Open AccessBook
Probability Theory: A Comprehensive Course
TLDR
Convergence Theorems are applied to the interpretation of Brownian Motion and the law of the Iterated Logarithm as well as to Martingales and Exchangeability.Abstract:
Basic Measure Theory.- Independence.- Generating Functions.- The Integral.- Moments and Laws of Large Numbers.- Convergence Theorems.- Lp-Spaces and Radon-Nikodym Theorem.- Conditional Expectations.- Martingales.- Optional Sampling Theorems.- Martingale Convergence Theorems and their Applications.- Backwards Martingales and Exchangeability.- Convergence of Measures.- Probability Measures on Product Spaces.- Characteristics Functions and Central Limit Theorem.- Infinitely Divisible Distributions.- Markov Chains.- Convergence of Markov Chains.- Markov Chains and Electrical Networks.- Ergodic Theory.- Brownian Motion.- Law of the Iterated Logarithm.- Large Deviations.- The Poisson Point Process.- The Ito Integral.- Stochastic Differential Equations.- References.- Notation Index.- Name Index.- Subject Index.read more
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Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients
TL;DR: In this article, an explicit and easily implementable numerical method for such an SDE was proposed, which converges strongly with the standard order one-half to the exact solution of the SDE.
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Demystifying MMD GANs
TL;DR: In this paper, the authors investigated the training and performance of GANs using the maximum mean discrepancy (MMD) as critic, termed MMD GAN, and clarified the situation with bias in GAN loss functions raised by recent work: they showed that gradient estimators used in the optimization process for both MMD and Wasserstein GAN are unbiased, but learning a discriminator based on samples leads to biased gradients for the generator parameters.
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A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics
TL;DR: Markov categories are developed as a framework for synthetic probability and statistics, following work of Golubtsov as well as Cho and Jacobs, and provides a uniform treatment of various types of probability theory.
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Loss of regularity for Kolmogorov equations
TL;DR: In this paper, the authors consider the intermediate regime of non-hypoelliptic second-order Kolmogorov PDEs with smooth coefficients and show that the standard Euler approximations may converge to the exact solution of the SDE in strong and numerically weak sense, but at a rate that is slower then any power law.
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Networked Control under Random and Malicious Packet Losses
TL;DR: Almost sure stabilization under an event-triggered control law is investigated, where Lyapunov-like functions are utilized to characterize the triggering times at which the plant and the controller attempt to exchange state and control data over the network.