On Renewal Matrices Connected with Branching Processes with Tails of Distributions of Different Orders

On Renewal Matrices Connected with Branching Processes with Tails of Distributions of Different... We study irreducible renewal matrices generated by matrices whose rows are proportional to various distribution functions. Such matrices arise in studies of multi-dimensional critical Bellman–Harris branching processes. Proofs of limit theorems for such branching processes are based on asymptotic properties of a chosen family of renewal matrices. In the theory of branching processes, unsolved problems are known that correspond to the case in which the tails of some of the above mentioned distribution functions are integrable, while the other distributions lack this property.We assume that the heaviest tails are regularly varying at the infinity with parameter −β ∈ [−1, 0) and asymptotically proportional, while the other tails are infinitesimal with respect to them. Under a series of additional conditions, we describe asymptotic properties of the first and second order increments for the renewal matrices. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Siberian Advances in Mathematics Springer Journals

On Renewal Matrices Connected with Branching Processes with Tails of Distributions of Different Orders

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Allerton Press, Inc.
Subject
Mathematics; Mathematics, general
ISSN
1055-1344
eISSN
1934-8126
D.O.I.
10.3103/S1055134418020037
Publisher site
See Article on Publisher Site

Abstract

We study irreducible renewal matrices generated by matrices whose rows are proportional to various distribution functions. Such matrices arise in studies of multi-dimensional critical Bellman–Harris branching processes. Proofs of limit theorems for such branching processes are based on asymptotic properties of a chosen family of renewal matrices. In the theory of branching processes, unsolved problems are known that correspond to the case in which the tails of some of the above mentioned distribution functions are integrable, while the other distributions lack this property.We assume that the heaviest tails are regularly varying at the infinity with parameter −β ∈ [−1, 0) and asymptotically proportional, while the other tails are infinitesimal with respect to them. Under a series of additional conditions, we describe asymptotic properties of the first and second order increments for the renewal matrices.

Journal

Siberian Advances in MathematicsSpringer Journals

Published: May 30, 2018

References

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