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.

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

1985 ◽  
Vol 22 (3) ◽  
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.

scholarly journals Skeletal stochastic differential equations for superprocesses

2020 ◽  
Vol 57 (4) ◽  
pp. 1111-1134
Author(s):  
Dorottya Fekete ◽  
Joaquin Fontbona ◽  
Andreas E. Kyprianou
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Subcritical Case ◽  
Markov Branching Process ◽  
Continuous State ◽  
Common Framework ◽  
Infinite Line ◽  
Current Article ◽  
Lines Of Descent ◽  
Skeletal Representation

AbstractIt is well understood that a supercritical superprocess is equal in law to a discrete Markov branching process whose genealogy is dressed in a Poissonian way with immigration which initiates subcritical superprocesses. The Markov branching process corresponds to the genealogical description of prolific individuals, that is, individuals who produce eternal genealogical lines of descent, and is often referred to as the skeleton or backbone of the original superprocess. The Poissonian dressing along the skeleton may be considered to be the remaining non-prolific genealogical mass in the superprocess. Such skeletal decompositions are equally well understood for continuous-state branching processes (CSBP).In a previous article [16] we developed an SDE approach to study the skeletal representation of CSBPs, which provided a common framework for the skeletal decompositions of supercritical and (sub)critical CSBPs. It also helped us to understand how the skeleton thins down onto one infinite line of descent when conditioning on survival until larger and larger times, and eventually forever.Here our main motivation is to show the robustness of the SDE approach by expanding it to the spatial setting of superprocesses. The current article only considers supercritical superprocesses, leaving the subcritical case open.

scholarly journals Information Length as a Useful Index to Understand Variability in the Global Circulation

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 299 ◽  
Author(s):  
Eun-jin Kim ◽  
James Heseltine ◽  
Hanli Liu
Keyword(s):  
Climate Model ◽  
Traditional Approach ◽  
Essential Feature ◽  
Complex Form ◽  
Time Dependent ◽  
Spatial Gradient ◽  
Mean Values ◽  
Global Circulation ◽  
New Perspective ◽  
Non Gaussian

With improved measurement and modelling technology, variability has emerged as an essential feature in non-equilibrium processes. While traditionally, mean values and variance have been heavily used, they are not appropriate in describing extreme events where a significant deviation from mean values often occurs. Furthermore, stationary Probability Density Functions (PDFs) miss crucial information about the dynamics associated with variability. It is thus critical to go beyond a traditional approach and deal with time-dependent PDFs. Here, we consider atmospheric data from the Whole Atmosphere Community Climate Model (WACCM) and calculate time-dependent PDFs and the information length from these PDFs, which is the total number of statistically different states that a system evolves through in time. Specifically, we consider the three cases of sampling data to investigate the distribution of information (information budget) along the altitude and longitude to gain a new perspective of understanding variabilities, correlation among different variables and regions. Time-dependent PDFs are shown to be non-Gaussian in general; the information length tends to increase with the altitude albeit in a complex form; this tendency is more robust for flows/shears than temperature. Much similarity among flows and shears in the information length is also found in comparison with the temperature. This means a strong correlation among flows/shears because of their coupling through gravity waves in this particular WACCM model. We also find the increase of the information length with the latitude and interesting hemispheric asymmetry for flows/shears/temperature, with the tendency of anti-correlation (correlation) between flows/shears and temperature at high (low) latitude. These results suggest the importance of high latitude/altitude in the information budget in the Earth’s atmosphere, the spatial gradient of the information length being a useful proxy for information flow.

scholarly journals Subcritical Sevastyanov branching processes with nonhomogeneous Poisson immigration

2017 ◽  
Vol 54 (2) ◽  
pp. 569-587 ◽  
Author(s):  
Ollivier Hyrien ◽  
Kosto V. Mitov ◽  
Nikolay M. Yanev
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Branching Processes ◽  
Asymptotic Properties ◽  
Cell Populations ◽  
Subcritical Case ◽  
Immigration Process ◽  
Large Numbers

Abstract We consider a class of Sevastyanov branching processes with nonhomogeneous Poisson immigration. These processes relax the assumption required by the Bellman–Harris process which imposes the lifespan and offspring of each individual to be independent. They find applications in studies of the dynamics of cell populations. In this paper we focus on the subcritical case and examine asymptotic properties of the process. We establish limit theorems, which generalize classical results due to Sevastyanov and others. Our key findings include a novel law of large numbers and a central limit theorem which emerge from the nonhomogeneity of the immigration process.

