Sur une procédure de branchement déterministe et ses dérivées aléatoires

1994 ◽  
Vol 31 (02) ◽  
pp. 333-347
Author(s):  
Thierry Huillet ◽  
Andrzej Kłopotowski
Keyword(s):  
Branching Processes ◽  
Asymptotic Results ◽  
Renewal Equations ◽  
Age Dependent ◽  
Probabilistic Version

This paper is concerned with the description of both a deterministic and stochastic branching procedure. The renewal equations for the deterministic branching population are first derived which allow for asymptotic results on the ‘number' and ‘generation' processes. A probabilistic version of these processes is then studied which presents some discrepancy with the standard Harris age-dependent branching processes.

Sur une procédure de branchement déterministe et ses dérivées aléatoires

1994 ◽  
Vol 31 (2) ◽  
pp. 333-347 ◽  
Author(s):  
Thierry Huillet ◽  
Andrzej Kłopotowski
Keyword(s):  
Branching Processes ◽  
Asymptotic Results ◽  
Renewal Equations ◽  
Age Dependent ◽  
Probabilistic Version

This paper is concerned with the description of both a deterministic and stochastic branching procedure. The renewal equations for the deterministic branching population are first derived which allow for asymptotic results on the ‘number' and ‘generation' processes. A probabilistic version of these processes is then studied which presents some discrepancy with the standard Harris age-dependent branching processes.

Strict supercritical generation-dependent Crump–Mode–Jagers branching processes

1978 ◽  
Vol 10 (04) ◽  
pp. 744-763 ◽  
Author(s):  
L. Edler
Keyword(s):  
Generating Functions ◽  
Positive Real Number ◽  
Branching Processes ◽  
Positive Real ◽  
Quadratic Mean ◽  
Malthusian Parameter ◽  
Renewal Equations ◽  
Branching Model ◽  
Age Dependent ◽  
Number Of Individuals

The general age-dependent branching model of Crump, Mode and Jagers will be generalized towards generation-dependent varying lifespan and reproduction distributions. A system of integral and renewal equations is established for the generating functions and the first two moments of Zi (t) (the number of individuals alive at time t), if the population was initiated at time 0 by one ancestor of age 0 from generation i. Convergence in quadratic mean of Zi (t)/EZi (t) as t tends to infinity is obtained if the generation-dependent reproduction functions converge to a supercritical one. In particular, if this convergence is slow enough t γ exp (αt) is the asymptotic behavior of EZi (t) for t tending to infinity, where γ is a positive real number and α the Malthusian parameter of growth of the limiting reproduction function.

Lundberg inequalities for renewal equations

2001 ◽  
Vol 33 (03) ◽  
pp. 674-689 ◽  
Author(s):  
Gordon E. Willmot ◽  
Jun Cai ◽  
X. Sheldon Lin
Keyword(s):  
Branching Processes ◽  
Geiger Counter ◽  
Upper And Lower Bounds ◽  
Exponential Inequalities ◽  
Renewal Equations ◽  
Additional Information ◽  
Special Cases ◽  
Age Dependent ◽  
Defective Renewal Equations ◽  
Generalized Type

Sharp upper and lower bounds are derived for the solution of renewal equations. These include as special cases exponential inequalities, some of which have been derived for specific renewal equations. Together with the well-known Cramér-Lundberg asymptotic estimate, these bounds give additional information about the behaviour of the solution. Nonexponential bounds, which are of use in connection with defective renewal equations, are also obtained. The results are then applied in examples involving the severity of insurance ruin, age-dependent branching processes, and a generalized type II Geiger counter.

