Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

Harmonic Analysis and the Theory of Probability. By S. Bochner pp. 176. 35s. 1955. (California University Press and Cambridge University Press)

Deep Neural Networks trained as image auto-encoders have recently emerged as a promising direction for advancing the state-of-the-art in image compression. The key challenge in learning such networks is twofold: To deal with quantization, and to control the trade-off between reconstruction error (distortion) and entropy (rate) of the latent image representation. In this paper, we focus on the latter challenge and propose a new technique to navigate the rate-distortion trade-off for an image compression auto-encoder. The main idea is to directly model the entropy of the latent representation by using a context model: A 3D-CNN which learns a conditional probability model of the latent distribution of the auto-encoder. During training, the auto-encoder makes use of the context model to estimate the entropy of its representation, and the context model is concurrently updated to learn the dependencies between the symbols in the latent representation. Our experiments show that this approach, when measured in MS-SSIM, yields a state-of-the-art image compression system based on a simple convolutional auto-encoder.

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Conditional Probability Models for Deep Image Compression

Введение в теорию формальных языков

This is lecture notes on the course "Stochastic Processes". In this format, the course was taught in the spring semesters 2017 and 2018 for third-year bachelor students of the Department of Control and Applied Mathematics, School of Applied Mathematics and Informatics at Moscow Institute of Physics and Technology. The base of this course was formed and taught for decades by professors from the Department of Mathematical Foundations of Control A.A. Natan, S.A. Guz, and O.G. Gorbachev. 
Besides standard chapters of stochastic processes theory (correlation theory, Markov processes) in this book (and lectures) the following chapters are included: von Neumann-Birkhoff-Khinchin ergodic theorem, macrosystem equilibrium concept, Markov Chain Monte Carlo, Markov decision processes and the secretary problem.

Lecture Notes on Stochastic Processes

Brownian motion with “green” noise in a periodic potential

We consider the behavior of stochastic systems driven by noise with a zero value of spectral density at zero frequency (“green” noise). For this purpose we propose the version of the Krylov–Bogoliubov averaging method to study the systems which are not stationary in the case of an external white noise. We use the ergodicity of a nonlinear random function in the method, and obtain equations for any approximation of the theory. In particular, it is shown in the first approximation that there is an effective potential to describe the averaged motion of the system. We consider a phase-locked loop as an example and show that metastable states are possible. The lifetime of these states essentially increases if the form of a green noise spectrum becomes sharper in the low-frequency region. The high stability of the system driven by green noise is confirmed by numerical simulation. It is important that the theoretical result obtained by the averaging method and the one obtained in the simulation coincide with sufficient accuracy. In conclusion, we discuss some of the unsolved green noise problems.

S. A. Guz

Papers

Lecture Notes on Stochastic Processes

Brownian motion with “green” noise in a periodic potential

"Green" noise in quasistationary stochastic systems.

Brownian particle in a solitary potential well under the influence of `green' noise

Catastrophes in Brownian motion