Critical markov branching process limit theorems allowing infinite variance

Pakes, Anthony G.

doi:10.1017/s0001867800004158

Cited by 11 publications

(26 citation statements)

References 21 publications

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“…For limit theorems of critical branching processes (single type or multitype) without the finite second moment condition, one can see, for instance [20,44,51,52,53,54] and the references therein.…”

Section: Motivationmentioning

confidence: 99%

Limit theorems for some critical superprocesses

Ren

Song²,

Zhang³

2015

Illinois J. Math.

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Let X = {X t , t ≥ 0; P µ } be a critical superprocess starting from a finite measure µ. Under some conditions, we first prove that lim t→∞ tP µ ( X t = 0) = ν −1 φ 0 , µ , where φ 0 is the eigenfunction corresponding to the first eigenvalue of the infinitesimal generator L of the mean semigroup of X, and ν is a positive constant. Then we show that, for a large class of functions f , conditioning on X t = 0, t −1 f, X t converges in distribution to f, ψ 0 m W , where W is an exponential random variable, and ψ 0 is the eigenfunction corresponding to the first eigenvalue of the dual of L. Finally, if f, ψ 0 m = 0, we prove that, conditioning on X t = 0,) is a normal random variable, and W and G(f ) are independent.

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Section: Motivationmentioning

confidence: 99%

Limit theorems for some critical superprocesses

Ren

Song²,

Zhang³

2015

Illinois J. Math.

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“…[6] proved a conditional limit theorem for f (s) satisfying (1.4) with α = 0. Recently, Pakes [8] generalized the above results to continuous time Markov branching process. The proofs given in [8], based on Karamata's theory for regular varying functions, are much easier.…”

Section: Introductionmentioning

confidence: 88%

“…When α = 1, it gives the well-known standard exponential law. For more information on Linnik Law, we refer readers to [8,Section 4] and references therein.…”

Section: Remark 1 the Stationary-excess Operation Onmentioning

confidence: 99%

Conditional limit theorems for critical continuous-state branching processes

2014

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In this paper we study the conditional limit theorems for critical continuousstate branching processes with branching mechanism ψ(λ) = λ 1+α L(1/λ) where α ∈ [0, 1] and L is slowly varying at ∞. We prove that if α ∈ (0, 1], there are norming constants Q t → 0 (as t ↑ +∞) such that for every x > 0, P x (Q t X t ∈ ·|X t > 0) converges weakly to a non-degenerate limit. The converse assertion is also true provided the regularity of ψ at 0. We give a conditional limit theorem for the case α = 0. The limit theorems we obtain in this paper allow infinite variance of the branching process.

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“…(For a detailed account on the critical Markov branching processes under weaker moment conditions see [11]. )…”

Section: Lemma 18mentioning

confidence: 99%

Tail generating functions for extendable branching processes

Sagitov

2017

Stochastic Processes and their Applications

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We study branching processes of independently splitting particles in the continuous time setting. If time is calibrated such that particles live on average one unit of time, the corresponding transition rates are fully determined by the generating function f for the offspring number of a single particle. We are interested in the deffective case f (1) = 1 − , where each splitting particle with probability is able to terminate the whole branching process. A branching process {Zt} t≥0 will be called extendable if f (q) = q and f (r) = r for some 0 ≤ q < r < ∞. Specializing on the extendable case we derive an integral equation for Ft(s) = Es Z t . This equation is expressed in terms of what we call, tail generating functions. With help of this equation, we obtain limit theorems for the time to termination as → 0. We find that conditioned on non-extinction, the typical values of the termination time follow an exponential distribution in the nearly subcritical case, and require different scalings depending on whether the reproduction regime is asymptotically critical or supercritical. Using the tail generating function approach we also obtain new refined asymptotic results for the regular branching processes with f (1) = 1.

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Critical markov branching process limit theorems allowing infinite variance

Cited by 11 publications

References 21 publications

Limit theorems for some critical superprocesses

Limit theorems for some critical superprocesses

Conditional limit theorems for critical continuous-state branching processes

Tail generating functions for extendable branching processes

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