scholarly journals Limit theorems for some branching measure-valued processes

2017 ◽  
Vol 49 (2) ◽  
pp. 549-580 ◽  
Author(s):  
Bertrand Cloez
Keyword(s):  
Continuous Time ◽  
Limit Theorems ◽  
Particle System ◽  
Large Population ◽  
Empirical Measure ◽  
Spatial Motion ◽  
Auxiliary Process ◽  
Discrete Population ◽  
A Cell ◽  
Fragmentation Equation

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.

scholarly journals Limit theorems for some continuous-time random walks

2011 ◽  
Vol 43 (3) ◽  
pp. 782-813 ◽  
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.

scholarly journals How linear reinforcement affects Donsker’s theorem for empirical processes

2020 ◽  
Vol 178 (3-4) ◽  
pp. 1173-1192 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Long Range ◽  
Limit Theorems ◽  
Brownian Bridge ◽  
Empirical Processes ◽  
Constant Factor ◽  
Empirical Measure ◽  
Reinforced Random Walks ◽  
Bernoulli Processes

Abstract A reinforcement algorithm introduced by Simon (Biometrika 42(3/4):425–440, 1955) produces a sequence of uniform random variables with long range memory as follows. At each step, with a fixed probability $$p\in (0,1)$$ p ∈ ( 0 , 1 ) , $${\hat{U}}_{n+1}$$ U ^ n + 1 is sampled uniformly from $${\hat{U}}_1, \ldots , {\hat{U}}_n$$ U ^ 1 , … , U ^ n , and with complementary probability $$1-p$$ 1 - p , $${\hat{U}}_{n+1}$$ U ^ n + 1 is a new independent uniform variable. The Glivenko–Cantelli theorem remains valid for the reinforced empirical measure, but not the Donsker theorem. Specifically, we show that the sequence of empirical processes converges in law to a Brownian bridge only up to a constant factor when $$p<1/2$$ p < 1 / 2 , and that a further rescaling is needed when $$p>1/2$$ p > 1 / 2 and the limit is then a bridge with exchangeable increments and discontinuous paths. This is related to earlier limit theorems for correlated Bernoulli processes, the so-called elephant random walk, and more generally step reinforced random walks.

Limit theorems for processes such as the Markov branching process

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&gt; 0 and for suitableq(·), these theorems can be translated into limit theorems forX(t), by using the continuous mapping theorem.

scholarly journals Limit theorems for additive functionals of continuous time random walks

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’.

scholarly journals The evolution of the cancer stem cell state in glioblastoma: emerging insights into the next generation of functional interactions

2020 ◽  
Author(s):  
Kelly Mitchell ◽  
Katie Troike ◽  
Daniel J Silver ◽  
Justin D Lathia
Keyword(s):  
Cellular Heterogeneity ◽  
Next Generation ◽  
Cell State ◽  
Functional Interactions ◽  
Current State ◽  
Key Concepts ◽  
Discrete Population ◽  
A Cell ◽  
Stem Cell State ◽  
Primary Malignant Brain Tumor

Abstract Cellular heterogeneity is a hallmark of advanced cancers and has been ascribed in part to a population of self-renewing, therapeutically resistant cancer stem cells (CSCs). Glioblastoma (GBM), the most common primary malignant brain tumor, has served as a platform for the study of CSCs. In addition to illustrating the complexities of CSC biology, these investigations have led to a deeper understanding of GBM pathogenesis, revealed novel therapeutic targets, and driven innovation towards the development of next-generation therapies. While there continues to be an expansion in our knowledge of how CSCs contribute to GBM progression, opportunities have emerged to revisit this conceptual framework. In this review, we will summarize the current state of CSCs in GBM using key concepts of evolution as a paradigm (variation, inheritance, selection, and time) to describe how the CSC state is subject to alterations of cell intrinsic and extrinsic interactions that shape their evolutionarily trajectory. We identify emerging areas for future consideration, including appreciating CSCs as a cell state that is subject to plasticity, as opposed to a discrete population. These future considerations will not only have an impact on our understanding of this ever-expanding field but will also provide an opportunity to inform future therapies to effectively treat this complex and devastating disease.

The principle of the diffusion of arbitrary constants

1972 ◽  
Vol 9 (3) ◽  
pp. 519-541 ◽  
Author(s):  
Andrew D. Barbour
Keyword(s):  
Central Limit ◽  
Continuous Time ◽  
Large Population ◽  
Biological Models ◽  
Deterministic Solution ◽  
Lattice Processes

Equations are derived describing a central limit type large population approximation for continuous time Markov lattice processes in one or more dimensions, such as are commonly encountered in biological models. A method of solving the equations using only the deterministic solution of the process is explained, and it is extended by the use of a martingale argument to provide more detailed information about the process.

Interacting particle systems approximations of the Kushner-Stratonovitch equation

1999 ◽  
Vol 31 (3) ◽  
pp. 819-838 ◽  
Author(s):  
D. Crişan ◽  
P. Del Moral ◽  
T. J. Lyons
Keyword(s):  
Continuous Time ◽  
Order Of Convergence ◽  
Interacting Particle Systems ◽  
Particle System ◽  
Particle Systems ◽  
Interacting Particle System ◽  
Order Convergence ◽  
Interacting Particle ◽  
Convergence Results ◽  
Filtering Problem

In this paper we consider the continuous-time filtering problem and we estimate the order of convergence of an interacting particle system scheme presented by the authors in previous works. We will discuss how the discrete time approximating model of the Kushner-Stratonovitch equation and the genetic type interacting particle system approximation combine. We present quenched error bounds as well as mean order convergence results.

Continuous-time branching processes with decreasing state-dependent immigration

1984 ◽  
Vol 16 (4) ◽  
pp. 697-714 ◽  
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.

Limit theorems for continuous-time random walks in the double-array limit scheme

2006 ◽  
Vol 138 (1) ◽  
pp. 5348-5365 ◽  
Author(s):  
V. E. Bening ◽  
V. Yu. Korolev ◽  
V. N. Kolokoltsov
Keyword(s):  
Random Walks ◽  
Continuous Time ◽  
Limit Theorems ◽  
Double Array ◽  
Continuous Time Random Walks
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