Moment Formulas for Multitype Continuous State and Continuous Time Branching Process with Immigration

2015 ◽  
Vol 29 (3) ◽  
pp. 958-995 ◽  
Author(s):  
Mátyás Barczy ◽  
Zenghu Li ◽  
Gyula Pap
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Continuous State ◽  
Branching Process With Immigration

scholarly journals Asymptotic behavior of critical, irreducible multi-type continuous state and continuous time branching processes with immigration

2016 ◽  
Vol 16 (04) ◽  
pp. 1650008 ◽  
Author(s):  
Mátyás Barczy ◽  
Gyula Pap
Keyword(s):  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Bessel Process ◽  
Step Functions ◽  
Squared Bessel Process ◽  
Continuous State ◽  
Random Step ◽  
Branching Process With Immigration ◽  
Continuous Time Branching Processes

Under natural assumptions, a Feller type diffusion approximation is derived for critical, irreducible multi-type continuous state and continuous time branching processes with immigration. Namely, it is proved that a sequence of appropriately scaled random step functions formed from a critical, irreducible multi-type continuous state and continuous time branching process with immigration converges weakly towards a squared Bessel process supported by a ray determined by the Perron vector of a matrix related to the branching mechanism of the branching process in question.

scholarly journals Asymptotic behavior of projections of supercritical multi-type continuous-state and continuous-time branching processes with immigration

2021 ◽  
Vol 53 (4) ◽  
pp. 1023-1060
Author(s):  
Mátyás Barczy ◽  
Sandra Palau ◽  
Gyula Pap
Keyword(s):  
Asymptotic Behavior ◽  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Order Moment ◽  
Moment Condition ◽  
First Order ◽  
Continuous State ◽  
Branching Process With Immigration ◽  
Continuous Time Branching Processes

AbstractUnder a fourth-order moment condition on the branching and a second-order moment condition on the immigration mechanisms, we show that an appropriately scaled projection of a supercritical and irreducible continuous-state and continuous-time branching process with immigration on certain left non-Perron eigenvectors of the branching mean matrix is asymptotically mixed normal. With an appropriate random scaling, under some conditional probability measure, we prove asymptotic normality as well. In the case of a non-trivial process, under a first-order moment condition on the immigration mechanism, we also prove the convergence of the relative frequencies of distinct types of individuals on a suitable event; for instance, if the immigration mechanism does not vanish, then this convergence holds almost surely.

Distributions of jumps in a continuous-state branching process with immigration

2016 ◽  
Vol 53 (4) ◽  
pp. 1166-1177 ◽  
Author(s):  
Xin He ◽  
Zenghu Li
Keyword(s):  
Branching Process ◽  
Lévy Measure ◽  
Borel Set ◽  
Jump Size ◽  
Distributional Properties ◽  
Continuous State ◽  
Continuous State Branching Process ◽  
Jump Time ◽  
Branching Process With Immigration

Abstract We study the distributional properties of jumps in a continuous-state branching process with immigration. In particular, a representation is given for the distribution of the first jump time of the process with jump size in a given Borel set. From this result we derive a characterization for the distribution of the local maximal jump of the process. The equivalence of this distribution and the total Lévy measure is then studied. For the continuous-state branching process without immigration, we also study similar problems for its global maximal jump.

On the identification of continuous-time Markov chains with a given invariant measure

1994 ◽  
Vol 31 (04) ◽  
pp. 897-910
Author(s):  
P. K. Pollett
Keyword(s):  
Invariant Measure ◽  
Continuous Time ◽  
Branching Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Birth Process ◽  
Continuous Time Markov Chains ◽  
Branching Process With Immigration ◽  
Birth Processes ◽  
Necessary And Sufficient

In [14] a necessary and sufficient condition was obtained for there to exist uniquely a Q-process with a specified invariant measure, under the assumption that Q is a stable, conservative, single-exit matrix. The purpose of this note is to demonstrate that, for an arbitrary stable and conservative q-matrix, the same condition suffices for the existence of a suitable Q-process, but that this process might not be unique. A range of examples is considered, including pure-birth processes, a birth process with catastrophes, birth-death processes and the Markov branching process with immigration.

