On application of slowly varying functions with remainder in the theory of Markov branching processes with mean one and infinite variance

  • A. Imomov Karshi State Univ., Uzbekistan
  • A. Meyliyev Karshi State Univ., Uzbekistan
Keywords: .


UDC 519.218.2

We investigate an application of slowly varying functions (in sense of Karamata) in the theory of Markov branching processes. We treat the critical case so that the infinitesimal generating function of the process has the infinite second moment, but it regularly varies with the remainder. We improve the basic lemma of the theory of critical Markov branching processes and refine known limit results.


S. Asmussen, H. Hering, Branching processes, Birkhauser, Boston (1983), https://doi.org/10.1007/978-1-4615-8155-0 DOI: https://doi.org/10.1007/978-1-4615-8155-0

K. B. Athreya, P. E. Ney, Branching processes, Springer, New York (1972). DOI: https://doi.org/10.1007/978-3-642-65371-1

N. H. Bingham, C. M. Goldie, J. L. Teugels, Regular variation, Univ. Press, Cambridge (1987), https://doi.org/10.1017/CBO9780511721434 DOI: https://doi.org/10.1017/CBO9780511721434

W. Feller, An introduction to probability theory and its applications, vol. 2. Mir, Moscow (1967).

T. E. Harris, The theory of branching processes, Springer-Verlag, Berlin (1963). DOI: https://doi.org/10.1007/978-3-642-51866-9

A. A. Imomov, On conditioned limit structure of the Markov branching process without finite second moment, Malays. J. Math. Sci., 11, № 3, 393 – 422 (2017).

A. A. Imomov, On Markov analogue of $Q$-processes with continuous time, Theory Probab. and Math. Statist., 84, 57 – 64 (2012), https://doi.org/10.1090/S0094-9000-2012-00853-3 DOI: https://doi.org/10.1090/S0094-9000-2012-00853-3

A. N. Kolmogorov, N. A. Dmitriev, Branching stochastic process, Rep. Acad. Sci. USSR, 61, 55 – 62 (1947).

A. G. Pakes, Critical Markov branching process limit theorems allowing infinite variance, Adv. Appl. Probab., 42, 460 – 488 (2010), https://doi.org/10.1239/aap/1275055238 DOI: https://doi.org/10.1017/S0001867800004158

E. Seneta, Regularly varying functions, Springer, Berlin (1976). DOI: https://doi.org/10.1007/BFb0079658

B. A. Sevastyanov, The theory of branching stochastic processes (in Russian), Uspekhi Math. Nauk, 6(46), 47 – 99 (1951).

V. M. Zolotarev, More exact statements of several theorems in the theory of branching processes, Theory Probab. and Appl., 2, 245 – 253 (1957). DOI: https://doi.org/10.1137/1102016

How to Cite
Imomov, A., and A. Meyliyev. “On Application of Slowly Varying Functions With Remainder in the Theory of Markov Branching Processes With Mean One and Infinite Variance”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 8, Aug. 2021, pp. 1056 -6, doi:10.37863/umzh.v73i8.684.
Research articles