Accessibility percolation on random rooted labeled trees

2019 ◽  
Vol 56 (2) ◽  
pp. 533-545
Author(s):  
Zhishui Hu ◽  
Zheng Li ◽  
Qunqiang Feng
Keyword(s):  
Weak Convergence ◽  
Explicit Form ◽  
Random Vector ◽  
Generating Function ◽  
Probability Generating Function ◽  
Percolation Model ◽  
Asymptotic Distributions ◽  
Labeled Trees ◽  
Local Weak Convergence ◽  
Labeled Tree

AbstractThe accessibility percolation model is investigated on random rooted labeled trees. More precisely, the number of accessible leaves (i.e. increasing paths) Zn and the number of accessible vertices Cn in a random rooted labeled tree of size n are jointly considered in this work. As n → ∞, we prove that (Zn, Cn) converges in distribution to a random vector whose probability generating function is given in an explicit form. In particular, we obtain that the asymptotic distributions of Zn + 1 and Cn are geometric distributions with parameters e/(1 + e) and 1/e, respectively. Much of our analysis is performed in the context of local weak convergence of random rooted labeled trees.

On the properties of processes associated with a markov branching process

1991 ◽  
Vol 28 (03) ◽  
pp. 520-528
Author(s):  
V. G. Gadag ◽  
R. P. Gupta
Keyword(s):  
Generating Function ◽  
Finite Time ◽  
Branching Process ◽  
Asymptotic Properties ◽  
Probability Generating Function ◽  
Asymptotic Distributions ◽  
Time Interval ◽  
Original Process ◽  
Markov Branching Process ◽  
Branching Property

Consider a time-homogeneous Markov branching process. We construct reduced processes, based on whether the length of line of descent of particles of this process are (a) greater than or (b) at most equal to, τ units of time, for some fixed τ ≧ 0. We show that in both cases the reduced processes retain the branching property, but the latter does not retain the time homogeneity. We investigate finite-time and asymptotic properties of the reduced processes. Based on a realization of the original process and a realization of a reduced process, observed continuously over a time interval [0, T] for T > 0, we propose estimators for the different parameters involved, including qτ , the probability that the original process becomes extinct before τ units of time, and f (j)(qτ ), the jth derivative of the offspring probability generating function f(s) at q τ when q τ is known. We study the properties of these estimators and derive their asymptotic distributions, under the assumption that the original process is supercritical.

On the properties of processes associated with a markov branching process

1991 ◽  
Vol 28 (3) ◽  
pp. 520-528
Author(s):  
V. G. Gadag ◽  
R. P. Gupta
Keyword(s):  
Generating Function ◽  
Finite Time ◽  
Branching Process ◽  
Asymptotic Properties ◽  
Probability Generating Function ◽  
Asymptotic Distributions ◽  
Time Interval ◽  
Original Process ◽  
Markov Branching Process ◽  
Branching Property

Consider a time-homogeneous Markov branching process. We construct reduced processes, based on whether the length of line of descent of particles of this process are (a) greater than or (b) at most equal to, τ units of time, for some fixed τ ≧ 0. We show that in both cases the reduced processes retain the branching property, but the latter does not retain the time homogeneity. We investigate finite-time and asymptotic properties of the reduced processes. Based on a realization of the original process and a realization of a reduced process, observed continuously over a time interval [0, T] for T > 0, we propose estimators for the different parameters involved, including qτ, the probability that the original process becomes extinct before τ units of time, and f(j)(qτ), the jth derivative of the offspring probability generating function f(s) at qτ when qτ is known. We study the properties of these estimators and derive their asymptotic distributions, under the assumption that the original process is supercritical.

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

scholarly journals Period-Life of a Branching Process with Migration and Continuous Time

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 868
Author(s):  
Khrystyna Prysyazhnyk ◽  
Iryna Bazylevych ◽  
Ludmila Mitkova ◽  
Iryna Ivanochko
Keyword(s):  
Random Process ◽  
Generating Function ◽  
Continuous Time ◽  
Branching Process ◽  
Probability Generating Function ◽  
Time Interval ◽  
The Moment ◽  
First Time ◽  
Critical Branching Process ◽  
Number Of Particles

The homogeneous branching process with migration and continuous time is considered. We investigated the distribution of the period-life τ, i.e., the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time. The probability generating function of the random process, which describes the behavior of the process within the period-life, was obtained. The boundary theorem for the period-life of the subcritical or critical branching process with migration was found.

