Open AccessBook
Foundations of modern probability
Olav Kallenberg
- 01 Jan 1997
TL;DR: In this article, the authors discuss the relationship between Markov Processes and Ergodic properties of Markov processes and their relation with PDEs and potential theory. But their main focus is on the convergence of random processes, measures, and sets.
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Abstract: * Measure Theory-Basic Notions * Measure Theory-Key Results * Processes, Distributions, and Independence * Random Sequences, Series, and Averages * Characteristic Functions and Classical Limit Theorems * Conditioning and Disintegration * Martingales and Optional Times * Markov Processes and Discrete-Time Chains * Random Walks and Renewal Theory * Stationary Processes and Ergodic Theory * Special Notions of Symmetry and Invariance * Poisson and Pure Jump-Type Markov Processes * Gaussian Processes and Brownian Motion * Skorohod Embedding and Invariance Principles * Independent Increments and Infinite Divisibility * Convergence of Random Processes, Measures, and Sets * Stochastic Integrals and Quadratic Variation * Continuous Martingales and Brownian Motion * Feller Processes and Semigroups * Ergodic Properties of Markov Processes * Stochastic Differential Equations and Martingale Problems * Local Time, Excursions, and Additive Functionals * One-Dimensional SDEs and Diffusions * Connections with PDEs and Potential Theory * Predictability, Compensation, and Excessive Functions * Semimartingales and General Stochastic Integration * Large Deviations * Appendix 1: Advanced Measure Theory * Appendix 2: Some Special Spaces * Historical and Bibliographical Notes * Bibliography * Indices
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Citations
Distributed average consensus via gossip algorithm with real-valued and quantized data for 0<q<1
TL;DR: For a gossip consensus algorithm with quantized data, it is proved that it can achieve the consensus almost surely, and the expected value of the final states is equal to the average of the initial states.
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On the Lamperti stable processes
TL;DR: In this paper, a new family of IR d -valued L evy processes, called Lamperti stable processes, is introduced, which shares many properties with the tempered stable and the layered stable processes.
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Weak and strong uniform consistency of a kernel error density estimator in nonparametric regression
TL;DR: In this paper, the problem of estimating the kernel error density function in nonparametric regression models is considered and sufficient conditions are given under which the kernel estimation based on non-parametric residuals is uniformly weakly and strongly consistent.
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Variation analysis of uncertain stationary independent increment processes
TL;DR: It is proved that the total variation of stationary independent increment process with finite variance is bounded almost surely, and the quadratic variation is 0 almost surely and in mean.
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Distributed Kalman Filtering With Quantized Sensing State
TL;DR: It is shown that, in the countable infinite quantization alphabet case, the network can still achieve weak consensus with the information loss due to quantization, i.e., the estimation error variance sequence at a randomly selected sensor can converge weakly (in distribution) to a unique invariant measure.
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References
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Brownian Motion and Stochastic Calculus
Ioannis Karatzas,Steven E. Shreve +1 more
- 01 Jan 1987
TL;DR: In this paper, the authors present a characterization of continuous local martingales with respect to Brownian motion in terms of Markov properties, including the strong Markov property, and a generalized version of the Ito rule.
9.2K
•Book
Continuous martingales and Brownian motion
Daniel Revuz,Marc Yor +1 more
- 01 Jan 1990
TL;DR: In this article, the authors present a comprehensive survey of the literature on limit theorems in distribution in function spaces, including Girsanov's Theorem, Bessel Processes, and Ray-Knight Theorem.
8.4K
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Limit Theorems for Stochastic Processes
Jean Jacod,Albert N. Shiryaev +1 more
- 01 Jan 1987
TL;DR: In this article, the General Theory of Stochastic Processes, Semimartingales, and Stochastically Integrals is discussed and the convergence of Processes with Independent Increments is discussed.
6.4K
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Stochastic integration and differential equations
Philip Protter
- 01 Jan 1990
TL;DR: In this article, the authors propose a method for general stochastic integration and local times, which they call Stochastic Differential Equations (SDEs), and expand the expansion of Filtrations.
6.3K
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Markov Processes: Characterization and Convergence
Stewart N. Ethier,Thomas G. Kurtz +1 more
- 04 Apr 1986
TL;DR: In this paper, the authors present a flowchart of generator and Markov Processes, and show that the flowchart can be viewed as a branching process of a generator.
6.2K
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