scholarly journals Asymptotic Bahavior of the Moran Particle System

2013 ◽  
Vol 45 (02) ◽  
pp. 379-397
Author(s):  
Shui Feng ◽  
Jie Xiong
Keyword(s):  
Asymptotic Behavior ◽  
Large Deviations ◽  
Random Sampling ◽  
Limit Theorems ◽  
Empirical Process ◽  
Particle System ◽  
Sampling Rate ◽  
Interacting Particle ◽  
Independent Motion ◽  
Number Of Particles

The asymptotic behavior is studied for an interacting particle system that involves independent motion and random sampling. For a fixed sampling rate, the empirical process of the particle system converges to the Fleming-Viot process when the number of particles approaches∞. If the sampling rate approaches 0 as the number of particles becomes large, the corresponding empirical process will converge to the deterministic flow of the motion. In the main results of this paper, we study the corresponding central limit theorems and large deviations. Both the Gaussian limits and the large deviations depend on the sampling scales explicitly.

scholarly journals Asymptotic Bahavior of the Moran Particle System

2013 ◽  
Vol 45 (2) ◽  
pp. 379-397
Author(s):  
Shui Feng ◽  
Jie Xiong
Keyword(s):  
Asymptotic Behavior ◽  
Large Deviations ◽  
Random Sampling ◽  
Limit Theorems ◽  
Empirical Process ◽  
Particle System ◽  
Sampling Rate ◽  
Interacting Particle ◽  
Independent Motion ◽  
Number Of Particles

The asymptotic behavior is studied for an interacting particle system that involves independent motion and random sampling. For a fixed sampling rate, the empirical process of the particle system converges to the Fleming-Viot process when the number of particles approaches ∞. If the sampling rate approaches 0 as the number of particles becomes large, the corresponding empirical process will converge to the deterministic flow of the motion. In the main results of this paper, we study the corresponding central limit theorems and large deviations. Both the Gaussian limits and the large deviations depend on the sampling scales explicitly.

Explicit sufficient invariants for an interacting particle system

1998 ◽  
Vol 35 (3) ◽  
pp. 633-641 ◽  
Author(s):  
Yoshiaki Itoh ◽  
Colin Mallows ◽  
Larry Shepp
Keyword(s):  
Markov Chain ◽  
Finite Time ◽  
Interacting Particle Systems ◽  
Particle System ◽  
Particle Systems ◽  
The Other ◽  
New Class ◽  
Interacting Particle ◽  
Death State ◽  
Number Of Particles

We introduce a new class of interacting particle systems on a graph G. Suppose initially there are Ni(0) particles at each vertex i of G, and that the particles interact to form a Markov chain: at each instant two particles are chosen at random, and if these are at adjacent vertices of G, one particle jumps to the other particle's vertex, each with probability 1/2. The process N enters a death state after a finite time when all the particles are in some independent subset of the vertices of G, i.e. a set of vertices with no edges between any two of them. The problem is to find the distribution of the death state, ηi = Ni(∞), as a function of Ni(0).We are able to obtain, for some special graphs, the limiting distribution of Ni if the total number of particles N → ∞ in such a way that the fraction, Ni(0)/S = ξi, at each vertex is held fixed as N → ∞. In particular we can obtain the limit law for the graph S2, the two-leaf star which has three vertices and two edges.

An infinite particle system with continual input and random death

1978 ◽  
Vol 10 (04) ◽  
pp. 764-787
Author(s):  
J. N. McDonald ◽  
N. A. Weiss
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Limit Theorems ◽  
Particle System ◽  
Countable State Space ◽  
Infinite Particle System ◽  
Large Numbers ◽  
Countable State ◽  
Infinite Particle ◽  
Number Of Particles

At times n = 0, 1, 2, · · · a Poisson number of particles enter each state of a countable state space. The particles then move independently according to the transition law of a Markov chain, until their death which occurs at a random time. Several limit theorems are then proved for various functionals of this infinite particle system. In particular, laws of large numbers and central limit theorems are proved.

Explicit sufficient invariants for an interacting particle system

1998 ◽  
Vol 35 (03) ◽  
pp. 633-641 ◽  
Author(s):  
Yoshiaki Itoh ◽  
Colin Mallows ◽  
Larry Shepp
Keyword(s):  
Markov Chain ◽  
Finite Time ◽  
Interacting Particle Systems ◽  
Particle System ◽  
Particle Systems ◽  
The Other ◽  
New Class ◽  
Interacting Particle ◽  
Death State ◽  
Number Of Particles

We introduce a new class of interacting particle systems on a graph G. Suppose initially there are N i (0) particles at each vertex i of G, and that the particles interact to form a Markov chain: at each instant two particles are chosen at random, and if these are at adjacent vertices of G, one particle jumps to the other particle's vertex, each with probability 1/2. The process N enters a death state after a finite time when all the particles are in some independent subset of the vertices of G, i.e. a set of vertices with no edges between any two of them. The problem is to find the distribution of the death state, η i = N i (∞), as a function of N i (0). We are able to obtain, for some special graphs, the limiting distribution of N i if the total number of particles N → ∞ in such a way that the fraction, N i (0)/S = ξ i , at each vertex is held fixed as N → ∞. In particular we can obtain the limit law for the graph S 2, the two-leaf star which has three vertices and two edges.