Polynomial bounds for probability generating functions

1975 ◽  
Vol 12 (3) ◽  
pp. 507-514 ◽  
Author(s):  
Henry Braun
Keyword(s):  
Lower Bound ◽  
Generating Function ◽  
Generating Functions ◽  
Branching Process ◽  
Probability Generating Function ◽  
Extinction Time ◽  
Probability Generating Functions ◽  
Extinction Probabilities ◽  
Polynomial Bounds

The problem of approximating an arbitrary probability generating function (p.g.f.) by a polynomial is considered. It is shown that if the coefficients rj are chosen so that LN(·) agrees with g(·) to k derivatives at s = 1 and to (N – k) derivatives at s = 0, then LN is in fact an upper or lower bound to g; the nature of the bound depends only on k and not on N. Application of the results to the problems of finding bounds for extinction probabilities, extinction time distributions and moments of branching process distributions are examined.

Interconnected population processes

1973 ◽  
Vol 10 (01) ◽  
pp. 1-14 ◽  
Author(s):  
E. Renshaw
Keyword(s):  
Generating Function ◽  
Probability Generating Function ◽  
Approximate Solutions ◽  
Death Process ◽  
Birth And Death Process ◽  
Population Processes

This paper investigates the effect of migration between two colonies each of which undergoes a simple birth and death process. Expressions are obtained for the first two moments and approximate solutions are developed for the probability generating function of the colony sizes.

scholarly journals A New Family of Discrete Distributions with Mathematical Properties, Characterizations, Bayesian and Non-Bayesian Estimation Methods

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1648
Author(s):  
Mohamed Aboraya ◽  
Haitham M. Yousof ◽  
G.G. Hamedani ◽  
Mohamed Ibrahim
Keyword(s):  
Bayesian Estimation ◽  
Generating Function ◽  
Bayesian Methods ◽  
Probability Generating Function ◽  
Real Data ◽  
Discrete Distributions ◽  
Estimation Methods ◽  
Squared Error Loss Function ◽  
New Family ◽  
Mathematical Properties

In this work, we propose and study a new family of discrete distributions. Many useful mathematical properties, such as ordinary moments, moment generating function, cumulant generating function, probability generating function, central moment, and dispersion index are derived. Some special discrete versions are presented. A certain special case is discussed graphically and numerically. The hazard rate function of the new class can be “decreasing”, “upside down”, “increasing”, and “decreasing-constant-increasing (U-shape)”. Some useful characterization results based on the conditional expectation of certain function of the random variable and in terms of the hazard function are derived and presented. Bayesian and non-Bayesian methods of estimation are considered. The Bayesian estimation procedure under the squared error loss function is discussed. Markov chain Monte Carlo simulation studies for comparing non-Bayesian and Bayesian estimations are performed using the Gibbs sampler and Metropolis–Hastings algorithm. Four applications to real data sets are employed for comparing the Bayesian and non-Bayesian methods. The importance and flexibility of the new discrete class is illustrated by means of four real data applications.

A carrier-borne epidemic with multiple stages of infection

1991 ◽  
Vol 28 (01) ◽  
pp. 1-8 ◽  
Author(s):  
J. Gani ◽  
Gy. Michaletzky
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Markov Chain ◽  
Generating Function ◽  
Continuous Time ◽  
Probability Generating Function ◽  
Death Process ◽  
Partial Differential ◽  
Multiple Stages

This paper considers a carrier-borne epidemic in continuous time with m + 1 > 2 stages of infection. The carriers U(t) follow a pure death process, mixing homogeneously with susceptibles X 0(t), and infectives Xi (t) in stages 1≦i≦m of infection. The infectives progress through consecutive stages of infection after each contact with the carriers. It is shown that under certain conditions {X 0(t), X 1(t), · ··, Xm (t) U(t); t≧0} is an (m + 2)-variate Markov chain, and the partial differential equation for its probability generating function derived. This can be solved after a transfomation of variables, and the probability of survivors at the end of the epidemic found.

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