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
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Data-dependent PAC-Bayes priors via differential privacy
TL;DR: This paper showed that a Gaussian prior mean chosen via stochastic gradient Langevin dynamics (SGLD) leads to a valid PAC-Bayes bound given control of the 2-Wasserstein distance to an ε-differentially private stationary distribution.
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Reaching the best possible rate of convergence to equilibrium for solutions of Kac's equation via central limit theorem
TL;DR: In this article, the authors proved that the total variation distance between the Gaussian density function and the probability density function of the solution of Kac's equation goes to zero, with an exponential rate equal to -1/4.
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Outliers in the Single Ring Theorem
TL;DR: It is proved that outliers can here have very various rates of convergence to their limits and that some correlations can appear between outliers at a macroscopic distance from each other (a fact already noticed by Knowles and Yin in (Ann Probab 42:1980–2031, 2014)
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Subgame-Perfect Equilibria in Stochastic Timing Games
TL;DR: In this article, the authors develop a notion of subgames and a related subgame-perfect equilibrium for stochastic timing games, possibly in mixed strategies, and provide sufficient conditions for equilibrium existence.
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Universality of the REM for dynamics of mean-field spin glasses
TL;DR: In this article, a version of a Glauber dynamics for a p-spin Sherrington-Kirkpatrick model of a spin glass is considered, where the dynamics exhibits aging at these time scales with time-time correlation function converging to the arcsine law of this subordinator.
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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
•Book
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
•Book
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
•Book
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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