Explosive Markov branching processes: entrance laws and limiting behaviour

1993 ◽  
Vol 25 (4) ◽  
pp. 737-756 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Recent Work ◽  
Markov Processes ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
General Construction ◽  
Discrete State ◽  
Construction Problem ◽  
Markov Branching Process ◽  
Limiting Behaviour

The supercritical Markov branching process is examined in the case where the minimal version of the process has strictly substochastic transition laws. This provides a nice example of the general construction theory for discrete-state Markov processes.Entrance laws corresponding to the minimal process are characterised. Limit properties of the processes constructed from these entrance laws are examined. All such processes which are honest and cannot hit zero are ergodic. Otherwise these processes are λ-positive and limit theorems conditional on not having left the positive states are given.A connection is made with recent work on the general construction problem when a λ-subinvariant measure is given. The case where immigration is allowed is mentioned.

Explosive Markov branching processes: entrance laws and limiting behaviour

1993 ◽  
Vol 25 (04) ◽  
pp. 737-756 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Recent Work ◽  
Markov Processes ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
General Construction ◽  
Discrete State ◽  
Construction Problem ◽  
Markov Branching Process ◽  
Limiting Behaviour

The supercritical Markov branching process is examined in the case where the minimal version of the process has strictly substochastic transition laws. This provides a nice example of the general construction theory for discrete-state Markov processes. Entrance laws corresponding to the minimal process are characterised. Limit properties of the processes constructed from these entrance laws are examined. All such processes which are honest and cannot hit zero are ergodic. Otherwise these processes are λ-positive and limit theorems conditional on not having left the positive states are given. A connection is made with recent work on the general construction problem when a λ-subinvariant measure is given. The case where immigration is allowed is mentioned.

scholarly journals Limit Properties of Transition Functions of Continuous-Time Markov Branching Processes

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Azam A. Imomov
Keyword(s):  
Continuous Time ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Local Limit ◽  
Markov Branching Process ◽  
Transition Functions ◽  
Basic Lemma ◽  
Invariant Properties ◽  
Local Limit Theorems

Consider the Markov Branching Process with continuous time. Our focus is on the limit properties of transition functions of this process. Using differential analogue of the Basic Lemma we prove local limit theorems for all cases and observe invariant properties of considering process.

scholarly journals Convergence Rates and Limit Theorems for the Dual Markov Branching Process

2017 ◽  
Vol 2017 ◽  
pp. 1-13
Author(s):  
Anthony G. Pakes
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Convergence Rates ◽  
Branching Processes ◽  
Transition Function ◽  
Optimal Convergence ◽  
Markov Branching Process ◽  
Optimal Convergence Rates ◽  
Positive Recurrent

This paper studies aspects of the Siegmund dual of the Markov branching process. The principal results are optimal convergence rates of its transition function and limit theorems in the case that it is not positive recurrent. Additional discussion is given about specifications of the Markov branching process and its dual. The dualising Markov branching processes need not be regular or even conservative.

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 A note on Markov branching processes

1985 ◽  
Vol 17 (02) ◽  
pp. 463-464
Author(s):  
Fred M. Hoppe
Keyword(s):  
Laplace Transform ◽  
Simple Proof ◽  
Branching Process ◽  
Branching Processes ◽  
Markov Branching Process ◽  
The Laplace Transform

We present a simple proof of Zolotarev’s representation for the Laplace transform of the normalized limit of a Markov branching process and relate it to the Harris representation.

Integer-valued branching processes with immigration

1983 ◽  
Vol 15 (04) ◽  
pp. 713-725 ◽  
Author(s):  
F. W. Steutel ◽  
W. Vervaat ◽  
S. J. Wolfe
Keyword(s):  
Difference Equation ◽  
Continuous Time ◽  
Queueing Theory ◽  
Branching Processes ◽  
Random Variables ◽  
Discrete State ◽  
Limiting Behaviour ◽  
Supercritical Branching Processes

The notion of self-decomposability for -valued random variables as introduced by Steutel and van Harn [10] and its generalization by van Harn, Steutel and Vervaat [5], are used to study the limiting behaviour of continuous-time Markov branching processes with immigration. This behaviour provides analogues to the behaviour of sequences of random variables obeying a certain difference equation as studied by Vervaat [12] and their continuous-time counterpart considered by Wolfe [13]. An application in queueing theory is indicated. Furthermore, discrete-state analogues are given for results on stability in the processes studied by Wolfe, and for results on self-decomposability in supercritical branching processes by Yamazato [14].

scholarly journals Extinction Times in Multitype Markov Branching Processes

2009 ◽  
Vol 46 (01) ◽  
pp. 296-307 ◽  
Author(s):  
Dominik Heinzmann
Keyword(s):  
Limit Theorem ◽  
Continuous Time ◽  
Branching Process ◽  
Branching Processes ◽  
Markov Branching Process ◽  
Time To Extinction

In this paper, a distributional approximation to the time to extinction in a subcritical continuous-time Markov branching process is derived. A limit theorem for this distribution is established and the error in the approximation is quantified. The accuracy of the approximation is illustrated in an epidemiological example. Since Markov branching processes serve as approximations to nonlinear epidemic processes in the initial and final stages, our results can also be used to describe the time to extinction for such processes.

scholarly journals Alternating branching processes

2005 ◽  
Vol 42 (04) ◽  
pp. 1095-1108 ◽  
Author(s):  
Penka Mayster
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Extinction Probability ◽  
Thinning Operator

We introduce the idea of controlling branching processes by means of another branching process, using the fractional thinning operator of Steutel and van Harn. This idea is then adapted to the model of alternating branching, where two Markov branching processes act alternately at random observation and treatment times. We study the extinction probability and limit theorems for reproduction by n cycles, as n → ∞.

Functional limit theorems for high-level subcritical branching processes in random environment

2014 ◽  
Vol 24 (5) ◽  
Author(s):  
Valeriy I. Afanasyev
Keyword(s):  
Random Environment ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Moment Generating Function ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
The Moment ◽  
High Level ◽  
Passage Times

AbstractThe paper is concerned with subcritical branching process in random environment. It is assumed that the moment-generating function of steps of the associated random walk is equal to 1 for some positive value of the argument. Functional limit theorems for sizes of various generations and passage times to various levels are put forward.

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