Ergodic Theory

Petersen, Karl

doi:10.1017/cbo9780511608728

Cited by 513 publications

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“…Corollary 2.10 is slightly more general than Kac's formula [7], many proofs of which are known: see, e.g., Petersen [16] or Pollicott and Yuri [17].…”

Section: Proof Note Thatmentioning

confidence: 98%

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Stationary flows and uniqueness of invariant measures

Baccelli

Konstantopoulos

2010

Quart. Appl. Math.

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Abstract. We consider a quadruple (Ω, A , ϑ, μ), where A is a σ-algebra of subsets of Ω, and ϑ is a measurable bijection from Ω into itself that preserves a finite measure μ. For each B ∈ A , we define and study the measure μ B obtained by integrating on B the number of visits to a set of the trajectory of a point of Ω before returning to B. In particular, we obtain a generalization of Kac's formula and discuss its relation to discretetime Palm theory. Although classical in appearance, its use in obtaining uniqueness of invariant measures of various stochastic models seems to be new. We apply the concept to countable Markov chains and Harris processes in general state space.

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“…Corollary 2.10 is slightly more general than Kac's formula [7], many proofs of which are known: see, e.g., Petersen [16] or Pollicott and Yuri [17].…”

Section: Proof Note Thatmentioning

confidence: 98%

“…As in ergodic theory [see, e.g., Petersen [16]], we shall consider the group (ϑ n , n ∈ Z) acting on Ω.…”

Section: The Master Formula Let (ωmentioning

confidence: 99%

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Stationary flows and uniqueness of invariant measures

Baccelli

Konstantopoulos

2010

Quart. Appl. Math.

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“…For this, it should be noted that the pointwise convergence in (4.18) with i = 0 means weak convergence in C(X) , and implies the strong convergence (see [25] p.72) which is precisely (4.19). For more information in this direction see [25] (p.12) and [40] (p.137).…”

Section: Proofmentioning

confidence: 99%

The Uniform Ergodic Theorem for Dynamical Systems

Peškir¹,

Weber²

1996

Convergence in Ergodic Theory and Probability

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Necessary and sufficient conditions are given for the uniform convergence over an arbitrary index set in von Neumann's mean and Birkhoff's pointwise ergodic theorem. Three different types of conditions already known from probability theory are investigated. Firstly it is shown that the property of being eventually totally bounded in the mean is necessary and sufficient. This condition involves as a particular case Blum-DeHardt's theorem which offers the best known sufficient condition for the uniform law of large numbers in the independent case. Secondly it is shown that eventual tightness is necessary and sufficient. In this way a link with a weak convergence is obtained. Finally it is shown that the existence of some particular totally bounded pseudo-metrics is necessary and sufficient. The conditions derived are of Lipschitz type, while the method of proof relies upon a result of independent interest, called the uniform ergodic lemma. This result considerably extends Hopf-Yosida-Kakutani's maximal ergodic lemma to a form more suitable for examinations of the uniform convergence under consideration. From this lemma an inequality is also derived which extends the classical (weak) maximal ergodic inequality to the uniform case. In addition, a uniform approximation by means of a dense family of maps satisfying the uniform ergodic theorem in a trivial way is investigated, and a particular result of this type is established. This approach is in the spirit of the classical Hilbert space method for the mean ergodic theorem of von Neumann, and therefore from the ergodic theory point of view it could be seen as the natural one. After this, a simple characterization is obtained for the uniform convergence of moving averages. Finally, a counter-example is constructed for a symmetrization inequality in the stationary ergodic case. This inequality is known to be of vital importance to support the Vapnik-Chervonenkis random entropy approach in the independent case. Further developments in this direction are indicated.

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“…This implies the existence of a time-invariant probability measure on the state-space [8], since the process dynamics have a weak temporal structure. Operationally, this property allows for the time average of some quantity to be calculated by averaging over the state space.…”

Section: Analysis Of Individual Processesmentioning

confidence: 99%

Principled Synthesis for Large-Scale Systems: Task Sequencing

Shell

Matarić

Distributed Autonomous Robotic Systems 7

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Summary. This paper describes ongoing work toward a principled controller synthesis methodology for large-scale, minimalist multi-robot systems. The work's key objectives is to establish a set of programming primitives (processes) for which macroscopic behavior can be formally predicted. Such prediction is made possible by statistical physics techniques that use properties of time-invariant processes while exploiting the system's large size. This paper's focus is on the use of numerical and simulation methods during construction of the primitive process set. A computational method, developed by physicists, is used as a high-level simulation to characterize individual process behavior. The output, when interpreted qualitatively, guides distributed system design. In order to validate the approach, we consider a sequential inspection domain with a swarm of 400+ simulated robots. Synchronization is achieved through processes analyzed with the methods described, and predictions are compared with behavior exhibited in a traditional multi-robot simulation. The two simulation tools play different roles in characterizing collective behavior, and these differences shed new light on the multi-robot controller synthesis problem.

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Ergodic Theory

Cited by 513 publications

References 0 publications

Stationary flows and uniqueness of invariant measures

Stationary flows and uniqueness of invariant measures

The Uniform Ergodic Theorem for Dynamical Systems

Principled Synthesis for Large-Scale Systems: Task Sequencing

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