. Asadullin and S. V. Nagaev, Limit theorems for a critical branching process with immigration, Mat. Zametki 32 (1982), no. 4, 537–548, 591. 10.1007/BF01152384Search in Google Scholar

[2] K. B. Athreya and P. E. Ney, Branching Processes, Grundlehren Math. Wiss. 196, Springer, New York, 1972. 10.1007/978-3-642-65371-1Search in Google Scholar

[3] I. S. Badalbaev, Limit theorems for multitype critical Markov branching processes with immigration of decreasing intensity, Limit Theorems for Random Processes and Statistical Inference, “Fan”, Tashkent (1981), 6–19, 217. Search in Google Scholar

[4] I. S. Badalbaev, Limit theorems for multi-type critical branching processes with discrete time and with immigration of decreasing intensity, Limit Theorems for Random Processes and Related Problems, “Fan”, Tashkent (1982), 41–54, 190. Search in Google Scholar

[5] I. S. Badalbaev, Limit theorems for multidimensional branching processes with immigration of growing intensity, Dokl. Akad. Nauk UzSSR (1983), no. 2, 3–5. Search in Google Scholar

[6] I. S. Badalbaev, Limit theorems for multidimensional branching processes with immigration of growing intensity, Asymptotic Problems for Probability Distributions, “Fan”, Tashkent (1984), 30–44, 162. Search in Google Scholar

[7] I. S. Badalbaev and I. Rahimov, Critical branching processes with immigration of decreasing intensity, Theory Probab. Appl. 23 (1978), no. 2, 259–268. 10.1137/1123030Search in Google Scholar

[8] I. S. Badalbaev and I. Rakhimov, Further results on branching random processes with immigration of decreasing intensity, Theory Probab. Appl. 28 (1983), no. 4, 775–780. 10.1137/1128080Search in Google Scholar

[9] I. S. Badalbaev and I. Rakhimov, New limit theorems for multitype branching processes with immigration of decreasing intensity, Izv. Akad. Nauk UzSSR (1985), no. 2, 17–22, 93. Search in Google Scholar

[10] I. S. Badalbaev and I. U. Rakhimov, Non-Homogeneous Flows of Branching Processes, “Fan”, Tashkent, 1993. Search in Google Scholar

[11] I. S. Badalbaev and A. M. Zubkov, Limit theorems for a sequence of branching processes with immigration, Theory Probab. Appl. 28 (1983), 404–409. 10.1137/1128034Search in Google Scholar

[12] O. A. Butkovsky, Limit behavior of a critical branching process with immigration, Mat. Zametki 92 (2012), no. 5, 670–677. 10.1134/S000143461211003XSearch in Google Scholar

[13] V. P. Čistjakov, Certain limit theorems for branching processes with final type, Theory Probab. Appl. 15 (1970), no. 3, 515–521. 10.1137/1115055Search in Google Scholar

[14] S. D. Durham, A problem concerning generalized age-dependent branching processes with immigration, Ann. Math. Statist. 42 (1971), 1121–1123. 10.1214/aoms/1177693344Search in Google Scholar

[15] J. H. Foster, Branching processes involving immigration, Ph.D. Thesis, University of Wisconsin, 1969. Search in Google Scholar

[16] J. H. Foster, A limit theorem for a branching process with state-dependent immigration, Ann. Math. Statist. 42 (1971), 1773–1776. 10.1214/aoms/1177693182Search in Google Scholar

[17] J. H. Foster and J. A. Williamson, Limit theorems for the Galton–Watson process with time-dependent immigration, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 20 (1971), 227–235. 10.1007/BF00534904Search in Google Scholar

[18] M. Gonzalez, I. del Puerto and G. P. Yanev, Controlled Branching Processes, John Wiley and Sons, London, 2018. 10.1002/9781119452973Search in Google Scholar

[19] H. Guo and M. Zhang, A fluctuation limit theorem for a critical branching process with dependent immigration, Statist. Probab. Lett. 94 (2014), 29–38. 10.1016/j.spl.2014.06.026Search in Google Scholar

[20] T. E. Harris, The Theory of Branching Processes, Grundlehren Math. Wiss. 119, Springer, Berlin, 1963. 10.1007/978-3-642-51866-9Search in Google Scholar

[21] C. R. Heathcote, A branching process allowing immigration, J. Roy. Statist. Soc. Ser. B 27 (1965), 138–143. 10.1111/j.2517-6161.1965.tb00596.xSearch in Google Scholar

[22] C. C. Heyde and E. Seneta, Estimation theory for growth and immigration rates in a multiplicative process, J. Appl. Probab. 9 (1972), 235–256. 10.1007/978-1-4419-5823-5_31Search in Google Scholar

