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Some central limit analogues for supercritical Galton-Watson processes

Published online by Cambridge University Press:  14 July 2016

C. C. Heyde*
Affiliation:
Australian National University

Extract

It is possible to interpret the classical central limit theorem for sums of independent random variables as a convergence rate result for the law of large numbers. For example, if Xi, i = 1, 2, 3, ··· are independent and identically distributed random variables with EXi = μ, var Xi = σ2 < ∞ and then the central limit theorem can be written in the form This provides information on the rate of convergence in the strong law as . (“a.s.” denotes almost sure convergence.) It is our object in this paper to discuss analogues for the super-critical Galton-Watson process.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1971 

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References

[1] Gnedenko, B. V. and Kolmogorov, A. N. (1954) Limit Distributions for Sums of Independent Random Variables. Addison Wesley, Reading, Mass.Google Scholar
[2] Heyde, C. C. (1970) Extension of a result of Seneta for the super-critical Galton-Watson process. Ann. Math. Statist. 41, 739742.CrossRefGoogle Scholar
[3] Heyde, C. C. (1970) A rate of convergence result for the super-critical Galton-Watson process. J. Appl. Prob. 7, 451454.Google Scholar
[4] Kesten, H. and Stigum, B. P. (1966) A limit theorem for multi-dimensional Galton-Watson processes. Ann. Math. Statist. 37, 12111223.Google Scholar
[5] Lamperti, J. (1967) Limiting distributions for branching processes. Proc. 5th Berkeley Symposium on Math. Statist. and Prob. II, 225241.Google Scholar
[6] Levinson, N. (1959) Limiting theorems for Galton-Watson branching process. Illinois J. Math. 3, 554565.CrossRefGoogle Scholar
[7] Seneta, E. (1969) Functional equations and the Galton-Watson process. Adv. Appl. Prob. 1, 142.Google Scholar
[8] Stigum, B. P. (1966) A theorem on the Galton-Watson process. Ann. Math. Statist. 37, 695698.Google Scholar