Remarks on the maxima of a martingale sequence with application to the simple critical branching process

1987 ◽  
Vol 24 (3) ◽  
pp. 768-772 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Branching Process ◽  
Alternative Proof ◽  
Watson Process ◽  
Image Position ◽  
Critical Branching Process

Let where {Zn, ℱn} is a non-negative submartingale satisfying Ei(Zn log Zn) →∞. It is shown that When {Zn} is a simple critical Galton–Watson process and is slowly varying at∞, conditions are given ensuring that This gives an alternative proof of a result recently established by Kammerle and Schuh.

Remarks on the maxima of a martingale sequence with application to the simple critical branching process

1987 ◽  
Vol 24 (03) ◽  
pp. 768-772 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Branching Process ◽  
Alternative Proof ◽  
Watson Process ◽  
Image Position ◽  
Critical Branching Process

Let where {Zn, ℱn } is a non-negative submartingale satisfying Ei (Zn log Zn ) →∞. It is shown that When {Zn } is a simple critical Galton–Watson process and is slowly varying at∞, conditions are given ensuring that This gives an alternative proof of a result recently established by Kammerle and Schuh.

scholarly journals The Critical Galton-Watson Process Without Further Power Moments

2007 ◽  
Vol 44 (03) ◽  
pp. 753-769 ◽  
Author(s):  
S. V. Nagaev ◽  
V. Wachtel
Keyword(s):  
Asymptotic Behavior ◽  
Limit Theorem ◽  
Generating Function ◽  
Branching Process ◽  
Alternative Proof ◽  
Conditional Limit Theorem ◽  
Power Moments ◽  
Watson Process ◽  
Conditional Limit

In this paper we prove a conditional limit theorem for a critical Galton-Watson branching process {Z n ; n ≥ 0} with offspring generating function s + (1 − s)L((1 − s)−1), where L(x) is slowly varying. In contrast to a well-known theorem of Slack (1968), (1972) we use a functional normalization, which gives an exponential limit. We also give an alternative proof of Sze's (1976) result on the asymptotic behavior of the nonextinction probability.

Moments of the maximum in a critical branching process

1984 ◽  
Vol 21 (04) ◽  
pp. 920-923 ◽  
Author(s):  
Howard Weiner
Keyword(s):  
Discrete Time ◽  
Branching Process ◽  
Image Position ◽  
Number Of Cells ◽  
Critical Branching Process

Let Zn denote the number of cells at time n in a critical discrete-time Galton–Watson branching process with finite offspring variance. Let Martingale arguments are used to show that for some 0<a≦b<∞

Moments of the maximum in a critical branching process

1984 ◽  
Vol 21 (4) ◽  
pp. 920-923 ◽  
Author(s):  
Howard Weiner
Keyword(s):  
Discrete Time ◽  
Branching Process ◽  
Image Position ◽  
Number Of Cells ◽  
Critical Branching Process

Let Zn denote the number of cells at time n in a critical discrete-time Galton–Watson branching process with finite offspring variance. Let Martingale arguments are used to show that for some 0<a≦b<∞

Domains of attraction for the subcritical Galton-Watson branching process

1968 ◽  
Vol 5 (01) ◽  
pp. 216-219 ◽  
Author(s):  
H. Rubin ◽  
D. Vere-Jones
Keyword(s):  
Generating Function ◽  
Branching Process ◽  
Domains Of Attraction ◽  
Offspring Distribution ◽  
Watson Process ◽  
Image Position

Let F(z) = σ fjzj be the generating function for the offspring distribution {fj } from a single ancestor in the usual Galton-Watson process. It is well-known (see Harris [1]) that if Π(z) is the generating function of the distribution of ancestors in the 0th generation, the distribution of offspring at the nth generation has generating function where F n (z), the nth functional iterate of F(z), gives the distribution of offspring at the nth generation from a single ancestor.