[23] O. Hyrien, K. V. Mitov and N. M. Yanev, Supercritical Sevastyanov branching processes with non-homogeneous Poisson immigration, Branching Processes and Their Applications, Lect. Notes Stat. 219, Springer, Cham (2016), 151–166. 10.1007/978-3-319-31641-3_9Search in Google Scholar

[24] O. Hyrien, K. V. Mitov and N. M. Yanev, Subcritical Sevastyanov branching processes with nonhomogeneous Poisson immigration, J. Appl. Probab. 54 (2017), no. 2, 569–587. 10.1017/jpr.2017.18Search in Google Scholar

[25] R. Ibragimov, Limit theorems for branching random processes, Random Processes and Statistical Inference, “Fan”, Tashkent (1972), 67–72, 95. Search in Google Scholar

[26] M. Ispány, G. Pap and M. C. A. van Zuijlen, Fluctuation limit of branching processes with immigration and estimation of the means, Adv. in Appl. Probab. 37 (2005), no. 2, 523–538. 10.1239/aap/1118858637Search in Google Scholar

[27] N. Kaplan and A. G. Pakes, Supercritical age-dependent branching processes with immigration, Stochastic Process. Appl. 2 (1974), no. 4, 371–389. 10.1016/0304-4149(74)90005-2Search in Google Scholar

[28] A. V. Karpenko and S. V. Nagaev, Limit theorems for the complete number of descendants in a Galton–Watson branching process, Theory Probab. Appl. 38 (1994), 433–455. 10.1137/1138041Search in Google Scholar

[29] Y. M. Khusanbaev, On the asymptotics of branching processes with immigration, Discrete Math. 28 (2016), no. 1, 113–122. 10.1515/dma-2017-0009Search in Google Scholar

[30] Y. M. Khusanbaev and S. O. Sharipov, A functional limit theorem for a critical branching process with dependent immigration, Uzbek. Mat. J. (2017), no. 3, 149–158. Search in Google Scholar

[31] Y. M. Khusanbaev and S. O. Sharipov, A functional limit theorem for nearly critical branching process with immigration, Uzbek Math. J. (2020), no. 2, 109–118. 10.29229/uzmj.2020-2-11Search in Google Scholar

[32] Y. M. Khusanbaev, S. O. Sharipov and V. Golomoziy, The Berry–Esseen bound for nearly critical branching processes with immigration, Bull. Taras Shevchenko National Univ. Phys. Math. 4 (2019), 42–49. 10.17721/1812-5409.2019/4.5Search in Google Scholar

[33] M. V. Kulkarni and A. G. Pakes, The total progeny of a simple branching process with state-dependent immigration, J. Appl. Probab. 20 (1983), no. 3, 472–481. 10.2307/3213885Search in Google Scholar

[34] K. V. Mitov, V. A. Vatutin and N. M. Yanev, Continuous-time branching processes with decreasing state-dependent immigration, Adv. in Appl. Probab. 16 (1984), no. 4, 697–714. 10.2307/1427337Search in Google Scholar

[35] K. V. Mitov and N. M. Yanev, Critical Galton–Watson processes with decreasing state-dependent immigration, J. Appl. Probab. 21 (1984), no. 1, 22–39. 10.2307/3213661Search in Google Scholar

[36] K. V. Mitov and N. M. Yanev, Sevastyanov branching processes with non-homogeneous Poisson immigration, Proc. Steklov Inst. Math. 282 (2013), 172–185. 10.1134/S0081543813060151Search in Google Scholar

[37] K. V. Mitov, N. M. Yanev and O. Hyrien, Multitype branching processes with non-homogeneous Poisson immigration, Adv. in Appl. Probab. 50 (2018), no. A, 211–228. 10.1017/apr.2018.81Search in Google Scholar

[38] S. V. Nagaev, A limit theorem for branching processes with immigration, Theory Probab. Appl. 20 (1975), 178–180. 10.1137/1120019Search in Google Scholar

[39] S. V. Nagaev and L. V. Han, Limit theorems for a critical Galton–Watson branching process with migration, Theory Probab. Appl. 25 (1980), no. 3, 514–525. 10.1137/1125063Search in Google Scholar

[40] A. G. Pakes, A branching process with a state-dependent immigration component, Adv. in Appl. Probab. 3 (1971), 301–314. 10.2307/1426173Search in Google Scholar

[41] A. G. Pakes, Branching processes with immigration, J. Appl. Probab. 8 (1971), 32–42. 10.2307/3211835Search in Google Scholar

