Exponential Approximation for the Nearly Critical Galton-Watson Process and Occupation Times of Markov Chains

Erol A. Peköz (Boston University)
Adrian Röllin (National University of Singapore)


In this article we provide new applications for exponential approximation using the framework of Peköz and Röllin (2011), which is based on Stein's method. We give error bounds for the nearly critical Galton-Watson process conditioned on non-extinction, and for the occupation times of Markov chains; for the latter, in particular, we give a new exponential approximation rate for the number of revisits to the origin for general two dimensional random walk, also known as the Erdös-Taylor theorem.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1381-1393

Publication Date: August 10, 2011

DOI: 10.1214/EJP.v16-914


  1. Arratia, R.; Goldstein, L. Size bias, sampling, the waiting time paradox, and infinite divisibility: when is the increment independent? Preprint, 2010. arXiv:1007.3910v1
  2. Barbour, A. D.; Holst, L.; Janson, S. Poisson approximation. Oxford Studies in Probability, 2. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1992. x+277 pp. ISBN: 0-19-852235-5 MR1163825 (93g:60043)
  3. Bingham, N. H. On the limit of a supercritical branching process. A celebration of applied probability. J. Appl. Probab. 1988, Special Vol. 25A, 215--228. MR0974583 (90a:60150)
  4. Brown, M. Exploiting the waiting time paradox: applications of the size-biasing transformation. Probab. Engrg. Inform. Sci. 20 (2006), no. 2, 195--230. MR2261286 (2008b:60186)
  5. Darling, D. A.; Kac, M. On occupation times for Markoff processes. Trans. Amer. Math. Soc. 84 (1957), 444--458. MR0084222 (18,832a)
  6. Erdős, P.; Taylor, S. J. Some problems concerning the structure of random walk paths. Acta Math. Acad. Sci. Hungar. 11 1960 137--162. (unbound insert). MR0121870 (22 #12599)
  7. Fahady, K. S.; Quine, M. P.; Vere-Jones, D. Heavy traffic approximations for the Galton-Watson process. Advances in Appl. Probability 3 1971 282--300. MR0288858 (44 #6053)
  8. Fujimagari, T. On the extinction time distribution of a branching process in varying environments. Adv. in Appl. Probab. 12 (1980), no. 2, 350--366. MR0569432 (81d:60086)
  9. Gärtner, J.; Sun, R. A quenched limit theorem for the local time of random walks on $Bbb Zsp 2$. Stochastic Process. Appl. 119 (2009), no. 4, 1198--1215. MR2508570 (2010f:60143)
  10. Gibbs, A. L., Su, F. E. On choosing and bounding probability metrics. International Statistical Review / Revue Internationale de Statistique 70 , 419--435.
  11. Lawler, G. F.; Limic, V. Random walk: a modern introduction. Cambridge Studies in Advanced Mathematics, 123. Cambridge University Press, Cambridge, 2010. xii+364 pp. ISBN: 978-0-521-51918-2 MR2677157
  12. Lyons, R.; Pemantle, R.; Peres, Y. Conceptual proofs of $Llog L$ criteria for mean behavior of branching processes. Ann. Probab. 23 (1995), no. 3, 1125--1138. MR1349164 (96m:60194)
  13. Peköz, E.; Röllin, A. New rates for exponential approximation and the theorems of Rényi and Yaglom. Ann. Probab 39 (2011), no. 2, 587--608.
  14. Peköz, E.; Röllin, A.; Čekanavičius, V.; Shwartz, M. A three-parameter binomial approximation. J. Appl. Probab. 46 (2009), no. 4, 1073--1085. MR2582707 (2011c:62029)
  15. Peköz, E.; Ross, S. A second course in probability. www.ProbabilityBookstore.com, Boston, MA. 2007.
  16. Yaglom, A. M. Certain limit theorems of the theory of branching random processes. (Russian) Doklady Akad. Nauk SSSR (N.S.) 56, (1947). 795--798. MR0022045 (9,149e)

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.