Rahimov, I. Approximation of exceedance processes in large populations. (English) Zbl 0986.60085 Stoch. Models 17, No. 2, 147-156 (2001). Consider a population of \(n\) individuals each of which generates a discrete-time stochastic branching process. The author studies the number of ancestors \(S(n,t)\) whose offspring at time \(t\) exceed a level \(\theta (t)\), where \(\theta (t)\) is a positive function. He describes conditions on the growth rate of \(\theta (t)\) and \(n=n(t)\) as \(t\to\infty\) under which \(S(n,t)\) converges in Skorokhod topology to stochastic processes with independent increments. Reviewer: Vladimir Vatutin (Moskva) Cited in 2 Documents MSC: 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) Keywords:population; ancestor; branching process; Poisson process; Brownian motion; binomial process; exceedance; Skorokhod topology PDFBibTeX XMLCite \textit{I. Rahimov}, Stoch. Models 17, No. 2, 147--156 (2001; Zbl 0986.60085) Full Text: DOI References: [1] Roberts, F. S. 1984.Applied Combinatorics606New Jersey, USA: Prentice-Hall Inc. [2] Borovkov K. A., J. Appl. Probab. 33 pp 614– (1996) · Zbl 0869.60077 [3] Pakes A., Adv. Apl. Probab. 30 pp 740– (1998) · Zbl 0917.60085 [4] Arnold B. C., Statistical Theory and Applications pp 81– (1996) · Zbl 0853.60066 [5] Rahimov I., J. Appl. Probab. 36 (3) pp 632– (1999) · Zbl 0947.60085 [6] Rahimov I., Malaysian Math. Soc. 21 pp 37– (1998) [7] Athreya, K. and Ney, P. 1972.Branching Processes287New York, USA: Springer-Verlag. · Zbl 0259.60002 [8] Gikhman, I. I. and Skorohod, A. V. 1975.The Theory of Stochastic ProcessesVol. I, 570New York: Springer-Verlag. [9] Billingsley, P. 1968.Convegence of Probability Measures253New York, USA: Wiley. · Zbl 0172.21201 [10] Sevastyanov, B. A. 1971.Branching Processes436Moscow: Nauka. [11] Pollard, D. 1984.Convergence of Staochastic Processes215New York, USA: Springer Ser. Statistics, Sprivger-Verlag. [12] Shiryaev, A. N. 1996.Probability, second edition 621New York, USA: Springer-Verlag. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.