scholarly journals On asymptotic joint distributions of cherries and pitchforks for random phylogenetic trees

2021 ◽  
Vol 83 (4) ◽  
Author(s):  
Kwok Pui Choi ◽  
Gursharn Kaur ◽  
Taoyang Wu

AbstractTree shape statistics provide valuable quantitative insights into evolutionary mechanisms underpinning phylogenetic trees, a commonly used graph representation of evolutionary relationships among taxonomic units ranging from viruses to species. We study two subtree counting statistics, the number of cherries and the number of pitchforks, for random phylogenetic trees generated by two widely used null tree models: the proportional to distinguishable arrangements (PDA) and the Yule-Harding-Kingman (YHK) models. By developing limit theorems for a version of extended Pólya urn models in which negative entries are permitted for their replacement matrices, we deduce the strong laws of large numbers and the central limit theorems for the joint distributions of these two counting statistics for the PDA and the YHK models. Our results indicate that the limiting behaviour of these two statistics, when appropriately scaled using the number of leaves in the underlying trees, is independent of the initial tree used in the tree generating process.

2020 ◽  
Vol 52 (4) ◽  
pp. 1127-1163
Author(s):  
Jie Yen Fan ◽  
Kais Hamza ◽  
Peter Jagers ◽  
Fima C. Klebaner

AbstractA general multi-type population model is considered, where individuals live and reproduce according to their age and type, but also under the influence of the size and composition of the entire population. We describe the dynamics of the population as a measure-valued process and obtain its asymptotics as the population grows with the environmental carrying capacity. Thus, a deterministic approximation is given, in the form of a law of large numbers, as well as a central limit theorem. This general framework is then adapted to model sexual reproduction, with a special section on serial monogamic mating systems.


2017 ◽  
Vol 54 (2) ◽  
pp. 569-587 ◽  
Author(s):  
Ollivier Hyrien ◽  
Kosto V. Mitov ◽  
Nikolay M. Yanev

Abstract We consider a class of Sevastyanov branching processes with nonhomogeneous Poisson immigration. These processes relax the assumption required by the Bellman–Harris process which imposes the lifespan and offspring of each individual to be independent. They find applications in studies of the dynamics of cell populations. In this paper we focus on the subcritical case and examine asymptotic properties of the process. We establish limit theorems, which generalize classical results due to Sevastyanov and others. Our key findings include a novel law of large numbers and a central limit theorem which emerge from the nonhomogeneity of the immigration process.


2021 ◽  
Vol 58 (2) ◽  
pp. 263-273
Author(s):  
Andrei N. Frolov

Fifty years ago P. Erdős and A. Rényi published their famous paper on the new law of large numbers. In this survey, we describe numerous results and achievements which are related with this paper or motivated by it during these years.


1989 ◽  
Vol 21 (04) ◽  
pp. 804-830 ◽  
Author(s):  
I. B. Ziedins ◽  
F.P. Kelly

Loss networks have long been of interest to telephone engineers, and are becoming increasingly important as models of computer and information systems. In this paper we present an asymptotic analysis of loss networks exhibiting various forms of symmetry. Our aim is to better understand the behaviour of networks involving very large numbers of links and routes, where an exact analysis is not possible and approximations are required.


Author(s):  
Paolo Giudici

Several classes of computational and statistical methods for data mining are available. Each class can be parameterised so that models within the class differ in terms of such parameters (see, for instance, Giudici, 2003; Hastie et al., 2001; Han & Kamber, 2000; Hand et al., 2001; Witten & Frank, 1999): for example, the class of linear regression models, which differ in the number of explanatory variables; the class of Bayesian networks, which differ in the number of conditional dependencies (links in the graph); the class of tree models, which differ in the number of leaves; and the class multi-layer perceptrons, which differ in terms of the number of hidden strata and nodes. Once a class of models has been established the problem is to choose the “best” model from it.


Genes ◽  
2019 ◽  
Vol 10 (7) ◽  
pp. 490 ◽  
Author(s):  
Sharma ◽  
Gupta

The class Hematozoa encompasses several clinically important genera, including Plasmodium, whose members cause the major life-threating disease malaria. Hence, a good understanding of the interrelationships of organisms from this class and reliable means for distinguishing them are of much importance. This study reports comprehensive phylogenetic and comparative analyses on protein sequences on the genomes of 28 hematozoa species to understand their interrelationships. In addition to phylogenetic trees based on two large datasets of protein sequences, detailed comparative analyses were carried out on the genomes of hematozoa species to identify novel molecular synapomorphies consisting of conserved signature indels (CSIs) in protein sequences. These studies have identified 79 CSIs that are exclusively present in specific groups of Hematozoa/Plasmodium species, also supported by phylogenetic analysis, providing reliable means for the identification of these species groups and understanding their interrelationships. Of these CSIs, six CSIs are specifically shared by all hematozoa species, two CSIs serve to distinguish members of the order Piroplasmida, five CSIs are uniquely found in all Piroplasmida species except B. microti and two CSIs are specific for the genus Theileria. Additionally, we also describe 23 CSIs that are exclusively present in all genome-sequenced Plasmodium species and two, nine, ten and eight CSIs which are specific for members of the Plasmodium subgenera Haemamoeba, Laverania, Vinckeia and Plasmodium (excluding P. ovale and P. malariae), respectively. Additionally, our work has identified several CSIs that support species relationships which are not evident from phylogenetic analysis. Of these CSIs, one CSI supports the ancestral nature of the avian-Plasmodium species in comparison to the mammalian-infecting groups of Plasmodium species, four CSIs strongly support a specific relationship of species between the subgenera Plasmodium and Vinckeia and three CSIs each that reliably group P. malariae with members of the subgenus Plasmodium and P. ovale within the subgenus Vinckeia, respectively. These results provide a reliable framework for understanding the evolutionary relationships among the Plasmodium/Piroplasmida species. Further, in view of the exclusivity of the described molecular markers for the indicated groups of hematozoa species, particularly large numbers of unique characteristics that are specific for all Plasmodium species, they provide important molecular tools for biochemical/genetic studies and for developing novel diagnostics and therapeutics for these organisms.


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