[42] A. G. Pakes, Some limit theorems for the total progeny of a branching process, Adv. in Appl. Probab. 3 (1971), 176–192. 10.2307/1426333Search in Google Scholar

[43] A. G. Pakes, Limit theorems for an age-dependent branching process with immigration, Math. Biosci. 14 (1972), 221–234. 10.1016/0025-5564(72)90076-4Search in Google Scholar

[44] A. G. Pakes, Some limit theorems for Markov chains with applications to branching processes, Studies in Probability and Statistics, Jerusalem Academic Press, Jerusalem (1974), 21–39. Search in Google Scholar

[45] A. G. Pakes, Revisiting conditional limit theorems for the mortal simple branching process, Bernoulli 5 (1999), no. 6, 969–998. 10.2307/3318555Search in Google Scholar

[46] L. Rabehasaina and J.-K. Woo, Multitype branching process with non-homogeneous Poisson and generalized Polya immigration, preprint (2020), https://arxiv.org/abs/1909.03684. Search in Google Scholar

[47] L. Rabehasaina and J.-K. Woo, Multitype branching process with non-homogeneous Poisson and contagious Poisson immigration, J. Appl. Probab. 58 (2021), no. 4, 1007–1042. 10.1017/jpr.2021.19Search in Google Scholar

[48] I. Rahimov, On branching processes with increasing immigration, Dokl. Akad. Nauk UzSSR (1981), no. 1, 3–5. Search in Google Scholar

[49] 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

[50] I. Rahimov, Multitype processes with reproduction-dependent immigration, J. Appl. Probab. 35 (1998), no. 2, 281–292. 10.1239/jap/1032192847Search in Google Scholar

[51] I. Rahimov, Functional limit theorems for critical processes with immigration, Adv. in Appl. Probab. 39 (2007), no. 4, 1054–1069. 10.1239/aap/1198177239Search in Google Scholar

[52] I. Rahimov, Asymptotic distribution of the CLSE in a critical process with immigration, Stochastic Process. Appl. 118 (2008), no. 10, 1892–1908. 10.1016/j.spa.2007.11.004Search in Google Scholar

[53] I. Rahimov, Deterministic approximation of a sequence of nearly critical branching processes, Stoch. Anal. Appl. 26 (2008), no. 5, 1013–1024. 10.1080/07362990802286442Search in Google Scholar

[54] I. Rahimov, Approximation of fluctuations in a sequence of nearly critical branching processes, Stoch. Models 25 (2009), no. 2, 348–373. 10.1080/15326340902870158Search in Google Scholar

[55] I. Rahimov, Asymptotic distributions for weighted estimators of the offspring mean in a branching process, TEST 18 (2009), no. 3, 568–583. 10.1007/s11749-008-0124-8Search in Google Scholar

[56] I. Rahimov, Asymptotically normal estimators for the offspring mean in the branching process with immigration, Comm. Statist. Theory Methods 38 (2009), no. 1–2, 13–28. 10.1080/03610920802155445Search in Google Scholar

[57] I. Rahimov, Bootstrap of the offspring mean in the critical process with a non-stationary immigration, Stochastic Process. Appl. 119 (2009), no. 11, 3939–3954. 10.1016/j.spa.2009.09.003Search in Google Scholar

[58] I. Rahimov, Estimation of the offspring mean in a supercritical branching process with non-stationary immigration, Statist. Probab. Lett. 81 (2011), no. 8, 907–914. 10.1016/j.spl.2011.03.038Search in Google Scholar

[59] I. Rahimov, Conditional least squares estimators for the offspring mean in a subcritical branching process with immigration, Comm. Statist. Theory Methods 41 (2012), no. 12, 2096–2110. 10.1080/03610926.2011.558658Search in Google Scholar

[60] I. Rahimov and A. Teshabaev, Decomposable branching processes with decreasing immigration, J. Appl. Statist. Sci. 3 (1996), no. 2–3, 169–189. Search in Google Scholar

[61] I. Rahimov and A. Teshabaev, Some limit theorems for decomposable branching processes with decreasing immigration, İstatistik 1 (1998), no. 1, 29–41. Search in Google Scholar

[62] I. Rahimov and G. P. Yanev, Variance estimators in critical branching processes with non-homogeneous immigration, Math. Popul. Stud. 19 (2012), no. 4, 188–199. 10.1080/08898480.2012.718941Search in Google Scholar

[63] I. Rahimov, F. Yildirim and A. Teshabaev, Nonhomogeneous decomposable branching processes, Branching Processes (Varna 1993), Lect. Notes Stat. 99, Springer, New York (1995), 67–76. 10.1007/978-1-4612-2558-4_8Search in Google Scholar

