Abstract
This paper investigates large deviation results for the supercritical multi-type and the critical single type branching processes when conditioned on non extinction thus extending the results in [1] and [2] in two directions. We show for example that in the multitype supercritical case the probability of large deviation between the empirical population proportion and its stable limit decays geometrically. Similarly in the critical single type case the (large deviation) probability that the ratio of the population at time (n + 1) to that at time n deviates from one by more than decays at an algebraic rate. A number of similar results are presented here. Some open problems are indicated.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Athreya K.B. (1994): Large deviation rates for Branching Processes - I the single type case Annals of Applied Probability 4 No. 3 779–790.
Athreya K.B. and Ney P.E. (1972): Branching processes Springer-Verlag New York.
Athreya K.B. and Vidyashankar A.N. (1995): Large deviation results for branchÂing processes -II the multitype case Annals of Applied Probability 5 No. 2 566–576.
Biggins J.D. and Shanbhag D.N. (1981): Some divisbility problems in branching processes Math. Proc. Camb. Phil. Soc. 90 321–330.
Chow Y.S. and Teicher H. (1988): Probability Theory Independence InterchageÂability Martingale Second Edition Springer-Verlag N.Y.
Karp R. and Zhang Y. (1994): Finite Branching Processes and AND/OR Tree Evaluation TR93–043 International Computer Science Institute Berkeley CA.
Vidyashankar A.N. (1994): Large deviations rates for branching processes in fixed and random environments Ph.D. Thesis Departments of Mathematics and Statistics Iowa State University Ames IA.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer Science+Business Media New York
About this chapter
Cite this chapter
Athreya, K.B., Vidyashankar, A.N. (1997). Large Deviation Rates for Supercritical and Critical Branching Processes. In: Athreya, K.B., Jagers, P. (eds) Classical and Modern Branching Processes. The IMA Volumes in Mathematics and its Applications, vol 84. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1862-3_1
Download citation
DOI: https://doi.org/10.1007/978-1-4612-1862-3_1
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-7315-8
Online ISBN: 978-1-4612-1862-3
eBook Packages: Springer Book Archive