An infinite particle system with continual input and random death

1978 ◽  
Vol 10 (4) ◽  
pp. 764-787 ◽  
Author(s):  
J. N. McDonald ◽  
N. A. Weiss
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Limit Theorems ◽  
Particle System ◽  
Countable State Space ◽  
Infinite Particle System ◽  
Large Numbers ◽  
Countable State ◽  
Infinite Particle ◽  
Number Of Particles

At times n = 0, 1, 2, · · · a Poisson number of particles enter each state of a countable state space. The particles then move independently according to the transition law of a Markov chain, until their death which occurs at a random time. Several limit theorems are then proved for various functionals of this infinite particle system. In particular, laws of large numbers and central limit theorems are proved.

scholarly journals The Microscopic Derivation and Well-Posedness of the Stochastic Keller–Segel Equation

2020 ◽  
Vol 31 (1) ◽  
Author(s):  
Hui Huang ◽  
Jinniao Qiu
Keyword(s):  
Existence Of Solutions ◽  
Particle System ◽  
Mean Field ◽  
Field Limit ◽  
Mean Field Limit ◽  
Unique Existence ◽  
Interacting Particle ◽  
Microscopic Derivation ◽  
The Mean ◽  
Limit Result

AbstractIn this paper, we propose and study a stochastic aggregation–diffusion equation of the Keller–Segel (KS) type for modeling the chemotaxis in dimensions $$d=2,3$$ d = 2 , 3 . Unlike the classical deterministic KS system, which only allows for idiosyncratic noises, the stochastic KS equation is derived from an interacting particle system subject to both idiosyncratic and common noises. Both the unique existence of solutions to the stochastic KS equation and the mean-field limit result are addressed.

scholarly journals A Local Barycentric Version of the Bak–Sneppen Model

2021 ◽  
Vol 182 (2) ◽  
Author(s):  
Philip Kennerberg ◽  
Stanislav Volkov
Keyword(s):  
Real Number ◽  
Uniform Distribution ◽  
Particle System ◽  
Interacting Particle System ◽  
Interacting Particle

AbstractWe study the behaviour of an interacting particle system, related to the Bak–Sneppen model and Jante’s law process defined in Kennerberg and Volkov (Adv Appl Probab 50:414–439, 2018). Let $$N\ge 3$$ N ≥ 3 vertices be placed on a circle, such that each vertex has exactly two neighbours. To each vertex assign a real number, called fitness (we use this term, as it is quite standard for Bak–Sneppen models). Now find the vertex which fitness deviates most from the average of the fitnesses of its two immediate neighbours (in case of a tie, draw uniformly among such vertices), and replace it by a random value drawn independently according to some distribution $$\zeta $$ ζ . We show that in case where $$\zeta $$ ζ is a finitely supported or continuous uniform distribution, all the fitnesses except one converge to the same value.

Large deviations of generalized renewal process

2020 ◽  
Vol 30 (4) ◽  
pp. 215-241
Author(s):  
Gavriil A. Bakay ◽  
Aleksandr V. Shklyaev
Keyword(s):  
Large Deviations ◽  
Renewal Process ◽  
Limit Theorems ◽  
Local Limit ◽  
Asymptotic Formulas ◽  
Additive Functionals ◽  
Wide Range ◽  
Finite Markov Chains ◽  
Independent Identically Distributed ◽  
Local Limit Theorems

AbstractLet (ξ(i), η(i)) ∈ ℝd+1, 1 ≤ i < ∞, be independent identically distributed random vectors, η(i) be nonnegative random variables, the vector (ξ(1), η(1)) satisfy the Cramer condition. On the base of renewal process, NT = max{k : η(1) + … + η(k) ≤ T} we define the generalized renewal process ZT = $\begin{array}{} \sum_{i=1}^{N_T} \end{array}$ξ(i). Put IΔT(x) = {y ∈ ℝd : xj ≤ yj < xj + ΔT, j = 1, …, d}. We find asymptotic formulas for the probabilities P(ZT ∈ IΔT(x)) as ΔT → 0 and P(ZT = x) in non-lattice and arithmetic cases, respectively, in a wide range of x values, including normal, moderate, and large deviations. The analogous results were obtained for a process with delay in which the distribution of (ξ(1), η(1)) differs from the distribution on the other steps. Using these results, we prove local limit theorems for processes with regeneration and for additive functionals of finite Markov chains, including normal, moderate, and large deviations.

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