Abstract
This paper studies the asymptotic behaviour of extreme order statistics of i.i.d. random scores ascribed to each individual in a Galton-Watson family tree. Of interest is the asymptotic behaviour of the order statistics within thenth generation, or up to and including thenth generation, and the index of the generation up to thenth which contains the largest observation.
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References
Arnold BC, Villaseñor JA (1996) The tallest man in the world. In: Nagaraja HN, Sen PK, Morrison DF (eds.) Statistical Theory and Applications. Papers in Honor of Herbert A. David, Springer, New York, pp. 81–88
Athreya KB, Ney PE (1972) Branching processes. Springer-Verlag, Berlin
Bingham NH, Doney RA (1974) Asymptotic properties of super-critical branching processes. I: The Galton-Watson process. Adv. Appl. Prob. 6: 711–731
Bru B, Jongmans F, Seneta E (1992) I.J. Bienaymé: Family information and proof of the criticality theorem. Intl. Statist. Review 60: 177–183
de Meyer A (1982) On a theorem of Bingham and Doney. J. Appl. Prob. 19: 217–220
David HA (1981) Order statistics, 2nd ed. Wiley, New York
Galambos J (1978) The asymptotic theory of extreme order statistics. Wiley, New York
Guttorp P (1991) Statistical inference for branching processes. Wiley, New York
Guttorp P (1995) Three papers on the history of branching processes. Intl. Statist. Review 63: 233–245
Harris TE (1951) Some mathematical models for branching processes. In: Neyman J (ed.) Second Symposium on Probability and Statistics, University of California Press, Berkeley, pp. 305–328
Jagers P (1975) Branching processes with biological applications. Wiley, London
Kaplan N (1983) The limiting behaviour of record values for an increasing population. Unpublished report, Dept. Statistics, Univ California, Berkeley
Pakes AG (1971) Some limit theorems for the total progeny of a branching process. Adv. Appl. Prob. 3: 176–192
Pakes AG (1974) Some limit theorems for Markov chains with applications to branching processes. In: Williams EJ (ed.) Studies in Probability and Statistics, North-Holland, Amsterdam, pp. 21–39
Pakes AG (1983) Remarks on a model of competitive bidding for employment. J. Appl. Prob. 20: 349–357
Pakes AG (1998) A limit theorem for the maxima of the para-critical simple branching process. Adv. Appl. Prob., To appear
Pakes AG (1997) Revisiting conditional limit theorems for the mortal simple branching process. Bernoulli, submitted
Papangelou F (1968) A lemma on the Galton-Watson process and some of its consequences. Proc. Amer. Math. Soc. 19: 1169–1179
Phatarfod RP (1983) Private communication
Resnick S (1987) Extreme values, regular variation, and point processes. Springer-Verlag, New York
Slack RS (1968) A branching process with mean one and possibly infinite variance. Z. Wahrs. 9: 139–145
Yang MCK (1975) On the increasing distribution of the inter-record times in an increasing population. J. Appl. Prob. 12: 148–154
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Pakes, A.G. Extreme order statistics on Galton-Watson trees. Metrika 47, 95–117 (1998). https://doi.org/10.1007/BF02742867
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DOI: https://doi.org/10.1007/BF02742867