Skeletal stochastic differential equations for continuous-state branching processes

2019 ◽  
Vol 56 (4) ◽  
pp. 1122-1150 ◽  
Author(s):  
D. Fekete ◽  
J. Fontbona ◽  
A. E. Kyprianou
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Continuous State ◽  
Continuous State Branching Process ◽  
Common Framework ◽  
Watson Process ◽  
Skeletal Representation

AbstractIt is well understood that a supercritical continuous-state branching process (CSBP) is equal in law to a discrete continuous-time Galton–Watson process (the skeleton of prolific individuals) whose edges are dressed in a Poissonian way with immigration which initiates subcritical CSBPs (non-prolific mass). Equally well understood in the setting of CSBPs and superprocesses is the notion of a spine or immortal particle dressed in a Poissonian way with immigration which initiates copies of the original CSBP, which emerges when conditioning the process to survive eternally. In this article we revisit these notions for CSBPs and put them in a common framework using the well-established language of (coupled) stochastic differential equations (SDEs). In this way we are able to deal simultaneously with all types of CSBPs (supercritical, critical, and subcritical) as well as understanding how the skeletal representation becomes, in the sense of weak convergence, a spinal decomposition when conditioning on survival. We have two principal motivations. The first is to prepare the way to expand the SDE approach to the spatial setting of superprocesses, where recent results have increasingly sought the use of skeletal decompositions to transfer results from the branching particle setting to the setting of measure valued processes. The second is to provide a pathwise decomposition of CSBPs in the spirit of genealogical coding of CSBPs via Lévy excursions, albeit precisely where the aforesaid coding fails to work because the underlying CSBP is supercritical.

On the identification of continuous-time Markov chains with a given invariant measure

1994 ◽  
Vol 31 (4) ◽  
pp. 897-910 ◽  
Author(s):  
P. K. Pollett
Keyword(s):  
Invariant Measure ◽  
Continuous Time ◽  
Branching Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Birth Process ◽  
Continuous Time Markov Chains ◽  
Branching Process With Immigration ◽  
Birth Processes ◽  
Necessary And Sufficient

In [14] a necessary and sufficient condition was obtained for there to exist uniquely a Q-process with a specified invariant measure, under the assumption that Q is a stable, conservative, single-exit matrix. The purpose of this note is to demonstrate that, for an arbitrary stable and conservative q-matrix, the same condition suffices for the existence of a suitable Q-process, but that this process might not be unique. A range of examples is considered, including pure-birth processes, a birth process with catastrophes, birth-death processes and the Markov branching process with immigration.

Maximum likelihood estimation for branching processes with immigration

1981 ◽  
Vol 13 (3) ◽  
pp. 498-509 ◽  
Author(s):  
B. R. Bhat ◽  
S. R. Adke
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Continuous Time ◽  
Strong Consistency ◽  
Branching Process ◽  
Branching Processes ◽  
Maximum Likelihood Estimators ◽  
Likelihood Estimation ◽  
Asymptotic Distributions ◽  
Branching Process With Immigration

This paper establishes the strong consistency of the maximum likelihood estimators of the parameters of discrete- and continuous-time Markov branching processes with immigration. The asymptotic distributions of the maximum likelihood estimators of the parameters of a Galton–Watson branching process with immigration are also obtained.

Maximum likelihood estimation for branching processes with immigration

1981 ◽  
Vol 13 (03) ◽  
pp. 498-509 ◽  
Author(s):  
B. R. Bhat ◽  
S. R. Adke
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Continuous Time ◽  
Strong Consistency ◽  
Branching Process ◽  
Branching Processes ◽  
Maximum Likelihood Estimators ◽  
Likelihood Estimation ◽  
Asymptotic Distributions ◽  
Branching Process With Immigration

This paper establishes the strong consistency of the maximum likelihood estimators of the parameters of discrete- and continuous-time Markov branching processes with immigration. The asymptotic distributions of the maximum likelihood estimators of the parameters of a Galton–Watson branching process with immigration are also obtained.

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