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Transactions of the American Mathematical Society

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Limiting distributions for branching random fields


Author: Joseph Fleischman
Journal: Trans. Amer. Math. Soc. 239 (1978), 353-389
MSC: Primary 60J80
DOI: https://doi.org/10.1090/S0002-9947-1978-0478375-X
MathSciNet review: 0478375
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Abstract: In this paper we derive limiting distributions for branching Brownian motion. The cases considered are where the state space is (1) the line and (2) the plane where (a) initially there's but one particle and (b) where there's initially a random number of independent particles. In all cases, the branching process is critical and we obtain results for the growth of selectively neutral mutant types. We use moment generating functions to derive these results.


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  • [1] K. Athreya and P. Ney, Branching processes, Springer-Verlag, Berlin and New York, 1972. MR 0373040 (51:9242)
  • [2] L. Breiman, Probability, Addison-Wesley, Reading, Mass., 1968. MR 0229267 (37:4841)
  • [3] J. Crow and M. Kimura, An introduction to population genetics theory, Harper and Row, New York, 1970. MR 0274068 (42:8944)
  • [4] W. Ewens, Population genetics, Methuen, New York, 1969. MR 0270768 (42:5656)
  • [5] J. Felsenstein, A pain in the torus: Some difficulties with models of isolation by distance, Amer. Natur. 109 (1975), 359-368.
  • [6] B. V. Gnedenko and A. N. Kolmogorov, Limit distributions for sums of independent random variables, Addison-Wesley, Reading, Mass., 1954. MR 0062975 (16:52d)
  • [7] E. Hille, Analytic function theory, Vol. 2, Ginn, Waltham, Mass., 1962. MR 0201608 (34:1490)
  • [8] N. Ikeda, M. Nagasawa and S. Watanabe, Branching Markov processes, I, II, III, J. Math. Kyoto Univ. 8 (1968), 233-278, 365-410, 9 (1969), 95-160. MR 0232439 (38:764)
  • [9] S. Karlin, A first course in stochastic processes, Academic Press, New York, 1969. MR 0208657 (34:8466)
  • [10] E. Lukacs, Characteristic functions, Griffin, London, 1960.
  • [11] J. E. Moyal, Discontinuous Markov processes, Acta Math. 98 (1967), 221-264. MR 0093824 (20:344)
  • [12] T. W. Mullikan, Limiting distributions for critical multi-type branching processes with discrete time, Trans. Amer. Math. Soc. 106 (1963), 469-494. MR 0144386 (26:1931)
  • [13] Y. Ogura, Asymptotic behavior of multi-type Galton-Watson processes, J. Math. Kyoto Univ. 15 (1975), 251-302. MR 0383560 (52:4441)
  • [14] A. V. Skorokhod, Branching diffusion process, Theor. Probability Appl. 9 (1964), 492-497.
  • [15] A. A. Savin and V. P. Chistyakov, Some theorems for branching processes with several types of particles, Theor. Probability Appl. 7 (1962), 93-100.
  • [16] S. Sawyer, A formula for semi-groups with an application to branching diffusion processes, Trans. Amer. Math. Soc. 152 (1970), 1-38. MR 0266319 (42:1226)
  • [17] -, Branching diffusion processes in population genetics, Advances in Appl. Probability 8 (1976), 659-689. MR 0432250 (55:5239)
  • [18] S. Watanabe, On the branching processes for Brownian particles with an absorbing boundary, J. Math. Kyoto Univ. 4 (1965), 385-398. MR 0178505 (31:2762)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0478375-X
Keywords: Branching process, branching diffusion process, random field, mutation
Article copyright: © Copyright 1978 American Mathematical Society

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