Large deviations for i.i.d. replications of the total progeny of a Galton–Watson process
Volume 4, Issue 1 (2017), pp. 1–13
Pub. online: 11 January 2017
Type: Research Article
Open Access
Open Access
Received
27 September 2016
27 September 2016
Revised
15 December 2016
15 December 2016
Accepted
17 December 2016
17 December 2016
Published
11 January 2017
11 January 2017
Abstract
The Galton–Watson process is the simplest example of a branching process. The relationship between the offspring distribution, and, when the extinction occurs almost surely, the distribution of the total progeny is well known. In this paper, we illustrate the relationship between these two distributions when we consider the large deviation rate function (provided by Cramér’s theorem) for empirical means of i.i.d. random variables. We also consider the case with a random initial population. In the final part, we present large deviation results for sequences of estimators of the offspring mean based on i.i.d. replications of total progeny.
References
Asmussen, S., Hering, H.: Branching Processes. Birkhäuser, Boston (1983). MR0701538. doi:10.1007/978-1-4615-8155-0
Athreya, K.B.: Large deviation rates for branching processes. I. Single type case. Ann. Appl. Probab. 4, 779–790 (1994). MR1284985. doi:10.1214/aoap/1177004971
Athreya, K.B., Ney, P.E.: Branching Processes. Springer, New York, Heidelberg (1972). MR0373040
Athreya, K.B., Vidyashankar, A.N.: Large deviation rates for branching processes. II. The multitype case. Ann. Appl. Probab. 5, 566–576 (1995). MR1336883. doi:10.1214/aoap/1177004778
Biggins, J.D., Bingham, N.H.: Large deviations in the supercritical branching process. Adv. Appl. Probab. 25, 757–772 (1993). MR1241927. doi:10.2307/1427790
Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Springer, New York (1998). MR1619036. doi:10.1007/978-1-4612-5320-4
Dwass, M.: The total progeny in a branching process and a related random walk. J. Appl. Probab. 6, 682–686 (1969). MR0253433. doi:10.1017/S0021900200026711
Fleischmann, K., Wachtel, V.: Lower deviation probabilities for supercritical Galton–Watson processes. Ann. Inst. Henri Poincaré Probab. Stat. 43, 233–255 (2007). MR2303121. doi:10.1016/j.anihpb.2006.03.001
González, M., Molina, M.: On the partial and total progeny of a bisexual Galton–Watson branching process. Appl. Stoch. Models Data Anal. 13, 225–232 (1998). MR1628630. doi:10.1002/(SICI)1099-0747(199709/12)13:3/4<225::AID-ASM316>3.0.CO;2-9
Guttorp, P.: Statistical Inference for Branching Processes. John Wiley and Sons, New York (1991). MR1254434
Hamza, K., Klebaner, F.C.: How did we get here? J. Appl. Probab. A 51, 63–72 (2014). MR3317350. doi:10.1239/jap/1417528467
Harris, T.E.: The Theory of Branching Processes. Springer, Berlin (1963). MR0163361
Jagers, P.: Branching Processes with Biological Applications. John Wiley and Sons, London, New York, Sydney (1975). MR0488341
Kennedy, D.P.: The Galton–Watson process conditioned on the total progeny. J. Appl. Probab. 12, 800–806 (1975). MR0386042. doi:10.1017/S0021900200048750
Kimmel, M., Axelrod, D.E.: Branching Processes in Biology, 2nd edn. Springer, New York (2015). MR3310028. doi:10.1007/978-1-4939-1559-0
Klebaner, F.C., Liptser, R.: Likely path to extinction in simple branching models with large initial population. J. Appl. Math. Stoch. Anal. 23, 60376 (2006). MR2221000. doi:10.1155/JAMSA/2006/60376
Klebaner, F.C., Liptser, R.: Large deviations analysis of extinction in branching models. Math. Popul. Stud. 15, 55–69 (2008). MR2380468. doi:10.1080/08898480701792477
Mode, C.J.: Multitype Branching Processes. Theory and Applications. Modern Analytic and Computational Methods in Science and Mathematics, vol. 34. American Elsevier Publishing Co., Inc., New York (1971). MR0279901
Ney, P.E., Vidyashankar, A.N.: Harmonic moments and large deviation rates for supercritical branching processes. Ann. Appl. Probab. 13, 475–489 (2003). MR1970272. doi:10.1214/aoap/1050689589
Ney, P.E., Vidyashankar, A.N.: Local limit theory and large deviations for supercritical branching processes. Ann. Appl. Probab. 14, 1135–1166 (2004). MR2071418. doi:10.1214/105051604000000242
Ney, P.E., Vidyashankar, A.N.: Corrections and acknowledgment for: “Local limit theory and large deviations for supercritical branching processes”. Ann. Appl. Probab. 16, 2272–2272 (2006). MR2288722. doi:10.1214/105051606000000574
Pakes, A.G.: Some limit theorems for the total progeny of a branching process. Adv. Appl. Probab. 3, 176–192 (1971). MR0283892. doi:10.1017/S0001867800037629
Varadhan, S.R.S.: Large deviations and entropy. In: Greven, A., Keller, G., Warnecke, G. (eds.): Entropy, pp. 199–214. Princeton University Press, Princeton (2003). MR2035822
