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**An introduction to continuity, extrema, and related topics for general Gaussian processes.**
*(English)*
Zbl 0747.60039

Institute of Mathematical Statistics Lecture Notes - Monograph Series 12. Hayward, CA: Institute of Mathematical Statistics (ISBN 0-940600-17-X/pbk). vii, 160 p., open access (1990).

These notes, as the author writes, “are meant to be an introduction to what I call the modern theory of sample path Gaussian processes,…based on concepts such as entropy and majorising measures”.

Although the focus is mainly on three topics, i.e. boundedness, continuity and suprema distributions, “these notes contain almost everything a beginning researcher needs to know about the mathematical basis of Gaussian process”, except for the relation with the theory of Markov processes and the \({\mathcal L}^ 2\) space associated with a Gaussian process.

First, a collection of examples is introduced, in order to clarify the basic ideas in view of the abstract setting of the modern theory. Some basic tools, such as Borel’s and Slepian’s inequalities, are reviewed before the core of the text, which consists, as mentioned above, in a thorough treatment of boundedness, continuity and suprema distributions.

The style is lively, elegant and clear. Altogether, a beautiful work.

Although the focus is mainly on three topics, i.e. boundedness, continuity and suprema distributions, “these notes contain almost everything a beginning researcher needs to know about the mathematical basis of Gaussian process”, except for the relation with the theory of Markov processes and the \({\mathcal L}^ 2\) space associated with a Gaussian process.

First, a collection of examples is introduced, in order to clarify the basic ideas in view of the abstract setting of the modern theory. Some basic tools, such as Borel’s and Slepian’s inequalities, are reviewed before the core of the text, which consists, as mentioned above, in a thorough treatment of boundedness, continuity and suprema distributions.

The style is lively, elegant and clear. Altogether, a beautiful work.

Reviewer: B.Bassan (Roma)