Branching processes with varying and random geometric offspring distributions

1975 ◽  
Vol 12 (01) ◽  
pp. 135-141 ◽  
Author(s):  
Niels Keiding ◽  
John E. Nielsen
Keyword(s):  
Asymptotic Behavior ◽  
Generating Functions ◽  
Conditional Expectation ◽  
Elementary Proof ◽  
Branching Processes ◽  
Random Environments ◽  
Subcritical Case ◽  
Supercritical Case ◽  
Extinction Probabilities ◽  
Varying Environments

The class of fractional linear generating functions is used to illustrate various aspects of the theory of branching processes in varying and random environments. In particular, it is shown that Church's theorem on convergence of the varying environments process admits of an elementary proof in this particular case. For random environments, examples are given on the asymptotic behavior of extinction probabilities in the supercritical case and conditional expectation given non-extinction in the subcritical case.

Limit theorems for point processes generated in a general branching process

1981 ◽  
Vol 13 (04) ◽  
pp. 650-668 ◽  
Author(s):  
Martin Härnqvist
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Branching Processes ◽  
Special Problem ◽  
Convergence Theory ◽  
Regularity Conditions ◽  
Convergence Results ◽  
Mixed Poisson Process ◽  
Proper Scaling ◽  
Extra Point

With the general convergence theory for branching processes as basis a special problem is studied. An extra point process of events during life is assigned to each realised individual, and the behaviour of the superposition of such point processes in action is studied as the population grows. With the proper scaling and under some regularity conditions the superposition is shown to converge in distribution to a Poisson process. Another scaling gives rise to a mixed Poisson process as limit. Established weak convergence techniques for point processes are applied, together with some recent strong convergence results for branching processes.

Malthusian behaviour of the critical and subcritical age-dependent branching processes with arbitrary state space

1976 ◽  
Vol 13 (3) ◽  
pp. 455-465
Author(s):  
D. I. Saunders
Keyword(s):  
State Space ◽  
Branching Process ◽  
Branching Processes ◽  
Rates Of Convergence ◽  
Expected Number ◽  
Subcritical Case ◽  
Arbitrary State ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Number Of Individuals

For the age-dependent branching process with arbitrary state space let M(x, t, A) be the expected number of individuals alive at time t with states in A given an initial individual at x. Subject to various conditions it is shown that M(x, t, A)e–at converges to a non-trivial limit where α is the Malthusian parameter (α = 0 for the critical case, and is negative in the subcritical case). The method of proof also yields rates of convergence.

Stochastic modelling of age-structured population with time and size dependence of immigration rate

2018 ◽  
Vol 33 (5) ◽  
pp. 289-299 ◽  
Author(s):  
Boris J. Pichugin ◽  
Nikolai V. Pertsev ◽  
Valentin A. Topchii ◽  
Konstantin K. Loginov
Keyword(s):  
Population Size ◽  
Branching Processes ◽  
Stochastic Modelling ◽  
Random Variable ◽  
Structured Population Model ◽  
Immigration Rate ◽  
Structured Population ◽  
Mean And Variance ◽  
Age Structured ◽  
Age Structured Population

Abstract A stochastic age-structured population model with immigration of individuals is considered. We assume that the lifespan of each individual is a random variable with a distribution function which may differ fromthe exponential one. The immigration rate of individuals depends on the time and total population size. Upper estimates for the mean and variance of the population size are established based on the theory of branching processes with constant immigration rate. A Monte Carlo simulation algorithm of population dynamics is developed. The results of numerical experiments with the model are presented.

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.

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