Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T15:29:55.235Z Has data issue: false hasContentIssue false
  • English
  • Français

A limit theorem for the maxima of the para-critical simple branching process

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  01 July 2016

Anthony G. Pakes
Anthony G. Pakes*
Affiliation:
University of Western Australia
*
∗ Postal address: Department of Mathematics, University of Western Australia, Nedlands, WA, Australia 6907. Email address: pakes@maths.uwa.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

Let Mn denote the size of the largest amongst the first n generations of a simple branching process. It is shown for near critical processes with a finite offspring variance that the law of Mn/n, conditioned on no extinction at or before n, has a non-defective weak limit. The proof uses a conditioned functional limit theorem deriving from the Feller-Lindvall (CB) diffusion limit for branching processes descended from increasing numbers of ancestors. Subsidiary results are given about hitting time laws for CB diffusions and Bessel processes.

Keywords

Branching processmaximum generation sizediffusion limitBessel processspecial functions

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F17: Functional limit theorems; invariance principles 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Secondary: 33C10: Bessel and Airy functions, cylinder functions, $_0F_1$ 33C15: Confluent hypergeometric functions, Whittaker functions, $_1F_1$
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 3 , September 1998 , pp. 740 - 756
Copyright
Copyright © Applied Probability Trust 1998 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adke, S.R. (1964). The maximum population size in the first n generations of a branching process. Biometrics 20, 649–651.Google Scholar
Aliev, S.A. and Shurenkov, V.M. (1982). Transitional phenomena and the convergence of Galton–Watson processes to Jiřina process. Theory Prob. Appl. 27, 472–485.Google Scholar
Athreya, K.B. (1988). On the maximum sequence in a critical branching process. Ann. Prob. 16, 502–507.Google Scholar
Bingham, N.H. (1976). Continuous branching processes and spectral positivity. Stoch. Proc. Appl. 4, 217–242.Google Scholar
Bingham, N.H., Goldie, C.M. and Teugels, J.M. (1987). Regular Variation. Cambridge University Press, Cambridge.Google Scholar
Bishir, J. (1962). Maximum population size in a branching process. Biometrics 18, 394–403.CrossRefGoogle Scholar
Bondesson, L. (1992). Generalized Gamma Convolutions and Related Classes of Distributions and Densities. Lecture Notes in Statistics 76, Springer, New York.Google Scholar
Borovkov, K.A. and Vatutin, V.A. (1996). On distribution tails and expectations of maxima in critical branching processes. J. Appl. Prob. 33, 614–622.Google Scholar
Ciesielski, Z. and Taylor, S.J. (1962). First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path. Trans. Amer. Math. Soc. 103, 434–450.Google Scholar
Durrett, R. (1978). Conditioned limit theorems for null recurrent Markov processes. Ann. Prob. 6, 798–828.Google Scholar
Fahady, K.S., Quine, M.P. and Vere-Jones, D. (1971). Heavy traffic approximations for the Galton–Watson process. Adv. Appl. Prob. 3, 282–300.Google Scholar
Feller, W. (1951a). Diffusion processes in genetics. Second Symposium on Probability and Statistics. ed. Neyman, J.. University of California Press, Berkeley, pp. 227–246.Google Scholar
Feller, W. (1951b). Two singular diffusion problems. Ann. Math. 54, 173–182.Google Scholar
Gadag, V.G. (1981). First passage time for a supercritical Galton–Watson process restricted to the non-extinction set. J. Theor. Biol. 93, 585–589.Google Scholar
Gradshteyn, I.S. and Ryzhik, I.M. (1980). Table of Integrals, Series and Products. Academic Press, New York.Google Scholar
Harris, T.E. (1951). Some mathematical models for branching processes. Second Symposium on Probability and Statistics. ed. Neyman, J.. University of California Press, Berkeley, pp. 305–328.Google Scholar
Ismail, M.E. H. and Kelker, D.H. (1979). Special functions, Stieltjes transforms and infinite divisibility. SIAM J. Math. Anal. 10, 884–901.CrossRefGoogle Scholar
Ismail, M.E. H. and May, C.P. (1979). Special functions, infinite divisibility and transcendental equations. Math. Proc. Camb. Phil. Soc. 85, 453–464.Google Scholar
Jirina, M. (1969). On Feller's branching diffusion processes. Časopis Pěst. Mat. 94, 84–90.Google Scholar
Kämmerle, K. and Schuh, H.-J. (1986). The maximum in critical Galton–Watson and birth and death processes. J. Appl. Prob. 23, 601–613.Google Scholar
Karlin, S. and Taylor, H.M. (1981). A Second Course in Stochastic Processes. Academic Press, New York.Google Scholar
Kent, J. (1978). Some probabilistic properties of Bessel functions. Ann. Prob. 6, 760–770.CrossRefGoogle Scholar
Kent, J. (1980). Eigenvalue expansions for diffusion hitting times. Z. Wahrscheinlichkeitsth. 52, 309–319.CrossRefGoogle Scholar
Lamperti, J. (1962). Semi-stable stochastic processes. Trans. Amer. Math. Soc. 104, 62–78.Google Scholar
Lamperti, J. (1967a). The limit of a sequence of branching processes. Z. Wahrscheinlichkeitsth. 7, 271–288.Google Scholar
Lamperti, J. (1967b). Continuous state branching processes. Bull. Amer. Math. Soc. 73, 382–386.Google Scholar
Lamperti, J. and Ney, P. (1968). Conditioned branching processes and their limiting diffusions. Theory Prob. Appl. 13, 128–139.CrossRefGoogle Scholar
Lindvall, T. (1974). Limit theorems for some functionals of certain Galton–Watson branching process. Adv. Appl. Prob. 6, 309–321.Google Scholar
Lindvall, T. (1976). On the maximum of a branching process. Scand. J. Statist. 3, 209–214.Google Scholar
Nerman, O. (1977). On the maximal generation size of a non-critical Galton–Watson process. Scand. J. Statist. 4, 131–135.Google Scholar
Pakes, A.G. (1987). Remarks on the maxima of a martingale sequence with application to the simple critical branching process. J. Appl. Prob. 24, 768–772.Google Scholar
Pakes, A.G. (1988). Some limit theorems for continuous-state branching processes. J. Austral. Math. Soc. (Series A) 44, 71–87.Google Scholar
Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin.Google Scholar
Sevast'yanov, B.A. (1959). Transient phenomena in branching processes. Theory Prob. Appl. 4, 113–128.Google Scholar
Slater, L.J. (1960). Confluent Hypergeometric Functions. Cambridge University Press, Cambridge.Google Scholar
Topchij, V.A. and Vatutin, V.A. (1997). Left-continuous random walk and the maximum of the critical Galton–Watson processes. Theory Prob. Appl. 42, 17–27.Google Scholar
Weiner, H. (1984). Moments of the maximum in a critical branching process. J. Appl. Prob. 21, 920–923.Google Scholar
Yor, M. (1992). Some Aspects of Brownian Motion Part I: Some Special Functionals. Birkhäuser, Basel.Google Scholar