scholarly journals On the distribution of the number of vertices in layers of random trees

1991 ◽  
Vol 4 (3) ◽  
pp. 175-186 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Asymptotic Behavior ◽  
Branching Processes ◽  
Random Trees ◽  
Rooted Trees ◽  
Brownian Excursion

Denote by Sn the set of all distinct rooted trees with n labeled vertices. A tree is chosen at random in the set Sn, assuming that all the possible nn−1 choices are equally probable. Define τn(m) as the number of vertices in layer m, that is, the number of vertices at a distance m from the root of the tree. The distance of a vertex from the root is the number of edges in the path from the vertex to the root. This paper is concerned with the distribution and the moments of τn(m) and their asymptotic behavior in the case where m=[2αn], 0<α<∞ and n→∞. In addition, more random trees, branching processes, the Bernoulli excursion and the Brownian excursion are also considered.

scholarly journals The profile of unlabeled trees

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Bernhard Gittenberger
Keyword(s):  
Local Time ◽  
Joint Distribution ◽  
Random Trees ◽  
Rooted Trees ◽  
Brownian Excursion ◽  
Level Number ◽  
International Audience ◽  
The Times ◽  
Simply Generated Trees

International audience We consider the number of nodes in the levels of unlabeled rooted random trees and show that the joint distribution of several level sizes (where the level number is scaled by $\sqrt{n}$) weakly converges to the distribution of the local time of a Brownian excursion evaluated at the times corresponding to the level numbers. This extends existing results for simply generated trees and forests to the case of unlabeled rooted trees.

scholarly journals Interacting branching processes and linear file-sharing networks

2010 ◽  
Vol 42 (03) ◽  
pp. 834-854 ◽  
Author(s):  
Lasse Leskelä ◽  
Philippe Robert ◽  
Florian Simatos
Keyword(s):  
Communication Network ◽  
Asymptotic Behavior ◽  
Distributed Systems ◽  
Stochastic Model ◽  
Branching Processes ◽  
File Sharing ◽  
Independent Interest ◽  
Constant Flow

File-sharing networks are distributed systems used to disseminate files among nodes of a communication network. The general simple principle of these systems is that once a node has retrieved a file, it may become a server for this file. In this paper, the capacity of these networks is analyzed with a stochastic model when there is a constant flow of incoming requests for a given file. It is shown that the problem can be solved by analyzing the asymptotic behavior of a class of interacting branching processes. Several results of independent interest concerning these branching processes are derived and then used to study the file-sharing systems.

scholarly journals Genealogy for supercritical branching processes

2006 ◽  
Vol 43 (04) ◽  
pp. 1066-1076 ◽  
Author(s):  
Andreas Nordvall Lagerås ◽  
Anders Martin-Löf
Keyword(s):  
Marginal Distribution ◽  
Branching Processes ◽  
Limiting Distribution ◽  
Random Trees ◽  
Limiting Distributions ◽  
Marginal Distributions ◽  
Special Cases ◽  
Supercritical Branching Processes

We study the genealogy of so-called immortal branching processes, i.e. branching processes where each individual upon death is replaced by at least one new individual, and conclude that their marginal distributions are compound geometric. The result also implies that the limiting distributions of properly scaled supercritical branching processes are compound geometric. We exemplify our results with an expression for the marginal distribution for a class of branching processes that have recently appeared in the theory of coalescent processes and continuous stable random trees. The limiting distribution can be expressed in terms of the Fox H-function, and in special cases by the Meijer G-function.

Branching processes with varying and random geometric offspring distributions

1975 ◽  
Vol 12 (01) ◽  
pp. 135-141 ◽  
Author(s):  
Niels Keiding ◽  
John E. Nielsen
Keyword(s):  
Asymptotic Behavior ◽  
Generating Functions ◽  
Conditional Expectation ◽  
Elementary Proof ◽  
Branching Processes ◽  
Random Environments ◽  
Subcritical Case ◽  
Supercritical Case ◽  
Extinction Probabilities ◽  
Varying Environments

The class of fractional linear generating functions is used to illustrate various aspects of the theory of branching processes in varying and random environments. In particular, it is shown that Church's theorem on convergence of the varying environments process admits of an elementary proof in this particular case. For random environments, examples are given on the asymptotic behavior of extinction probabilities in the supercritical case and conditional expectation given non-extinction in the subcritical case.

scholarly journals Modeling Y-Linked Pedigrees through Branching Processes

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 256
Author(s):  
Miguel González ◽  
Cristina Gutiérrez ◽  
Rodrigo Martínez
Keyword(s):  
Hearing Loss ◽  
Asymptotic Behavior ◽  
Gibbs Sampler ◽  
Mutant Allele ◽  
Branching Process ◽  
Branching Processes ◽  
Dominant Allele ◽  
Linked Gene ◽  
Different Types ◽  
The Mean

A multidimensional two-sex branching process is introduced to model the evolution of a pedigree originating from the mutation of an allele of a Y-linked gene in a monogamous population. The study of the extinction of the mutant allele and the analysis of the dominant allele in the pedigree is addressed on the basis of the classical theory of multi-type branching processes. The asymptotic behavior of the number of couples of different types in the pedigree is also derived. Finally, using the estimates of the mean growth rates of the allele and its mutation provided by a Gibbs sampler, a real Y-linked pedigree associated with hearing loss is analyzed, concluding that this mutation will persist in the population although without dominating the pedigree.

Asymptotic behavior of near-critical multitype branching processes

1991 ◽  
Vol 28 (03) ◽  
pp. 512-519 ◽  
Author(s):  
Fima C. Klebaner
Keyword(s):  
Asymptotic Behavior ◽  
Population Size ◽  
Discrete Time ◽  
Growth Rates ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Limiting Distribution ◽  
Size Dependent ◽  
Generalized Gamma ◽  
Multitype Branching Processes

Sufficient conditions for survival and extinction of multitype population-size-dependent branching processes in discrete time are obtained. Growth rates are determined on the set of divergence to infinity. The limiting distribution of a properly normalized process can be generalized gamma, normal or degenerate.

scholarly journals Genealogy for supercritical branching processes

2006 ◽  
Vol 43 (4) ◽  
pp. 1066-1076 ◽  
Author(s):  
Andreas Nordvall Lagerås ◽  
Anders Martin-Löf
Keyword(s):  
Marginal Distribution ◽  
Branching Processes ◽  
Limiting Distribution ◽  
Random Trees ◽  
Limiting Distributions ◽  
Marginal Distributions ◽  
Special Cases ◽  
Supercritical Branching Processes

We study the genealogy of so-called immortal branching processes, i.e. branching processes where each individual upon death is replaced by at least one new individual, and conclude that their marginal distributions are compound geometric. The result also implies that the limiting distributions of properly scaled supercritical branching processes are compound geometric. We exemplify our results with an expression for the marginal distribution for a class of branching processes that have recently appeared in the theory of coalescent processes and continuous stable random trees. The limiting distribution can be expressed in terms of the Fox H-function, and in special cases by the Meijer G-function.

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