Lower deviation probabilities for supercritical Galton–Watson processes

Klaus Fleischmann; Vitali Wachtel

Annales de l'I.H.P. Probabilités et statistiques (2007)

  • Volume: 43, Issue: 2, page 233-255
  • ISSN: 0246-0203

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Fleischmann, Klaus, and Wachtel, Vitali. "Lower deviation probabilities for supercritical Galton–Watson processes." Annales de l'I.H.P. Probabilités et statistiques 43.2 (2007): 233-255. <http://eudml.org/doc/77933>.

@article{Fleischmann2007,
author = {Fleischmann, Klaus, Wachtel, Vitali},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {local limit theorem; large deviation; Cramér transform; concentration function; Schröder equation; Böttcher equation},
language = {eng},
number = {2},
pages = {233-255},
publisher = {Elsevier},
title = {Lower deviation probabilities for supercritical Galton–Watson processes},
url = {http://eudml.org/doc/77933},
volume = {43},
year = {2007},
}

TY - JOUR
AU - Fleischmann, Klaus
AU - Wachtel, Vitali
TI - Lower deviation probabilities for supercritical Galton–Watson processes
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2007
PB - Elsevier
VL - 43
IS - 2
SP - 233
EP - 255
LA - eng
KW - local limit theorem; large deviation; Cramér transform; concentration function; Schröder equation; Böttcher equation
UR - http://eudml.org/doc/77933
ER -

References

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  1. [1] A. Asmussen, H. Hering, Branching Processes, Progress in Probab. and Statistics, vol. 3, Birkhäuser Boston Inc., Boston, MA, 1983. Zbl0516.60095MR701538
  2. [2] K.B. Athreya, P.E. Ney, The local limit theorem and some related aspects of supercritical branching processes, Trans. Amer. Math. Soc.152 (2) (1970) 233-251. Zbl0214.16203MR268971
  3. [3] J.D. Biggins, N.H. Bingham, Near-constancy phenomena in branching processes, Math. Proc. Cambridge Philos. Soc.110 (3) (1991) 545-558. Zbl0749.60077MR1120488
  4. [4] J.D. Biggins, N.H. Bingham, Large deviations in the supercritical branching process, Adv. Appl. Probab.25 (4) (1993) 757-772. Zbl0796.60090MR1241927
  5. [5] N.H. Bingham, On the limit of a supercritical branching process, J. Appl. Probab.25A (1988) 215-228. Zbl0669.60078MR974583
  6. [6] S. Dubuc, La densite de la loi-limite d'un processus en cascade expansif, Z. Wahrsch. Verw. Gebiete19 (1971) 281-290. Zbl0215.25603MR300353
  7. [7] S. Dubuc, Etude theorique et numerique de la fonction de Karlin–McGregor, J. Analyse Math.42 (1982) 15-37. Zbl0545.65084
  8. [8] S. Dubuc, E. Seneta, The local limit theorem for the Galton–Watson process, Ann. Probab.4 (1976) 490-496. Zbl0332.60059
  9. [9] W. Feller, An Introduction to Probability Theory and its Applications, vol. II, second ed., John Wiley and Sons, New York, 1971. Zbl0219.60003MR270403
  10. [10] K. Fleischmann, V. Wachtel, Lower deviation probabilities for supercritical Galton–Watson processes, WIAS Berlin, Preprint No. 1025 of April 28, 2005, http://www.wias-berlin.de/publications/preprints/1025/. 
  11. [11] K. Fleischmann, V. Wachtel, Large deviations for sums defined on a Galton–Watson process, WIAS Berlin, Preprint No. 1135 of May 29, 2006. Zbl1141.60048
  12. [12] R. Höpfner, Local limit theorems for non-critical Galton–Watson processes with or without immigration, J. Appl. Probab.19 (1982) 262-271. Zbl0483.60081
  13. [13] M. Kuczma, Functional Equations in a Single Variable, PWN, Warszaw, 1968. Zbl0196.16403MR228862
  14. [14] P.E. Ney, A.N. Vidyashankar, Local limit theory and large deviations for supercritical branching processes, Ann. Appl. Probab.14 (2004) 1135-1166. Zbl1084.60542MR2071418
  15. [15] V.V. Petrov, Sums of Independent Random Variables, Springer-Verlag, Berlin, 1975. Zbl0322.60042MR388499
  16. [16] E. Seneta, Regularly Varying Functions, Lecture Notes in Math., vol. 508, Springer-Verlag, Berlin, 1976. Zbl0324.26002MR453936

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