Law of large numbers and central limit theorem for unbounded jump mean-field models

Dawson, Donald A.; Zheng, Xiangqi

doi:10.1016/0196-8858(91)90015-b

Cited by 34 publications

(34 citation statements)

References 12 publications

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“…Under the estimation in Lemma 11, together with (5), (7), (8), (9), (32) and (33), a similar argument to that in [7] implies that ) and (35) imply that the distribution P ∞ of U is a solution to the martingale problem with respect to the following operator G,…”

Section: Law Of Large Numbersmentioning

confidence: 80%

“…The main result of this section is stated in Theorem 12, the proof of which is similar to that in [7]. We first prove a lemma.…”

Section: Law Of Large Numbersmentioning

confidence: 94%

“…From the mean-field point of view, the performance analysis of a large system could be done by investigating the limiting behavior as N → ∞. The major advantage of this method is that the limit can often be found by solving a deterministic system [5][6][7]. In this paper, we will use this method to investigate a queueing system consisting of a large number of queues.…”

Section: Introductionmentioning

confidence: 99%

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Balancing Queues by Mean Field Interaction

2005

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Section: Law Of Large Numbersmentioning

confidence: 80%

“…The main result of this section is stated in Theorem 12, the proof of which is similar to that in [7]. We first prove a lemma.…”

Section: Law Of Large Numbersmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

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Balancing Queues by Mean Field Interaction

2005

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“…In fact, the mean-field method has many applications and is often considered to be an approximation to describe the complicated original models in statistical physics. Moreover, various mean-field models have been well investigated by many authors; see [6], [7], [9], [23], and the references therein. However, these investigations mainly dealt with limit theorems such as laws of large numbers and central limit theorems.…”

Section: X K (T))mentioning

confidence: 99%

“…The usefulness of such systems, the potential impact on many practical situations, and the challenges they present to both physicists and mathematicians have attracted much attention in recent years. Dawson [6] presented a detailed study on the cooperative behavior of such systems, and, subsequently, in [7] delineated the law 222 F. XI AND G. YIN of large numbers and central limit theorem for jump mean fields; see also the related works of Hitsuda and Mitoma [9] and Shiga and Tanaka [23], and the references therein. From a probabilistic point of view, one of the many important problems is that of gaining an in depth understanding of the ergodicity of such systems.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Properties of a Mean-Field Model with a Continuous-State-Dependent Switching Process

Yin

2009

J. Appl. Probab.

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This work is concerned with a class of mean-field models given by a switching diffusion with a continuous-state-dependent switching process. Focusing on asymptotic properties, the regularity or nonexplosiveness, Feller continuity, and strong Feller continuity are established by means of introducing certain auxiliary processes and by making use of the truncations. Based on these results, exponential ergodicity is obtained under the FosterLyapunov drift conditions. By virtue of the coupling methods, the strong ergodicity or uniform ergodicity in the sense of convergence in the variation norm is established for the mean-field model with a Markovian switching process. Besides this, several examples are presented for demonstration and illustration.

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Eigenvalues, Inequalities, and Ergodic Theory

Chen

2005

Probability and Its Applications

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This paper surveys the main results obtained during the period 1992-1999 on three aspects mentioned at the title. The first result is a new and general variational formula for the lower bound of spectral gap (i.e., the first non-trivial eigenvalue) of elliptic operators in Euclidean space, Laplacian on Riemannian manifolds or Markov chains ( §1). Here, a probabilistic method-coupling method is adopted. The new formula is a dual of the classical variational formula. The last formula is actually equivalent to Poincaré inequality. To which, there are closely related logarithmic Sobolev inequality, Nash inequality, Liggett inequality and so on. These inequalities are treated in a unified way by using Cheeger's method which comes from Riemannian geometry. This consists of §2. The results on these two aspects are mainly completed by the author joint with F. Y. Wang. Furthermore, a diagram of the inequalities and the traditional three types of ergodicity is presented ( §3). The diagram extends the ergodic theory of Markov processes. The details of the methods used in the paper will be explained in a subsequent paper under the same title.

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Law of large numbers and central limit theorem for unbounded jump mean-field models

Cited by 34 publications

References 12 publications

Balancing Queues by Mean Field Interaction

Balancing Queues by Mean Field Interaction

Asymptotic Properties of a Mean-Field Model with a Continuous-State-Dependent Switching Process

Eigenvalues, Inequalities, and Ergodic Theory

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