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.

scholarly journals Branching processes in a random environment with immigration stopped at zero

2020 ◽  
Vol 57 (1) ◽  
pp. 237-249 ◽  
Author(s):  
Elena Dyakonova ◽  
Doudou Li ◽  
Vladimir Vatutin ◽  
Mei Zhang
Keyword(s):  
Random Environment ◽  
Branching Process ◽  
Branching Processes ◽  
Time Interval ◽  
Tail Distribution ◽  
The Moment ◽  
Branching Process With Immigration ◽  
First Time ◽  
Critical Branching Process ◽  
Number Of Particles

AbstractA critical branching process with immigration which evolves in a random environment is considered. Assuming that immigration is not allowed when there are no individuals in the population, we investigate the tail distribution of the so-called life period of the process, 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.

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.

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.

A branching model by population size dependence

1975 ◽  
Vol 7 (03) ◽  
pp. 495-510
Author(s):  
Carla Lipow
Keyword(s):  
Limit Theorem ◽  
Population Size ◽  
Generating Function ◽  
Continuous Time ◽  
Branching Process ◽  
Size Dependence ◽  
Stationary Measure ◽  
Markov Branching Process ◽  
Branching Model

A continuous-time Markov branching process is modified to allow some dependence of offspring generating function on population size. The model involves a given population size M, below which the offspring generating function is supercritical and above which it is subcritical. Immigration is allowed when the population size is 0. The process has a stationary measure, and an expression for its generating function is found. A limit theorem for the stationary measure as M tends to ∞ is then obtained.

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.

A note on a functional equation arising in Galton-Watson branching processes

1971 ◽  
Vol 8 (3) ◽  
pp. 589-598 ◽  
Author(s):  
Krishna B. Athreya
Keyword(s):  
Functional Equation ◽  
Continuous Function ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Age Dependent ◽  
Watson Process ◽  
Natural Way

The functional equation ϕ(mu) = h(ϕ(u)) where is a probability generating function with 1 < m = h'(1 –) < ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) < 1 and F(+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.

A note on a functional equation arising in Galton-Watson branching processes

1971 ◽  
Vol 8 (03) ◽  
pp. 589-598 ◽  
Author(s):  
Krishna B. Athreya
Keyword(s):  
Functional Equation ◽  
Continuous Function ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Age Dependent ◽  
Watson Process ◽  
Natural Way

The functional equation ϕ(mu) = h(ϕ(u)) where is a probability generating function with 1 &lt; m = h'(1 –) &lt; ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) &lt; 1 and F(+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.

Amplitudes of Composition Fluctuations in the Simplest Chemical Reaction Equilibrium

2004 ◽  
Vol 218 (9) ◽  
pp. 1033-1040 ◽  
Author(s):  
M. Šolc ◽  
J. Hostomský
Keyword(s):  
Random Walk ◽  
Discrete Time ◽  
Continuous Time ◽  
Time Scale ◽  
Numerical Study ◽  
Mean Amplitude ◽  
Time Interval ◽  
The Mean ◽  
Composition Fluctuations ◽  
Number Of Particles

AbstractWe present a numerical study of equilibrium composition fluctuations in a system where the reaction X1 ⇔ X2 having the equilibrium constant equal to 1 takes place. The total number of reacting particles is N. On a discrete time scale, the amplitude of a fluctuation having the lifetime 2r reaction events is defined as the difference between the number of particles X1 in the microstate most distant from the microstate N/2 visited at least once during the fluctuation lifetime, and the equilibrium number of particles X1, N/2. On the discrete time scale, the mean value of this amplitude, m̅(r̅), is calculated in the random walk approximation. On a continuous time scale, the average amplitude of fluctuations chosen randomly and regardless of their lifetime from an ensemble of fluctuations occurring within the time interval (0,z), z → ∞, tends with increasing N to ~1.243 N0.25. Introducing a fraction of fluctuation lifetime during which the composition of the system spends below the mean amplitude m̅(r̅), we obtain a value of the mean amplitude of equilibrium fluctuations on the continuous time scale equal to ~1.19√N. The results suggest that using the random walk value m̅(r̅) and taking into account a) the exponential density of fluctuations lifetimes and b) the fact that the time sequence of reaction events represents the Poisson process, we obtain values of fluctuations amplitudes which differ only slightly from those derived for the Ehrenfest model.

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