The Annals of Applied Probability

Local limit theory and large deviations for supercritical Branching processes

Peter E. Ney and Anand N. Vidyashankar

Full-text: Open access

Abstract

In this paper we study several aspects of the growth of a supercritical Galton–Watson process {Zn:n1}, and bring out some criticality phenomena determined by the Schröder constant. We develop the local limit theory of Zn, that is, the behavior of P(Zn=vn) as vn, and use this to study conditional large deviations of {YZn:n1}, where Yn satisfies an LDP, particularly of {Zn1Zn+1:n1} conditioned on Znvn.

Article information

Source
Ann. Appl. Probab., Volume 14, Number 3 (2004), 1135-1166.

Dates
First available in Project Euclid: 13 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1089736280

Digital Object Identifier
doi:10.1214/105051604000000242

Mathematical Reviews number (MathSciNet)
MR2071418

Zentralblatt MATH identifier
1084.60542

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F10: Large deviations

Keywords
Branching processes large deviations local limit theorems.

Citation

Ney, Peter E.; Vidyashankar, Anand N. Local limit theory and large deviations for supercritical Branching processes. Ann. Appl. Probab. 14 (2004), no. 3, 1135--1166. doi:10.1214/105051604000000242. https://projecteuclid.org/euclid.aoap/1089736280


Export citation

References

  • Athreya, K. B. (1994). Large deviation rates for branching processes I. Single type case. Ann. Appl. Probab. 4 779–790.
  • Athreya, K. B. and Ney, P. E. (1970). The local limit theorem and some related aspects of super-critical branching processes. Trans. Amer. Math. Soc. 152 233–251.
  • Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, Berlin.
  • Athreya, K. B. and Vidyashankar, A. N. (1993). Large deviation results for branching processes. In Stochastic Processes (S. Cambanis, J. K. Ghosh, R. L. Karandikar and P. K. Sen, eds.) 7–12. Springer, New York.
  • Bahadur, R. R. and Ranga Rao, R. (1960). On deviations of the sample mean. Ann. Math. Statist. 31 1015–1027.
  • Basawa, I. V. (1981). Efficient conditional tests for mixture experiments with applications to the birth and branching processes. Biometrika 68 153–164.
  • Biggins, J. D. and Bingham, N. H. (1994). Large deviations in the supercritical branching process. Adv. in Appl. Probab. 25 757–772.
  • Bingham, N. H. (1988). On the limit of a supercritical branching process. J. Appl. Probab. 25A 215–228.
  • Blaisdell, B. E. (1985). A method of estimating from two-aligned present-day DNA sequences and their ancestral composition and subsequent rates of substitution, possibly different in the two lineages, corrected for multiple and parallel substitutions at the same site. J. Mol. Evol. 18 225–239.
  • Brown, W. M., Pager, E. M., Wang, A. and Wilson, A. C. (1982). Mitochondrial DNA sequences of primates: Tempo and mode of evolution. J. Mol. Evol. 18 225–239.
  • Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications. Springer, New York.
  • Dubuc, S. (1970). La fonction de Green d'un processus de Galton–Watson. Studia Math. 34 69–87.
  • Dubuc, S. (1971). La densitè de la loi-limite d'un processus en cascade expansif. Z. Wahrsch. Verw. Gebiete 19 281–290.
  • Dubuc, S. (1971). Problémes relatifs á l'itération de fonctions suggérés par les processus en cascade. Ann. Inst. Fourier 21 171–251.
  • Dubuc, S. and Senata, E. (1976). The local limit theorem for Galton–Watson process. Ann. Probab. 3 490–496.
  • Efron, B. and Hinkley, D. (1996). Assessing the accuracy of maximum likelihood estimator: Observed versus expected information. Biometrika 65 457–487.
  • Heyde, C. C. (1975). Remarks on efficiency in estimation for branching processes. Biometrika 62 49–55.
  • Heyde, C. C. (1977). An optimal property of maximum likelihood with application to branching process estimation. Bull. Inst. Internat. Statist. 47 407–416.
  • Joffe, A. and Waugh, W. A. O'N. (1982). Exact distributions of kin numbers in a Galton–Watson process. J. Appl. Probab. 19 767–775.
  • Joffe, A. and Waugh, W. A. O'N. (1985). Exact distributions of kin numbers in multitype Galton–Watson population. J. Appl. Probab. 22 37–47.
  • Karlin, S. and McGregor, J. (1968). Embeddability of discrete-time simple branching processes into continuous-time branching processes. Trans. Amer. Math. Soc. 132 115–136.
  • Karlin, S. and McGregor, J. (1968). Embedding iterates of analytic functions with two fixed points into continuous groups. Trans. Amer. Math. Soc. 132 137–145.
  • Kelly, C. (1994). A test for DNA evolutionary models. Biometrics 50 653–664.
  • Ney, P. E. and Vidyashankar, A. N. (2003). Harmonic moments and large deviation rates for supercritical branching processes. Ann. Appl. Probab. 13 475–489.
  • Pakes, A. G. (1975). Nonparametric estimation in the Galton–Watson process. Math. Biosci. 26 1–18.
  • Severini, T. A. (1996). Information and conditional inference. J. Amer. Statist. Assoc. 90 1341–1346.
  • Severini, T. A. (2000). Likelihood Methods in Statistics. Oxford Univ. Press.
  • Sweeting, T. (1978). On efficient tests for branching processes. Biometrika 65 123–127.
  • Sweeting, T. (1986). Asymptotic conditional inference for the offspring mean of a supercritical Galton–Watson process. Ann. Statist. 14 925–933.
  • Taïb, Z. (1992). Branching Processes and Neutral Evolution. Springer, Berlin.
  • Zygmund, A. (1988). Trigonometric Series, 2nd ed. Cambridge Univ. Press.