scholarly journals Large deviations for i.i.d. replications of the total progeny of a Galton–Watson process

2017 ◽  
Vol 4 (1) ◽  
pp. 1-13
Author(s):  
Claudio Macci ◽  
Barbara Pacchiarotti
Keyword(s):  
Large Deviations ◽  
Total Progeny ◽  
Watson Process

The Galton-Watson process conditioned on the total progeny

1975 ◽  
Vol 12 (04) ◽  
pp. 800-806 ◽  
Author(s):  
Douglas P. Kennedy
Keyword(s):  
Marked Contrast ◽  
Total Progeny ◽  
Limiting Behaviour ◽  
Watson Process

Let Zk denote the number in the kth generation of a Galton-Watson process initiated by one individual and let N be the total progeny, i.e., As n → ∞ the limiting behaviour of the process {Zk, 0 ≦ k ≦ n} conditioned on the event {N =n} is studied. The results obtained are of exactly the same form for the subcritical, critical and supercritical cases. This is in marked contrast to the analogous situation got by conditioning on non-extinction by the nth generation and letting n → ∞. In the latter case the limiting results differ in form for the critical and non-critical cases.

The Galton-Watson process conditioned on the total progeny

1975 ◽  
Vol 12 (4) ◽  
pp. 800-806 ◽  
Author(s):  
Douglas P. Kennedy
Keyword(s):  
Marked Contrast ◽  
Total Progeny ◽  
Limiting Behaviour ◽  
Watson Process

Let Zk denote the number in the kth generation of a Galton-Watson process initiated by one individual and let N be the total progeny, i.e., As n → ∞ the limiting behaviour of the process {Zk, 0 ≦ k ≦ n} conditioned on the event {N =n} is studied. The results obtained are of exactly the same form for the subcritical, critical and supercritical cases. This is in marked contrast to the analogous situation got by conditioning on non-extinction by the nth generation and letting n → ∞. In the latter case the limiting results differ in form for the critical and non-critical cases.

scholarly journals How did we get here?

2014 ◽  
Vol 51 (A) ◽  
pp. 63-72
Author(s):  
Kais Hamza ◽  
Fima C. Klebaner
Keyword(s):  
Large Deviations ◽  
Algebraic Equation ◽  
Existence Of Solutions ◽  
Rate Function ◽  
Initial Number ◽  
Watson Process ◽  
Density Process ◽  
Population Numbers

Looking at a large branching population we determine along which path the population that started at 1 at time 0 ended up in B at time N. The result describes the density process, that is, population numbers divided by the initial number K (where K is assumed to be large). The model considered is that of a Galton-Watson process. It is found that in some cases population paths exhibit the strange feature that population numbers go down and then increase. This phenomenon requires further investigation. The technique uses large deviations, and the rate function based on Cramer's theorem is given. It also involves analysis of existence of solutions of a certain algebraic equation.

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