Summary
In recent years several authors have obtained limit theorems for L n , the location of the rightmost particle in a supercritical branching random walk but all of these results have been proved under the assumption that the offspring distribution has ϕ(θ) = ∝ exp(θx)dF(x)<∞ for some θ>0. In this paper we investigate what happens when there is a slowly varying function K so that 1−F(x)∼x }-q K(x) as x → ∞ and log(−x)F(x)→0 as x→−∞. In this case we find that there is a sequence of constants a n , which grow exponentially, so that L n /a n converges weakly to a nondegenerate distribution. This result is in sharp contrast to the linear growth of L n observed in the case ϕ(θ)<∞.
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Athreya, K., Ney, P.: Branching Processes. Berlin-Heidelberg-New York: Springer 1972
Bahadur, R., Rao, Ranga: On deviations of the sample mean. Ann. Math. Statist. 31, 1015–1027 (1960)
Bramson, M.: Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math. 31, 531–582 (1978)
Bühler, W.: The distribution of generations and other aspects of the family structure of branching processes. Proc. Sixth Berkeley Sympos. Math. Statist. Probab. University of California. Vol. III, 463–480 (1972)
Durrett, R.: The genealogy of critical branching processes, Stoch. Proc. Appl. 8, 101–116 (1978)
Durrett, R.: Maxima of branching random walks vs. Independent Random Walks. Stoch. Proc. Appl. 9, 117–135 (1979)
Feller, W.: An Introduction to Probability Theory And Its Applications, Vol. II, second edition. New York: John Wiley 1971
Gnedenko, B.V., Kolmogorov, A.N.: Limit Theorems For Sums of Independent Random Variables. Reading, Mass.: Addison-Wesley 1954
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Durrett, R. Maxima of branching random walks. Z. Wahrscheinlichkeitstheorie verw Gebiete 62, 165–170 (1983). https://doi.org/10.1007/BF00538794
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DOI: https://doi.org/10.1007/BF00538794