Weighted Limit Theorems for Continuous-Time Vector Martingales with Explosive and Mixed Growth

AbstractWe consider a particle system in continuous time, a discrete population, with spatial motion, and nonlocal branching. The offspring's positions and their number may depend on the mother's position. Our setting captures, for instance, the processes indexed by a Galton–Watson tree. Using a size-biased auxiliary process for the empirical measure, we determine the asymptotic behaviour of the particle system. We also obtain a large population approximation as a weak solution of a growth-fragmentation equation. Several examples illustrate our results. The main one describes the behaviour of a mitosis model; the population is size structured. In this example, the sizes of the cells grow linearly and if a cell dies then it divides into two descendants.

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Limit theorems for some continuous-time random walks

Advances in Applied Probability ◽

10.1239/aap/1316792670 ◽

2011 ◽

Vol 43 (3) ◽

pp. 782-813 ◽

Cited By ~ 1

Author(s):

M. Jara ◽

T. Komorowski

Keyword(s):

Markov Chain ◽

Random Walk ◽

Random Walks ◽

Quantum Transport ◽

Continuous Time ◽

Limit Theorems ◽

Transport Theory ◽

Sufficient Conditions ◽

Continuous Time Random Walks ◽

Quantum Transport Theory

In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {Xn,n≥ 0} and two observables, τ(∙) andV(∙), corresponding to the renewal times and jump sizes. Assuming that these observables belong to the domains of attraction of some stable laws, we give sufficient conditions on the chain that guarantee the existence of the scaled limits for CTRWs. An application of the results to a process that arises in quantum transport theory is provided. The results obtained in this paper generalize earlier results contained in Becker-Kern, Meerschaert and Scheffler (2004) and Meerschaert and Scheffler (2008), and the recent results of Henry and Straka (2011) and Jurlewicz, Kern, Meerschaert and Scheffler (2010), where {Xn,n≥ 0} is a sequence of independent and identically distributed random variables.

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Limit theorems for randomly coarse grained continuous-time random walks

Dissertationes Mathematicae ◽

10.4064/dm431-0-1 ◽

2005 ◽

Vol 431 ◽

pp. 1-45 ◽

Cited By ~ 11

Author(s):

Agnieszka Jurlewicz

Keyword(s):

Random Walks ◽

Continuous Time ◽

Limit Theorems ◽

Coarse Grained ◽

Continuous Time Random Walks

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Limit theorems for processes such as the Markov branching process

Journal of Applied Probability ◽

10.1017/s0021900200047975 ◽

1975 ◽

Vol 12 (02) ◽

pp. 289-297

Author(s):

Andrew D. Barbour

Keyword(s):

Markov Process ◽

Residence Time ◽

Continuous Time ◽

Limit Theorems ◽

Branching Process ◽

Continuous Mapping ◽

Random Variables ◽

Time Parameter ◽

Image Position ◽

Independent Identically Distributed

LetX(t) be a continuous time Markov process on the integers such that, ifσis a time at whichXmakes a jump,X(σ)– X(σ–) is distributed independently ofX(σ–), and has finite meanμand variance. Letq(j) denote the residence time parameter for the statej.Iftndenotes the time of thenth jump andXn≡X(tb), it is easy to deduce limit theorems forfrom those for sums of independent identically distributed random variables. In this paper, it is shown how, forμ> 0 and for suitableq(·), these theorems can be translated into limit theorems forX(t), by using the continuous mapping theorem.

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Limit theorems for additive functionals of continuous time random walks

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2020.33 ◽

2020 ◽

pp. 1-22

Author(s):

Yuri Kondratiev ◽

Yuliya Mishura ◽

Georgiy Shevchenko

Keyword(s):

Continuous Time ◽

Limit Theorems ◽

Domain Of Attraction ◽

Weak Limit ◽

Additive Functional ◽

Stable Law ◽

Additive Functionals ◽

Lévy Motion ◽

Continuous Time Random Walks ◽

Poisson Shot Noise

Abstract For a continuous-time random walk X = {X t , t ⩾ 0} (in general non-Markov), we study the asymptotic behaviour, as t → ∞, of the normalized additive functional $c_t\int _0^{t} f(X_s)\,{\rm d}s$ , t⩾ 0. Similarly to the Markov situation, assuming that the distribution of jumps of X belongs to the domain of attraction to α-stable law with α > 1, we establish the convergence to the local time at zero of an α-stable Lévy motion. We further study a situation where X is delayed by a random environment given by the Poisson shot-noise potential: $\Lambda (x,\gamma )= {\rm e}^{-\sum _{y\in \gamma } \phi (x-y)},$ where $\phi \colon \mathbb R\to [0,\infty )$ is a bounded function decaying sufficiently fast, and γ is a homogeneous Poisson point process, independent of X. We find that in this case the weak limit has both ‘quenched’ component depending on Λ, and a component, where Λ is ‘averaged’.

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Continuous-time branching processes with decreasing state-dependent immigration

Advances in Applied Probability ◽

10.2307/1427337 ◽

1984 ◽

Vol 16 (4) ◽

pp. 697-714 ◽

Cited By ~ 8

Author(s):

K. V. Mitov ◽

V. A. Vatutin ◽

N. M. Yanev

Keyword(s):

Asymptotic Behaviour ◽

Population Size ◽

Continuous Time ◽

Limit Theorems ◽

Critical Case ◽

Branching Processes ◽

State Dependent ◽

Different Types ◽

Continuous Time Branching Processes

This paper deals with continuous-time branching processes which allow a temporally-decreasing immigration whenever the population size is 0. In the critical case the asymptotic behaviour of the probability of non-extinction and of the first two moments is investigated and different types of limit theorems are also proved.

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