本帖隐藏的内容

Gaussian processes can be viewed as a far-reaching infinite-dimensional extension
of classical normal random variables. Their theory 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 Briefs 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 first chapters introduce essentials of the classical theory of Gaussian processes
and measures with the core notions of reproducing kernel, integral representation,
isoperimetric property, large deviation principle. The brevity being a priority for
teaching and learning purposes, certain technical details and proofs are omitted.
The later chapters touch important recent issues not sufficiently reflected in the
literature, such as small deviations, expansions, and quantization of processes. In
university teaching, one can build a one-semester advanced course upon these
Briefs.
目录
1 Gaussian Vectors and Distributions . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Univariate Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Multivariate Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Gaussian Objects in ‘‘Arbitrary’’ Linear Spaces . . . . . . . . . . . 5
2 Examples of Gaussian Vectors, Processes and Distributions . . . . . . . 6
3 Gaussian White Noise and Integral Representations . . . . . . . . . . . . . 13
3.1 White Noise: Definition and Integration . . . . . . . . . . . . . . . . 13
3.2 Integral Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Measurable Functionals and the Kernel . . . . . . . . . . . . . . . . . . . . . 22
4.1 Main Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 Factorization Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 Alternative Approaches to Kernel Definition . . . . . . . . . . . . . 32
5 Cameron–Martin Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6 Isoperimetric Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38