Some limit theorems for Markov branching processes

1973 ◽  
Vol 10 (2) ◽  
pp. 299-306 ◽  
Author(s):  
J. R. Leslie
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Original Process ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
Logarithm Law ◽  
Increasing Process ◽  
Watson Process ◽  
Iterated Logarithm Law

Analogues of the central limit theorem and iterated logarithm law have recently been obtained for the Galton-Watson process; similar results are established in this paper for the temporally homogeneous Markov branching process and for the associated increasing process consisting of the number of splits in the original process up to time t.

Some limit theorems for Markov branching processes

1973 ◽  
Vol 10 (02) ◽  
pp. 299-306 ◽  
Author(s):  
J. R. Leslie
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Original Process ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
Logarithm Law ◽  
Increasing Process ◽  
Watson Process ◽  
Iterated Logarithm Law

Analogues of the central limit theorem and iterated logarithm law have recently been obtained for the Galton-Watson process; similar results are established in this paper for the temporally homogeneous Markov branching process and for the associated increasing process consisting of the number of splits in the original process up to time t.

scholarly journals On the central limit theorem and iterated logarithm law for stationary processes

1975 ◽  
Vol 12 (1) ◽  
pp. 1-8 ◽  
Author(s):  
C.C. Heyde
Keyword(s):  
Central Limit ◽  
Stationary Process ◽  
Stationary Processes ◽  
Martingale Differences ◽  
Stationary Increments ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
Logarithm Law ◽  
Functional Forms ◽  
Iterated Logarithm Law

It has recently emerged that a convenient way to establish central limit and iterated logarithm results for processes with stationary increments is to use approximating martingales with stationary increments. Functional forms of the limit results can be obtained via a representation for the increments of the stationary process in terms of stationary martingale differences plus other terms whose sum telescopes and disappears under suitable norming. Results based on the most general form of such a representation are here obtained.

scholarly journals Improved classical limit analogues for Galton-Watson processes with or without immigration

1971 ◽  
Vol 5 (2) ◽  
pp. 145-155 ◽  
Author(s):  
C.C. Heyde ◽  
J.R. Leslie
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Classical Limit ◽  
Moment Conditions ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
Logarithm Law ◽  
Iterated Logarithm Law

It has recently emerged that the central limit theorem and iterated logarithm law for random walk processes have natural counterparts for Galton-Watson processes with or without immigration. Much of the work on these counterparts has previously involved the imposition of supplementary moment conditions. In this paper we show how to dispense with these supplementary conditions and in so doing make the analogy with the random walk results complete.

Markov processes associated with critical Galton-Watson processes with application to extinction probabilities

1976 ◽  
Vol 8 (2) ◽  
pp. 278-295 ◽  
Author(s):  
Michael Sze
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Continuous Process ◽  
Regularly Varying Functions ◽  
Original Process ◽  
Additional Information ◽  
Embedding Technique ◽  
Extinction Probabilities ◽  
Watson Process ◽  
The Given

As an alternative to the embedding technique of T. E. Harris, S. Karlin and J. McGregor, we show that given a critical Galton–Watson process satisfying some mild assumptions, we can always construct a continuous-time Markov branching process having the same asymptotic behaviour as the given process. Thus, via the associated continuous process, additional information about the original process is obtained. We apply this technique to the study of extinction probabilities of a critical Galton–Watson process, and provide estimates for the extinction probabilities by regularly varying functions.

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.

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

A limit theorem for a class of supercritical branching processes

1972 ◽  
Vol 9 (04) ◽  
pp. 707-724 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
General Class ◽  
Branching Process ◽  
Branching Processes ◽  
Generating Functional ◽  
Number Of Children ◽  
Supercritical Case ◽  
Age Dependent ◽  
Watson Process ◽  
Supercritical Branching Processes ◽  
Convergence In Mean

In the Bellman-Harris (B-H) age-dependent branching process, the birth of a child can occur only at the time of its parent's death. A general class of branching process in which births can occur throughout the lifetime of a parent has been introduced by Crump and Mode. This class shares with the B-H process the property that the generation sizes {ξn } form a Galton-Watson process, and so may be classified into subcritical, critical or supercritical according to the value of m = E{ξ 1}. Crump and Mode showed that, as regards extinction probability, asymptotic behaviour, and for the supercritical case, convergence in mean square of Z(t)/E[Z(t)], as t → ∞, where Z(t) is the population size at time t given one ancestor at t = 0, properties of the B-H process can be extended to this general class. In this paper conditions are found for the convergence in distribution of Z(t)/E{Z(t)} in the supercritical case to a non-degenerate limit distribution. In contrast to the B-H process, these conditions are not the same as those for ξn /mn to have a non-degenerate limit. An integral equation is established for the generating function of Z(t), which is more complicated than the corresponding one for the B-H process and involves the conditional probability generating functional of N(x), x 0, ≧ the number of children born to an individual in the age interval [0, x].

A joint functional law for the Wiener process and principal value

2003 ◽  
Vol 40 (1-2) ◽  
pp. 213-241
Author(s):  
E. Csáki ◽  
A. Földes ◽  
Z. Shi
Keyword(s):  
Wiener Process ◽  
Local Times ◽  
Iterated Logarithm ◽  
Logarithm Law ◽  
Principal Value ◽  
Iterated Logarithm Law

We present a joint functional iterated logarithm law for the Wiener process and the principal value of its local times.

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 → ∞.

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