Sevastyanov branching processes with non-homogeneous Poisson immigration

Kosto V. Mitov ; Nikolay M. Yanev

Informatics and Automation, Branching processes, random walks, and related problems, Tome 282 (2013), pp. 181-194.

Résumé

Sevastyanov age-dependent branching processes allowing an immigration component are considered in the case when the moments of immigration form a non-homogeneous Poisson process with intensity $r(t)$. The asymptotic behavior of the expectation and of the probability of non-extinction is investigated in the critical case depending on the asymptotic rate of $r(t)$. Corresponding limit theorems are also proved using different types of normalization. Among them we obtained limiting distributions similar to the classical ones of Yaglom (1947) and Sevastyanov (1957) and also discovered new phenomena due to the non-homogeneity.

Bibliographie

[1] Asmussen S., Hering H., Branching processes, Birkhäuser, Boston, 1983 | MR | Zbl

[2] Athreya K.B., Ney P.E., Branching processes, Springer, Berlin, 1972 | MR | Zbl

[3] Bellman R., Harris T.E., “On the theory of age-dependent stochastic branching processes”, Proc. Natl. Acad. Sci. USA, 34 (1948), 601–604 | DOI | MR | Zbl

[4] Bellman R., Harris T., “On age-dependent binary branching processes”, Ann. Math. Ser. 2, 55 (1952), 280–295 | DOI | MR | Zbl

[5] Bingham N.H., Goldie C.M., Teugels J.L., Regular variation, Cambridge Univ. Press, Cambridge, 1987 | MR | Zbl

[6] Foster J.H., “A limit theorem for a branching process with state-dependent immigration”, Ann. Math. Stat., 42 (1971), 1773–1776 | DOI | MR | Zbl

[7] Haccou P., Jagers P., Vatutin V.A., Branching processes: Variation, growth and extinction of populations, Cambridge Univ. Press, Cambridge, 2005 | Zbl

[8] Harris T.E., The theory of branching processes, Springer, Berlin, 1963 | MR | Zbl

[9] Heathcote C.R., “A branching process allowing immigration”, J. R. Stat. Soc. B, 27 (1965), 138–143 | MR

[10] Heathcote C.R., “Corrections and comments on the paper “A branching process allowing immigration””, J. R. Stat. Soc. B, 28 (1966), 213–217 | MR | Zbl

[11] Hyrien O., Yanev N.M., “Age-dependent branching processes with non-homogeneous Poisson immigration as models of cell kinetics”, Modeling and inference in biomedical sciences: In memory of Andrei Yakovlev, IMS Collections Ser., eds. D. Oakes, W.J. Hall, A. Almudevar, Inst. Math. Stat., Beachwood, OH

[12] Theory Probab. Appl., 13 (1968), 225–236 | DOI | MR | Zbl

[13] Jagers P., Branching processes with biological applications, J. Wiley Sons, London, 1975 | MR | Zbl

[14] Kaplan N., Pakes A.G., “Supercritical age-dependent branching processes with immigration”, Stoch. Processes Appl., 2 (1974), 371–389 | DOI | MR | Zbl

[15] Kimmel M., Axelrod D.E., Branching processes in biology, Springer, New York, 2002 | MR

[16] Kolmogorov A.N., Dmitriev N.A., “Vetvyaschiesya sluchainye protsessy”, DAN SSSR, 56:1 (1947), 7–10 | MR

[17] Kolmogorov A.N., Sevastyanov B.A., “Vychislenie finalnykh veroyatnostei dlya vetvyaschikhsya sluchainykh protsessov”, DAN SSSR, 56:8 (1947), 783–786 | MR | Zbl

[18] Korn G.A., Korn T.M., Mathematical handbook for scientists and engineers, McGraw-Hill, New York, 1961 | MR | Zbl

[19] Mitov K.V., Vatutin V.A., Yanev N.M., “Continuous-time branching processes with decreasing state-dependent immigration”, Adv. Appl. Probab., 16 (1984), 697–714 | DOI | MR | Zbl

[20] Mitov K.V., Yanev N.M., “Critical Galton–Watson processes with decreasing state-dependent immigration”, J. Appl. Probab., 21 (1984), 22–39 | DOI | MR | Zbl

