Excursions above high levels for stationary Gaussian processes

A normal comparison inequality and its applications

“…Put ρ ij = max(|r 1 ij |, |r 0 ij |). Then as stated in Leadbetter, Lindgren and Rootzén (1983), page 81, and based on early works of Slepian (1962), Berman (1964Berman ( , 1971), Cramér and Leadbetter (1967),…”

Random Fields and Geometry

“…We shall see later that if f is stationary, 18 then the convergence of the entropy integral is also necessary for continuity and that continuity and boundedness always occur together (Theorem 1.5.4). Now, however, we shall prove Theorems 1.3.3 and 1.3.5 following the approach of Talagrand [154].…”

“…If f is a separable process on T and X a centered random variable (not necessarily independent of f ), then E sup t∈T (f t + X) = E sup t∈T f t . 18 We shall treat stationarity in detail in Chapter 5. For the moment all you need to know is that under stationarity the expectation E{f (t)} is constant, while the covariance E{f (t)f (s)} is a function of s − t only.…”

“…footnote 3). 18 1 Gaussian Fields E sup t∈T f t ≤ KS, (1.3.15) with K = 1 + 2 ∞ 1 2 −u 2 du. Thus, all that remains to complete the proof is to compute S.…”

“…Here, however, we shall give the proofs of Theorems 4.1.2 and 4.1.3. The central idea is old, and goes back at least to the works of Berman [18,19] and Pickands [123,124,125] in the mid to late 1960s, which themselves have roots in Cramér's classic paper [41].…”