Some limit theorems for the critical Galton–Watson branching processes

  • Kh. Kudratov National University of Uzbekistan, Tashkent
  • Ya. Khusanbaev Institute of Mathematics, Tashkent, Uzbekistan
Keywords: Critical Galton-Watson process, generating function, slowly varying function.

Abstract

UDC 519.21

We consider critical Galton–Watson processes starting from a random number of particles and determine the effect of the mean value of the initial state on the asymptotic state of the process. For processes starting from a large number of particles and satisfying the condition $(S),$ we prove the limit theorem similar to the result of W. Feller. We also prove the theorem under the condition $W(n)>0$ for critical processes satisfying the conditions $(S)$ and $(M).$

References

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

A. N. Kolmogorov, K resheniyu odnoi biologicheskoi zadachi, Izv. NII Mat. Mekh. Tomskogo Univ., 2, № 1 (1938) (in Russian).

A. M. Yaglom, Nekotorye predelnye teoremy teorii vetvyashchikhsya sluchainykh protsessov, Dokl. AN SSSR, 56, № 8, 795–798 (1947) (in Russian).

F. Spitzer, H. Kesten, P. Ney, The Galton–Watson process with mean one and finite variance, Theory Probab. and Appl., 11, № 4, 513–540 (1966). DOI: https://doi.org/10.1137/1111059

V. M. Zolotarev, More exact statements of several theorems in the theory of branching processes, Theory Probab. and Appl., 2, № 2, 245–253 (1957). DOI: https://doi.org/10.1137/1102016

R. S. Slack, A branching process with mean one and possibly infinite variance, Z. Wahrscheinlichkeitstheor. und verw. Geb., 9, 139–145 (1968). DOI: https://doi.org/10.1007/BF01851004

W. Feller, Diffusion processes in genetics, Proc. 2nd Berkeley Symp. Math. Statist. and Probab., 227–246 (1951).

K. V. Mitov, G. K. Mitov, N. M. Yanev, Limit theorems for critical randomly indexed branching processes, Workshop on Branching Processes and their Applications, Springer (2010). DOI: https://doi.org/10.1007/978-3-642-11156-3_7

S. V. Nagaev, V. Wachtel, The critical Galton–Watson process without further power moments, J. Appl. Probab., 44, № 3, 753–769 (2007). DOI: https://doi.org/10.1239/jap/1189717543

E. Seneta, Regularly varying functions, Lect. Notes Math., 508, Springer, Berlin (1976). DOI: https://doi.org/10.1007/BFb0079658

Published
10.05.2023
How to Cite
Kudratov, K., and Y. Khusanbaev. “Some Limit Theorems for the Critical Galton–Watson Branching Processes”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, no. 4, May 2023, pp. 467 -7, doi:10.37863/umzh.v75i4.6781.
Section
Research articles