[21] Mitov K.V., Yanev N.M., “Bellman–Harris branching processes with state-dependent immigration”, J. Appl. Probab., 22 (1985), 757–765 | DOI | MR | Zbl

[22] Mitov K.V., Yanev N.M., “Bellman–Harris branching processes with a special type of state-dependent immigration”, Adv. Appl. Probab., 21 (1989), 270–283 | DOI | MR | Zbl

[23] Mitov K.V., Yanev N.M., “Critical Bellman–Harris branching processes with infinite variance allowing state-dependent immigration”, Stoch. Models, 18 (2002), 281–300 | DOI | MR | Zbl

[24] Pakes A.G., “A branching process with a state dependent immigration component”, Adv. Appl. Probab., 3 (1971), 301–314 | DOI | MR | Zbl

[25] Pakes A.G., “Limit theorems for an age-dependent branching process with immigration”, Math. Biosci., 14 (1972), 221–234 | DOI | MR | Zbl

[26] Pakes A.G., Kaplan N., “On the subcritical Bellman–Harris process with immigration”, J. Appl. Probab., 11 (1974), 652–668 | DOI | MR | Zbl

[27] Radcliffe J., “The convergence of a super-critical age-dependent branching processes allowing immigration at the epochs of a renewal process”, Math. Biosci., 14 (1972), 37–44 | DOI | MR | Zbl

[28] Rahimov I., Random sums and branching stochastic processes, Springer, New York, 1995 | MR

[29] Sevastyanov B.A., “Limit theorems for branching stochastic processes of special form”, Theory Probab. Appl., 2 (1957), 321–331 | DOI | MR | Zbl

[30] Sevastyanov B.A., “Vetvyaschiesya protsessy s prevrascheniyami, zavisyaschimi ot vozrasta chastits”, Teoriya veroyatn. i ee primen., 9:4 (1964), 577–594 ; “Письмо в редакцию”, Теория вероятн. и ее примен., 11:2 (1966), 363–364 ; Sevastyanov B.A., “Age-dependent branching processes”, Theory Probab. Appl., 9 (1964), 521–537 ; “Addendum”, Theory Probab. Appl., 11 (1966), 321–322 | DOI | MR | DOI

[31] Sevastyanov B.A., “Renewal equations and moments of branching processes”, Math. Notes, 3 (1968), 3–10 | DOI | Zbl

[32] Sevastyanov B.A., “Limit theorems for age-dependent branching processes”, Theory Probab. Appl., 13 (1968), 237–259 | DOI | Zbl

[33] Sevastyanov B.A., Vetvyaschiesya protsessy, Nauka, M., 1971 | MR | Zbl

[34] Vatutin V.A., “Branching processes with infinite variance”, Proc. 4th Int. Summer School on Probability Theory and Mathematical Statistics, Varna (Bulgaria), 1982, Publ. House Bulg. Acad. Sci., Sofia, 1983, 9–38 | MR | Zbl

[35] Yaglom A.M., “Nekotorye predelnye teoremy teorii vetvyaschikhsya sluchainykh protsessov”, DAN SSSR, 56:8 (1947), 795–798 | MR | Zbl

[36] Yakovlev A.Yu., Yanev N.M., Transient processes in cell proliferation kinetics, Springer, Berlin, 1989 | MR | Zbl

[37] Yakovlev A., Yanev N., “Branching stochastic processes with immigration in analysis of renewing cell populations”, Math. Biosci., 203 (2006), 37–63 | DOI | MR | Zbl

[38] Yakovlev A.Yu., Yanev N.M., “Age and residual lifetime distributions for branching processes”, Stat. Probab. Lett., 77 (2007), 503–513 | DOI | MR | Zbl

[39] Yanev N.M., “Razklonyavaschi se sluchaini protsesi s imigratsiya”, Bull. Inst. Math. Acad. Bulg. Sci., 15 (1972), 71–88

[40] Yanev N.M., “Ob odnom klasse razlozhimykh vetvyaschikhsya protsessov, zavisyaschikh ot vozrasta chastits”, Math. Balk., 2 (1972), 58–75

[41] Yanev N.M., Mitov K.V., “Critical branching processes with nonhomogeneous migration”, Ann. Probab., 13 (1985), 923–933 | DOI | MR | Zbl