scholarly journals A functional limit law for the profile of plane-oriented recursive trees.

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Henning Sulzbach
Keyword(s):  
Continuous Time ◽  
Functional Limit ◽  
Convergence Theorems ◽  
International Audience ◽  
Recursive Trees ◽  
Martingale Convergence

International audience We give a functional limit law for the normalized profile of random plane-oriented recursive trees. The proof uses martingale convergence theorems in discrete and continuous-time. This complements results of Hwang (2007).

scholarly journals The Limiting Distribution for the Number of Symbol Comparisons Used by QuickSort is Nondegenerate (Extended Abstract)

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Patrick Bindjeme ◽  
james Allen fill
Keyword(s):  
Large Class ◽  
Continuous Time ◽  
Random Variable ◽  
Limiting Distribution ◽  
International Audience ◽  
The Mean

International audience In a continuous-time setting, Fill (2012) proved, for a large class of probabilistic sources, that the number of symbol comparisons used by $\texttt{QuickSort}$, when centered by subtracting the mean and scaled by dividing by time, has a limiting distribution, but proved little about that limiting random variable $Y$—not even that it is nondegenerate. We establish the nondegeneracy of $Y$. The proof is perhaps surprisingly difficult.

ON SEVERAL PROPERTIES OF A CLASS OF PREFERENTIAL ATTACHMENT TREES—PLANE-ORIENTED RECURSIVE TREES

2020 ◽  
pp. 1-19
Author(s):  
Panpan Zhang
Keyword(s):  
Continuous Time ◽  
Preferential Attachment ◽  
Mass Function ◽  
Probability Mass Function ◽  
Zagreb Index ◽  
Exact Probability ◽  
Probability Mass ◽  
Proper Scaling ◽  
Recursive Trees ◽  
Degree Variable

In this paper, several properties of a class of trees presenting preferential attachment phenomenon—plane-oriented recursive trees (PORTs) are uncovered. Specifically, we investigate the degree profile of a PORT by determining the exact probability mass function of the degree of a node with a fixed label. We compute the expectation and the variance of degree variable via a Pólya urn approach. In addition, we study a topological index, Zagreb index, of this class of trees. We calculate the exact first two moments of the Zagreb index (of PORTs) by using recurrence methods. Lastly, we determine the limiting degree distribution in PORTs that grow in continuous time, where the embedding is done in a Poissonization framework. We show that it is exponential after proper scaling.

NONSTATIONARY NONLINEARITY: A SURVEY ON PETER PHILLIPS’S CONTRIBUTIONS WITH A NEW PERSPECTIVE

2014 ◽  
Vol 30 (4) ◽  
pp. 894-922 ◽  
Author(s):  
Joon Y. Park
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Local Times ◽  
Nonstationary Processes ◽  
Functional Limit ◽  
Limit Theory ◽  
Nonparametric Models ◽  
New Perspective

In this paper, we provide a survey of Peter Phillips’s works on the econometrics for models with nonstationary nonlinearity, and some of the extensions that were made possible due to his original contributions. Parametric and nonparametric models are considered in both discrete time and continuous time setups. Although some of the asymptotics in the paper are applicable more generally for a wide variety of nonstationary models, we mainly analyze models with nonstationary processes that allow for the functional limit theory with limit processes having well defined local times.

scholarly journals Supercritical multitype branching processes: the ancestral types of typical individuals

2003 ◽  
Vol 35 (4) ◽  
pp. 1090-1110 ◽  
Author(s):  
Hans-Otto Georgii ◽  
Ellen Baake
Keyword(s):  
Continuous Time ◽  
Branching Processes ◽  
Almost Sure Convergence ◽  
Family Tree ◽  
Mutation Process ◽  
Convergence Theorems ◽  
Conceptual Proof ◽  
The Family ◽  
Multitype Branching Processes

For supercritical multitype Markov branching processes in continuous time, we investigate the evolution of types along those lineages that survive up to some time t. We establish almost-sure convergence theorems for both time and population averages of ancestral types (conditioned on nonextinction), and identify the mutation process describing the type evolution along typical lineages. An important tool is a representation of the family tree in terms of a suitable size-biased tree with trunk. As a by-product, this representation allows a ‘conceptual proof’ (in the sense of Kurtz et al.) of the continuous-time version of the Kesten-Stigum theorem.

scholarly journals Note on the weighted internal path length of b-ary trees

2007 ◽  
Vol Vol. 9 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Ludger Rüschendorf ◽  
Eva-Maria Schopp
Keyword(s):  
Limit Theorem ◽  
Continuous Time ◽  
Path Length ◽  
Analysis Of Algorithms ◽  
Random Variables ◽  
Time Parameter ◽  
Recursive Sequences ◽  
International Audience

Analysis of Algorithms International audience In a recent paper Broutin and Devroye (2005) have studied the height of a class of edge-weighted random trees.This is a class of trees growing in continuous time which includes many wellknown trees as examples. In this paper we derive a limit theorem for the internal path length for this class of trees.For the proof we extend a limit theorem in Neininger and Rüschendorf (2004) to recursive sequences of random variables with continuous time parameter.

Martingale Convergence Theorems

2000 ◽  
pp. 225-238
Author(s):  
Jean Jacod ◽  
Philip Protter
Keyword(s):  
Convergence Theorems ◽  
Martingale Convergence

Fluctuation theory in continuous time

1975 ◽  
Vol 7 (04) ◽  
pp. 705-766 ◽  
Author(s):  
N. H. Bingham
Keyword(s):  
Weak Convergence ◽  
Local Time ◽  
Continuous Time ◽  
Regular Variation ◽  
Heavy Traffic ◽  
Fluctuation Theory ◽  
Convergence Theorems ◽  
Time Fluctuation ◽  
Exit Problems ◽  
Time Theory

Our aim here is to give a survey of that part of continuous-time fluctuation theory which can be approached in terms of functionals of Lévy processes, our principal tools being Wiener-Hopf factorisation and local-time theory. Particular emphasis is given to one- and two-sided exit problems for spectrally negative and spectrally positive processes, and their applications to queues and dams. In addition, we give some weak-convergence theorems of heavy-traffic type, and some tail-estimates involving regular variation.

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