A branching-process solution of the polydisperse coagulation equation

1984 ◽  
Vol 16 (01) ◽  
pp. 56-69 ◽  
Author(s):  
John L. Spouge
Keyword(s):  
Discrete Time ◽  
Branching Process ◽  
Branching Processes ◽  
Critical Time ◽  
Coagulation Equation ◽  
Multitype Branching Processes ◽  
Irreversible Aggregation ◽  
Image Position ◽  
Supercritical Branching Processes ◽  
Critical Branching Process

The polydisperse coagulation equation models irreversible aggregation of particles with varying masses. This paper uses a one-parameter family of discrete-time continuous multitype branching processes to solve the polydisperse coagulation equation when The critical time tc when diverges corresponds to a critical branching process, while post-critical times t&gt; tc correspond to supercritical branching processes.

Domains of attraction for the subcritical Galton-Watson branching process

1968 ◽  
Vol 5 (1) ◽  
pp. 216-219 ◽  
Author(s):  
H. Rubin ◽  
D. Vere-Jones
Keyword(s):  
Generating Function ◽  
Branching Process ◽  
Domains Of Attraction ◽  
Offspring Distribution ◽  
Watson Process ◽  
Image Position

Let F(z) = σ fjzj be the generating function for the offspring distribution {fj} from a single ancestor in the usual Galton-Watson process. It is well-known (see Harris [1]) that if Π(z) is the generating function of the distribution of ancestors in the 0th generation, the distribution of offspring at the nth generation has generating function where Fn(z), the nth functional iterate of F(z), gives the distribution of offspring at the nth generation from a single ancestor.

The critical branching process with immigration stopped at zero

1985 ◽  
Vol 22 (01) ◽  
pp. 223-227 ◽  
Author(s):  
B. Gail Ivanoff ◽  
E. Seneta
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Critical Case ◽  
Initial Distribution ◽  
New Form ◽  
Watson Process ◽  
Branching Process With Immigration ◽  
Limit Result ◽  
Critical Branching Process

Limit theorems for the Galton–Watson process with immigration (BPI), where immigration is not permitted when the process is in state 0 (so that this state is absorbing), have been studied for the subcritical and supercritical cases by Seneta and Tavaré (1983). It is pointed out here that, apart from a change of context, the corresponding theorem in the critical case has been obtained by Vatutin (1977). Extensions which follow from a more general form of initial distribution are sketched, including a new form of limit result (7).

The progeny of a branching process

1971 ◽  
Vol 8 (2) ◽  
pp. 407-412 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Generating Function ◽  
Time Distribution ◽  
Branching Process ◽  
Life Time ◽  
Simple Relation ◽  
Age Dependent ◽  
Watson Process ◽  
Image Position

1. Let {Z(t), t ≧ 0} be an age-dependent branching process with offspring generating function and life-time distribution function G(t). Denote by N(t) the progeny of the process, that is the total number of objects which have been born in [0, t], counting the ancestor. (See Section 2 for definitions.) Then in the Galton-Watson process (i.e., when G(t) = 0 for t ≦ 1, G(t) = 1 for t > 1) we have the simple relation Nn = Z0 + Z1 + ··· + Zn, so that the asymptotic behaviour of Nn as n → ∞ follows from a knowledge of the asymptotic behaviour of Zn. In particular, if 1 < m = h'(1) < ∞ and Zn(ω)/E(Zn) → Z(ω) > 0 then also Nn(ω)/E(Nn) → Z(ω) > 0; since E(Zn)/E(Nn) → 1 – m–1 this means that

scholarly journals On the Maximal Offspring in a Critical Branching Process with Infinite Variance

2011 ◽  
Vol 48 (02) ◽  
pp. 576-582 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Branching Process ◽  
Shape Parameter ◽  
Scale Parameter ◽  
Infinite Variance ◽  
Regularly Varying ◽  
Watson Process ◽  
Critical Branching Process

We investigate the maximal number M k of offspring amongst all individuals in a critical Galton-Watson process started with k ancestors. We show that when the reproduction law has a regularly varying tail with index -α for 1 &lt; α &lt; 2, then k -1 M k converges in distribution to a Frechet law with shape parameter 1 and scale parameter depending only on α.

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