[64] I. Rakhimov, Critical Galton–Watson processes with increasing immigration, Izv. Akad. Nauk UzSSR (1978), no. 4, 22–31. Search in Google Scholar

[65] I. Rakhimov, Limit distributions for the total number of particles in critical Galton–Watson processes with immigration, Asymptotic Problems for Probability Distributions, “Fan”, Tashkent (1984), 106–119, 164. Search in Google Scholar

[66] I. Rakhimov, On limit theorems for a sequence of branching processes with non-homogeneous immigration, Theory Probab. Appl. 29 (1984), no. 4, 853–854. Search in Google Scholar

[67] I. Rakhimov, Uniform estimates in limit theorems for branching processes with immigration, Izv. Akad. Nauk UzSSR (1984), no. 3, 24–29. Search in Google Scholar

[68] I. Rakhimov, Convergence of a sequence of branching processes with immigration to processes with a continuous state-space, Limit Theorems for Probability Distributions, “Fan”, Tashkent (1985), 134–148, 229. Search in Google Scholar

[69] I. Rakhimov, Limit distributions for integrals of the Bellman–Harris process with non-homogeneous immigration, Izv. Akad. Nauk UzSSR (1985), no. 5, 20–25, 86. Search in Google Scholar

[70] I. Rakhimov, Asymptotic behavior of the probability of hitting a fixed state for Galton–Watson processes with decreasing immigration. I, II, Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk (1986), no. 2, 33–38; no. 3 (1986), 38-46. Search in Google Scholar

[71] I. Rakhimov, Critical branching processes with infinite variance and decreasing immigration, Theory Probab. Appl. 31 (1986), no. 1, 98–110. 10.1137/1131008Search in Google Scholar

[72] I. Rakhimov, Limit theorems for decomposable branching processes with immigration, Dokl. Akad. Nauk UzSSR (1987), no. 6, 5–7. Search in Google Scholar

[73] I. Rakhimov, Two limit theorems for age-dependent multitype branching processes with immigration, Asymptotic Methods in Mathematical Statistics, “Fan”, Tashkent (1987), 97–108. Search in Google Scholar

[74] I. Rakhimov, A local theorem for Galton–Watson processes with immigration, in the case of a uniform limit distribution, Serdica 14 (1988), no. 3, 234–244. Search in Google Scholar

[75] I. Rakhimov, Asymptotics of the probability of nonextinction of decomposable branching processes with decreasing immigration, Izv. Akad. Nauk UzSSR (1988), no. 2, 26–28, 84. Search in Google Scholar

[76] I. Rakhimov, Local limit theorems for critical Galton–Watson processes with decreasing immigration, Theory Probab. Appl. 33 (1988), no. 2, 387–392. 10.1137/1133057Search in Google Scholar

[77] I. Rakhimov, Branching random processes with generalized immigration, Izv. Akad. Nauk UzSSR (1989), no. 2, 35–40, 107. Search in Google Scholar

[78] I. Rakhimov, General branching processes with immigration that depends on reproduction, Theory Probab. Appl. 37 (1992), no. 3, 513–525. 10.1137/1137098Search in Google Scholar

[79] I. Rakhimov, Sample sums of dependent variables, mixtures of infinitely divisible laws, and branching random processes, Discrete Math. Appl. 2 (1992), no. 3, 337–356. 10.1515/dma.1992.2.3.337Search in Google Scholar

[80] I. Rakhimov, Critical processes with infinite variance and growing immigration, Math. Notes 53 (1993), no. 5–6, 628–634. 10.1007/BF01212600Search in Google Scholar

[81] I. Rakhimov and S. Kaverin, A class of limit distributions of critical branching processes with decreasing immigration depending on the state, Dokl. Akad. Nauk UzSSR (1986), no. 1, 4–6. Search in Google Scholar

[82] I. Rakhimov and S. Kaverin, A method for proving limit theorems for branching processes with state-dependent immigration, Probabilistic Models and Mathematical Statistics, “Fan”, Tashkent (1987), 61–76, 174. Search in Google Scholar

[83] I. Rakhimov and S. Kurbanov, Branching processes with inhomogeneous migration and infinite variance, Functionals of Random Processes, and Statistical Inferences, “Fan”, Tashkent (1989), 71–85, 150. Search in Google Scholar

[84] I. Rakhimov and S. Kurbanov, Critical Bellman–Harris branching processes with infinite variance and decreasing immigration, Uzbek. Mat. Zh. (1992), no. 2, 22–31. Search in Google Scholar

