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On best fractional linear generating function bounds

Published online by Cambridge University Press:  14 July 2016

Tea-Yuan Hwang  and
Nae-Sheng Wang
Tea-Yuan Hwang*
Affiliation:
National Tsing Hua University, Taiwan
Nae-Sheng Wang*
Affiliation:
National Tsing Hua University, Taiwan
*
∗Postal address: Institute of Applied Mathematics, National Tsing Hua University, Hsinchu, Taiwan 300, Republic of China.
∗Postal address: Institute of Applied Mathematics, National Tsing Hua University, Hsinchu, Taiwan 300, Republic of China.
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Abstract

Under weak conditions, this paper provides a best lower and a best upper bounding fractional linear generating function for any probability generating function when they have the same mean. These bounds can be used to obtain bounds for the expectation and the percentiles of the extinction-time distribution of a Galton-Watson branching process and other parameters of interest. For the special case of the four points probability generating function, the best bounds obtained are better than the bounds derived by Agresti (1974).

Keywords

GALTON-WATSON PROCESSFRACTIONAL LINEAR GENERATING FUNCTIONEXTINCTION TIMEPERCENTILE
Type
Short Communications
Information
Journal of Applied Probability , Volume 16 , Issue 2 , June 1979 , pp. 449 - 453
Copyright
Copyright © Applied Probability Trust 1979 

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References

Agresti, A. (1974) Bounds on the extinction time distribution of a branching process. Adv. Appl. Prob. 6, 322–335.Google Scholar
Heathcote, C. R. and Seneta, E. (1966) Inequalities for branching processes. J. Appl. Prob. 3, 261–267.CrossRefGoogle Scholar
Seneta, E. (1968) On asymptotic properties of subcritical branching processes. J. Austral. Math. Soc. 8, 671–682.Google Scholar