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B-stability

Part of: Distribution theory - Probability Limit theorems Markov processes

Published online by Cambridge University Press:  14 July 2016

Colin Mallows  and
Larry Shepp
Colin Mallows*
Affiliation:
Avaya Labs
Larry Shepp*
Affiliation:
Rutgers University
*
∗Postal address: Avaya Labs, 233 Mount Airy Road, Basking Ridge, NJ 07920, USA. Email address: colinm@research.avayalabs.com
∗∗Postal address: Department of Statistics, Rutgers University, Piscataway, NJ 08854, USA. Email address: shepp@stat.rutgers.edu
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Abstract

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A random variable Y is branching stable (B-stable) for a nonnegative integer-valued random variable J with E(J)>1 if Y*J∿cY for some scalar c, where Y*J is the sum of J independent copies of Y. We explore some aspects of this notion of stability and show that, for any Y0 with finite nonzero mean, if we define Yn+1=Yn*J/E(J) then the sequence Yn converges in law to a random variable Y∞ that is B-stable for J. Also Y∞ is the unique B-stable law with mean E(Y0). We also present results relating to random variables Y0 with zero means and infinite means. The notion of B-stability arose in a scheme for cataloguing a large network of computers.

Keywords

Stable lawbranching processcataloguing large networks of computers

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60E07: Infinitely divisible distributions; stable distributions 60E10: Characteristic functions; other transforms 60F17: Functional limit theorems; invariance principles
Type
Short Communications
Information
Journal of Applied Probability , Volume 42 , Issue 2 , June 2005 , pp. 581 - 586
Copyright
© Applied Probability Trust 2005 

References

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