On the asymptotic behaviour of subcritical branching processes with continuous state space

1968 ◽  
Vol 10 (3) ◽  
pp. 212-225 ◽  
Author(s):  
E. Seneta ◽  
D. Vere-Jones
Keyword(s):  
Asymptotic Behaviour ◽  
State Space ◽  
Branching Processes ◽  
Continuous State Space ◽  
Continuous State

Asymptotic behaviour of continuous time, continuous state-space branching processes

1974 ◽  
Vol 11 (04) ◽  
pp. 669-677 ◽  
Author(s):  
D. R. Grey
Keyword(s):  
Asymptotic Behaviour ◽  
State Space ◽  
Discrete Time ◽  
Continuous Time ◽  
Branching Processes ◽  
Discrete State ◽  
Space And Time ◽  
Continuous State Space ◽  
Continuous State ◽  
Discrete State Space

Results on the behaviour of Markov branching processes as time goes to infinity, hitherto obtained for models which assume a discrete state-space or discrete time or both, are here generalised to a model with both state-space and time continuous. The results are similar but the methods not always so.

Asymptotic behaviour of continuous time, continuous state-space branching processes

1974 ◽  
Vol 11 (4) ◽  
pp. 669-677 ◽  
Author(s):  
D. R. Grey
Keyword(s):  
Asymptotic Behaviour ◽  
State Space ◽  
Discrete Time ◽  
Continuous Time ◽  
Branching Processes ◽  
Discrete State ◽  
Space And Time ◽  
Continuous State Space ◽  
Continuous State ◽  
Discrete State Space

Results on the behaviour of Markov branching processes as time goes to infinity, hitherto obtained for models which assume a discrete state-space or discrete time or both, are here generalised to a model with both state-space and time continuous. The results are similar but the methods not always so.

Asymptotic properties of super-critical branching processes II: Crump-Mode and Jirina processes

1975 ◽  
Vol 7 (01) ◽  
pp. 66-82 ◽  
Author(s):  
N. H. Bingham ◽  
R. A. Doney
Keyword(s):  
State Space ◽  
Continuous Time ◽  
Regular Variation ◽  
Branching Processes ◽  
Asymptotic Properties ◽  
Random Variables ◽  
The Other ◽  
Continuous State Space ◽  
Continuous State ◽  
Integer Exponent

We obtain results connecting the distribution of the random variablesYandWin the supercritical generalized branching processes introduced by Crump and Mode. For example, if β > 1,EYβandEWβconverge or diverge together and regular variation of the tail of one ofY, Wwith non-integer exponent β > 1 is equivalent to regular variation of the other. We also prove analogous results for the continuous-time continuous state-space branching processes introduced by Jirina.

Asymptotic properties of super-critical branching processes II: Crump-Mode and Jirina processes

1975 ◽  
Vol 7 (1) ◽  
pp. 66-82 ◽  
Author(s):  
N. H. Bingham ◽  
R. A. Doney
Keyword(s):  
State Space ◽  
Continuous Time ◽  
Regular Variation ◽  
Branching Processes ◽  
Asymptotic Properties ◽  
Random Variables ◽  
The Other ◽  
Continuous State Space ◽  
Continuous State ◽  
Integer Exponent

We obtain results connecting the distribution of the random variables Y and W in the supercritical generalized branching processes introduced by Crump and Mode. For example, if β > 1, EYβ and EWβ converge or diverge together and regular variation of the tail of one of Y, W with non-integer exponent β > 1 is equivalent to regular variation of the other. We also prove analogous results for the continuous-time continuous state-space branching processes introduced by Jirina.

A note on some results of Schuh

1984 ◽  
Vol 21 (01) ◽  
pp. 192-196 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
State Space ◽  
Large Class ◽  
Branching Processes ◽  
Random Variables ◽  
Continuous State Space ◽  
Continuous State ◽  
Class Of Functions

We extend recent results of Schuh on the convergence of , where α ≧ 0 and Sn is the sum of n i.i.d. positive random variables to sums of the form for a large class of functions g, give simpler proofs than those of Schuh, and derive reformulations of the explosion criteria for Markov branching processes with discrete and continuous state space.

A note on some results of Schuh

1984 ◽  
Vol 21 (1) ◽  
pp. 192-196 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
State Space ◽  
Large Class ◽  
Branching Processes ◽  
Random Variables ◽  
Continuous State Space ◽  
Continuous State ◽  
Class Of Functions

We extend recent results of Schuh on the convergence of , where α ≧ 0 and Sn is the sum of n i.i.d. positive random variables to sums of the form for a large class of functions g, give simpler proofs than those of Schuh, and derive reformulations of the explosion criteria for Markov branching processes with discrete and continuous state space.

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