On a limit structure of the Galton-Watson branching processes with regularly varying generating functions

• Autorzy: Imomov Azam A.

Identyfikatory

DOI

10.19195/0208-4147.39.1.4

• Języki publikacji: EN

Abstrakty

EN

We investigate limit properties of discrete time branching processes with application of the theory of regularly varying functions in the sense of Karamata. In the critical situation we suppose that the offspring probability generating function has an infinite second moment but its tail regularly varies. In the noncritical case, the finite moment of type E [x ln x] is required. The lemma on the asymptotic representation of the generating function of the process and its differential analogue will underlie our conclusions.

Słowa kluczowe

EN

Galton-Watson branching processes; regularly varying functions; transition functions; invariant measures; limit theorems; Q-processes

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2019
• Tom: Vol. 39, Fasc. 1
• Strony: 61--73
• Opis fizyczny: Bibliogr. 16 poz.

Twórcy

autor

Imomov Azam A.

• State Testing Center under the Cabinet of Ministers of the Republic of Uzbekistan, 12, Bogishamol St., 100202, Tashkent, Uzbekistan
• Karshi State University, 17, Kuchabag St., 180100, Karshi City, Uzbekistan

Bibliografia

• [1] S. Asmussen and H. Hering, Branching Processes, Birkhäuser, Boston 1983.
• [2] K. B. Athreya and P. E. Ney, Branching Processes, Springer, New York 1972.
• [3] N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge 1987.
• [4] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, Mir, Moscow 1967 (Russian edition).
• [5] Sh. K. Formanov and Zh. B. Azimov, Markov branching processes with regularly varying generating function and immigration of a special form, Theory Probab. Math. Statist. 65 (2002), pp. 181-188.
• [6] T. E. Harris, The Theory of Branching Processes, Springer, Berlin 1963.
• [7] A. A. Imomov, A differential analog of the main lemma of the theory of Markov branching processes and its applications, Ukrainian Math. J. 57 (2) (2005), pp. 307-315.
• [8] J. Karamata, Sur un mode de croissance régulière: Théorèmes fondamentaux, Bull. Soc. Math. France 61 (1933), pp. 55-62.
• [9] A. G. Pakes, Some new limit theorems for the critical branching process allowing immigration, Stochastic Process. Appl. 3 (1975), pp. 175-185.
• [10] E. Seneta, On the invariant measures for simple branching processes, J. Appl. Probab. 8 (1971), pp. 43-51.
• [11] E. Seneta, Regularly Varying Functions, Springer, Berlin 1972.
• [12] E. Seneta, A Tauberian theorem of E. Landau and W. Feller, Ann. Probab. 1 (1973), pp. 1057-1058.
• [13] E. Seneta, Regularly varying functions in the theory of simple branching processes, Adv. in Appl. Probab. 6 (1974), pp. 408-420.
• [14] R. S. Slack, A branching process with mean one and possible infinite variance, Z. Wahrsch. Verw. Gebiete 9 (1968), pp. 139-145.
• [15] R. S. Slack, Further notes on branching processes with mean 1, Z. Wahrsch. Verw. Gebiete 25 (1972), pp. 31-38.
• [16] V. M. Zolotarev, More exact statements of several theorems in the theory of branching processes, Theory Probab. Appl. 2 (1957), pp. 245-253.

Uwagi

Dedicated to the fond memory of my Father.