Adler R.J. An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes

Institute of Mathematical Statistics, 1990. — 160 p. English. (OCR-слой).They are meant to Ье an introduction to what 1 саll the "modern" theory of sample path properties of Gaussian processes, where Ьу "modern" 1 mеan а theory based оп concepts such as entropy and majorising measures. They are directed at an audience that has а reasonable probability background, at the level of anу of the standard texts (Bi11ingsley, Breiman, Chung, etc.). It also helps if the reader already knows something about Gaussian processes, since the modern treatment is very general and thus rather abstract, and it is а substantial help to one's understanding to have some concrete examples to hang the theory оп. То help the novice get а feel for what we are talking about, Chapter 1 has а goodly collection of exaтples.The basic ideas.
The Brownian family of processes.
А collection of examples.
Exercises.
Two Basic Results.
Borell's inequality.
Slepian's inequality.
Applications in Banach spaces.
Exercises.
Prelude to Continuity.
Boundedness and continuity.
Zero -one laws and continuity.
The Karhunen -Loeve expansion.
Exercises.
Boundedness and Continuity.
Majorising measures.
Upper bound proof.
Lower bound proof.
Entropy.
Ultrametricity and discrete majorising measures.
Discontinuous processes.
Exercises.
Suprema Distributions.Some easy bounds.
Processes with а unique point of maximal variance.
General bounds.
The Brownian sheet оп the unit square.
Exercises.
Afterthoughts.
Two topics that were left out.
Directions for research.Indices.