[85] I. Rakhimov and S. Kurbanov, Subcritical processes with decreasing immigration and infinite variance, Uzbek. Math. J. (1994), no. 1, 51–57. Search in Google Scholar

[86] I. Rakhimov and S. K. Sirazhdinov, Approximation of the distribution of a sum in a scheme for the summation of independent random variables, Asymptotic Methods in Probability Theory and Mathematical Statistics. “Fan”, Tashkent (1988), 136–151. Search in Google Scholar

[87] I. Rakhimov and S. K. Sirazhdinov, Approximation of the distribution of a sum in a scheme for the summation of independent random variables, Soviet Math. Dokl. 38 (1989), no. 1, 23–27. Search in Google Scholar

[88] A. A. Savin and V. P. Chistyakov, Some limit theorems for branching processes with a few types of particles, Theory Probab. Appl. 7 (1962), no. 1, 95–104. 10.1137/1107008Search in Google Scholar

[89] E. Seneta, An explicit-limit theorem for the critical Galton–Watson process with immigration, J. Roy. Statist. Soc. Ser. B 32 (1970), 149–152. 10.1111/j.2517-6161.1970.tb00826.xSearch in Google Scholar

[90] E. Seneta, Regularly Varying Functions, “Nauka”, Moscow, 1985. Search in Google Scholar

[91] B. A. Sevastyanov, Limit theorems for branching stochastic processes of special form, Theory Probab. Appl. 2 (1957), 339–348. 10.1137/1102022Search in Google Scholar

[92] B. A. Sevastyanov, Branching Processes, “Nauka”, Moscow, 1971. Search in Google Scholar

[93] R. S. Slack, A branching process with mean one and possibly infinite variance, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 9 (1968), 139–145.10.1007/BF01851004Search in Google Scholar

[94] M. Slavtchova-Bojkova and N. M. Yanev, Poisson random measures and critical Sevastyanov branching processes, Stoch. Models 35 (2019), no. 2, 197–208. 10.1080/15326349.2019.1600411Search in Google Scholar

[95] T. N. Sriram, Invalidity of bootstrap for critical branching processes with immigration, Ann. Statist. 22 (1994), no. 2, 1013–1023. 10.1214/aos/1176325509Search in Google Scholar

[96] D. B. Stewart, A subcritical branching process with state dependent immigration, Math. Biosci. 31 (1976), no. 1–2, 175–190. 10.1016/0025-5564(76)90048-1Search in Google Scholar

[97] D. B. Stewart, A supercritical branching process with state dependent immigration, Math. Biosci. 32 (1976), no. 3–4, 187–202. 10.1016/0025-5564(76)90113-9Search in Google Scholar

[98] V. A. Vatutin, A critical Galton–Watson branching process with emigration, Theory Probab. Appl. 22 (1977), no. 3, 465–481. 10.1137/1122058Search in Google Scholar

[99] V. A. Vatutin and E. E. Dyakonova, Decomposable branching processes with two types of particles, Discrete Math. 30 (2018), no. 1, 3–18. 10.1515/dma-2018-0012Search in Google Scholar

[100] V. A. Vatutin and A. M. Zubkov, Branching processes. II, J. Soviet Math. 39 (1993), 2431–2475. 10.1007/BF01096272Search in Google Scholar

[101] O. V. Viskov, Several remarks on branching processes, Mat. Zametki 8 (1970), 409–418. Search in Google Scholar

[102] G. P. Yanev, Critical controlled branching processes and their relatives, Pliska Stud. Math. 24 (2015), 111–130. Search in Google Scholar

[103] 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

[104] N. M. Yanev and K. V. Mitov, Branching processes with decreasing migration, C. R. Acad. Bulgare Sci. 37 (1984), no. 4, 465–468. Search in Google Scholar

[105] N. M. Yanev and K. V. Mitov, Controllable branching processes with inhomogeneous migration, Pliska Stud. Math. Bulgar. 7 (1984), 90–96. Search in Google Scholar

[106] N. M. Yanev and K. V. Mitov, Critical branching processes with nonhomogeneous migration, Ann. Probab. 13 (1985), no. 3, 923–933. 10.1214/aop/1176992914Search in Google Scholar

[107] A. M. Zubkov, The life spans of a branching process with immigration, Theory Probab. Appl. 17 (1972), 179–188. 10.1137/1117018Search in Google Scholar

Received: 2021-08-10
Revised: 2021-12-09
Accepted: 2021-12-09
Published Online: 2022-01-08
Published in Print: 2022-01-01

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