Addendum: Limit Theorems for a Critical Galton–Watson Branching Process with Migration

1982 ◽  
Vol 26 (2) ◽  
pp. 434-434
Author(s):  
S. V. Nagaev ◽  
L. V. Khan
Keyword(s):  
Limit Theorems ◽  
Branching Process

Some properties of a branching process with group immigration and emigration

1986 ◽  
Vol 18 (3) ◽  
pp. 628-645 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Population Size ◽  
Renewal Process ◽  
Limit Theorems ◽  
Branching Process ◽  
Limiting Distribution ◽  
Immigration And Emigration ◽  
Event Times

Batches of immigrants arrive in a region at event times of a renewal process and individuals grow according to a Bellman-Harris branching process. Tribal emigration allows the possibility that all descendants of a group of immigrants collectively leave the region at some instant.A number of results are derived giving conditions for the existence of a limiting distribution for the population size. These conditions can be given either in terms of the immigration distribution or in terms of the distribution of emigration times. Some limit theorems are obtained when the latter conditions are not fulfilled.

Limit theorems for processes such as the Markov branching process

1975 ◽  
Vol 12 (02) ◽  
pp. 289-297
Author(s):  
Andrew D. Barbour
Keyword(s):  
Markov Process ◽  
Residence Time ◽  
Continuous Time ◽  
Limit Theorems ◽  
Branching Process ◽  
Continuous Mapping ◽  
Random Variables ◽  
Time Parameter ◽  
Image Position ◽  
Independent Identically Distributed

LetX(t) be a continuous time Markov process on the integers such that, ifσis a time at whichXmakes a jump,X(σ)– X(σ–) is distributed independently ofX(σ–), and has finite meanμand variance. Letq(j) denote the residence time parameter for the statej.Iftndenotes the time of thenth jump andXn≡X(tb), it is easy to deduce limit theorems forfrom those for sums of independent identically distributed random variables. In this paper, it is shown how, forμ> 0 and for suitableq(·), these theorems can be translated into limit theorems forX(t), by using the continuous mapping theorem.

Some limit theorems for the total progeny of a branching process

1971 ◽  
Vol 3 (1) ◽  
pp. 176-192 ◽  
Author(s):  
A. G. Pakes
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Total Progeny

We consider a branching process in which each individual reproduces independently of all others and has probability aj(j = 0, 1, · · ·) of giving rise to j progeny in the following generation. It is assumed, without further comment, that 0 < a0, a0 + a1 < 1.

Functional limit theorems for the decomposable branching process with two types of particles

2016 ◽  
Vol 26 (2) ◽  
Author(s):  
Valeriy I Afanasiev
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Random Numbers ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Watson Process

AbstractA decomposable Galton - Watson process with two types of particles is considered. Particles of the first type produce equal random numbers of particles of both types, particles of the second type produce particles of the second type only. Under the condition that the total number of the first type particles is equal to

scholarly journals Revisiting Conditional Limit Theorems for the Mortal Simple Branching Process

Bernoulli ◽  
1999 ◽  
Vol 5 (6) ◽  
pp. 969 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Conditional Limit Theorems ◽  
Conditional Limit

scholarly journals Alternating branching processes

2005 ◽  
Vol 42 (04) ◽  
pp. 1095-1108 ◽  
Author(s):  
Penka Mayster
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Extinction Probability ◽  
Thinning Operator

We introduce the idea of controlling branching processes by means of another branching process, using the fractional thinning operator of Steutel and van Harn. This idea is then adapted to the model of alternating branching, where two Markov branching processes act alternately at random observation and treatment times. We study the extinction probability and limit theorems for reproduction by n cycles, as n → ∞.

Functional limit theorems for high-level subcritical branching processes in random environment

2014 ◽  
Vol 24 (5) ◽  
Author(s):  
Valeriy I. Afanasyev
Keyword(s):  
Random Environment ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Moment Generating Function ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
The Moment ◽  
High Level ◽  
Passage Times

AbstractThe paper is concerned with subcritical branching process in random environment. It is assumed that the moment-generating function of steps of the associated random walk is equal to 1 for some positive value of the argument. Functional limit theorems for sizes of various generations and passage times to various levels are put forward.

Some limit theorems for a supercritical branching process allowing immigration

1976 ◽  
Vol 13 (1) ◽  
pp. 17-26 ◽  
Author(s):  
A. G. Pakes
Keyword(s):  
Population Growth ◽  
Limit Theorems ◽  
Branching Process ◽  
Random Variable ◽  
Positive Random Variable ◽  
Linear Limit ◽  
Non Linear ◽  
The Mean

We consider the Bienaymé–Galton–Watson model of population growth in which immigration is allowed. When the mean number of offspring per individual, α, satisfies 1 < α < ∞, a well-known result proves that a normalised version of the size of the n th generation converges to a finite, positive random variable iff a certain condition is satisfied by the immigration distribution. In this paper we obtain some non-linear limit theorems when this condition is not satisfied. Results are also given for the explosive case, α = ∞.

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