Strict supercritical generation-dependent Crump–Mode–Jagers branching processes

1978 ◽  
Vol 10 (4) ◽  
pp. 744-763 ◽  
Author(s):  
L. Edler
Keyword(s):  
Generating Functions ◽  
Positive Real Number ◽  
Branching Processes ◽  
Positive Real ◽  
Quadratic Mean ◽  
Malthusian Parameter ◽  
Renewal Equations ◽  
Branching Model ◽  
Age Dependent ◽  
Number Of Individuals

The general age-dependent branching model of Crump, Mode and Jagers will be generalized towards generation-dependent varying lifespan and reproduction distributions. A system of integral and renewal equations is established for the generating functions and the first two moments of Zi(t) (the number of individuals alive at time t), if the population was initiated at time 0 by one ancestor of age 0 from generation i. Convergence in quadratic mean of Zi(t)/EZi(t) as t tends to infinity is obtained if the generation-dependent reproduction functions converge to a supercritical one. In particular, if this convergence is slow enough tγ exp (αt) is the asymptotic behavior of EZi(t) for t tending to infinity, where γ is a positive real number and α the Malthusian parameter of growth of the limiting reproduction function.

Lundberg inequalities for renewal equations

2001 ◽  
Vol 33 (3) ◽  
pp. 674-689 ◽  
Author(s):  
Gordon E. Willmot ◽  
Jun Cai ◽  
X. Sheldon Lin
Keyword(s):  
Branching Processes ◽  
Geiger Counter ◽  
Upper And Lower Bounds ◽  
Exponential Inequalities ◽  
Renewal Equations ◽  
Additional Information ◽  
Special Cases ◽  
Age Dependent ◽  
Defective Renewal Equations ◽  
Generalized Type

Sharp upper and lower bounds are derived for the solution of renewal equations. These include as special cases exponential inequalities, some of which have been derived for specific renewal equations. Together with the well-known Cramér-Lundberg asymptotic estimate, these bounds give additional information about the behaviour of the solution. Nonexponential bounds, which are of use in connection with defective renewal equations, are also obtained. The results are then applied in examples involving the severity of insurance ruin, age-dependent branching processes, and a generalized type II Geiger counter.

Multi-type age-dependent branching processes as models of metastasis evolution

2019 ◽  
Vol 35 (3) ◽  
pp. 284-299
Author(s):  
Maroussia Slavtchova-Bojkova ◽  
Kaloyan Vitanov
Keyword(s):  
Branching Processes ◽  
Age Dependent

Parametric inference in Markov branching processes with time-dependent random immigration rate

1985 ◽  
Vol 22 (03) ◽  
pp. 503-517
Author(s):  
Helmut Pruscha
Keyword(s):  
Point Processes ◽  
Traditional Approach ◽  
Branching Processes ◽  
External Source ◽  
Time Dependent ◽  
Parametric Inference ◽  
Asymptotic Results ◽  
Subcritical Case ◽  
Immigration Rate ◽  
Pearson Type

The present paper deals with continuous-time Markov branching processes allowing immigration. The immigration rate is allowed to be random and time-dependent where randomness may stem from an external source or from state-dependence. Unlike the traditional approach, we base the analysis of these processes on the theory of multivariate point processes. Using the tools of this theory, asymptotic results on parametric inference are derived for the subcritical case. In particular, the limit distributions of some parametric estimators and of Pearson-type statistics for testing simple and composite hypotheses are established.

Stochastic models in cell kinetics

1988 ◽  
Vol 25 (A) ◽  
pp. 91-111
Author(s):  
Peter J. Brockwell
Keyword(s):  
Life Cycle ◽  
Stochastic Models ◽  
Death Rate ◽  
Cell Kinetics ◽  
Branching Processes ◽  
Experimental Studies ◽  
Renewal Theory ◽  
Biological Cell ◽  
Age Dependent

We discuss the role of stochastic processes in modelling the life-cycle of a biological cell and the growth of cell populations. Results for multiphase age-dependent branching processes have proved invaluable for the interpretation of many of the basic experimental studies of the life-cycle. Moreover problems from cell kinetics, in particular those related to diurnal rhythm in cell-growth, have stimulated research into ‘periodic' renewal theory, and the asymptotic behaviour of populations of cells with periodic death rate.

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