Lifshits M. Lectures on Gaussian Processes

Springer, 2012. — 133 p.Gaussian processes can be viewed as a far-reaching infinite-dimensional extension of classical normal random variables. Their theory is one of the most advanced fields in the probability science and presents a powerful range of tools for probabilistic modelling in various academic and technical domains such as Statistics, Forecasting, Finance, Information Transmission, Machine Learning—to mention just a few.
The objective of these lectures is to present a quick and condensed treatment of the core theory that a reader must understand in order to make his own independent contributions. The primary intended readership are Ph.D/Masters students and researchers working in pure or applied mathematics. The knowledge of basics in measure theory, functional analysis, and, of course, probability, is required for successful reading.
The first chapters introduce essentials of the classical theory of Gaussian processes and measures. The core notions of Gaussian measure, reproducing kernel, integral representation, isoperimetric property, large deviation principle are explained and illustrated by numerous thoroughly chosen examples. This part mainly follows my book ‘‘Gaussian Random Functions’’ but the chosen exposition style is different. The brevity being a priority for teaching and learning purposes, certain technical details and proofs are omitted, rendering approach less formal, more appropriate to the lecture notes than to a textbook.
Obviously, new issues that emerged during last decade are also present in the exposition. Inequalities related to correlation conjecture and to other extremal problems, the entropy approaches to evaluation of small deviation probabilities, expansions of Gaussian vectors, relations to the theory of linear operators, and links to quantization problems for random processes fit into this category.
The short lecture notes by no means aim to provide a complete account of immense research field in pure and applied mathematics related to Gaussian processes. A few indications on further possible reading are given in ‘‘Invitation to Further Reading’’Gaussian Vectors and Distributions.
Gaussian White Noise and Integral Representations.
Measurable Functionals and the Kernel.
Cameron–Martin Theorem.
Isoperimetric Inequality.
Measure Concavity and Other Inequalities.
Large Deviation Principle.
Functional Law of the Iterated Logarithm.
Metric Entropy and Sample Path Properties.
Small Deviations.
Expansions of Gaussian Vectors.
Quantization of Gaussian Vectors.
Invitation to Further Reading.