By Robert J. Adler
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1), that p2 (u) ≤ 2N u. 1 (cf. 3)) immediately yields the con1 tinuity and boundedness of W . 7) write u ∨ v for max(u, v) and note that N d(s, t) ≤ 2 N (si ∨ ti ) − 2 i=1 Set a = 2 N −1 i=1 (si ∨ ti ) and b = 2 (si ∧ ti ). i=1 N −1 i=1 (si ∧ ti ). Then 2 ≥ a > b and d(s, t) ≤ a(sN ∨ tN ) − b(sN ∧ tN ). 52 2. Gaussian fields If sN > tN the right-hand side is equal to asN − btN = a(sN − tN ) + tN (a − b) ≤ 2|sN − tN | + |a − b|. Similarly, if sN < tN the right-hand side equals atN − bsN = a(tN − sN ) + sN (a − b) ≤ 2|tN − sN | + |a − b|, so that N −1 d(s, t) ≤ 2|tN − sN | + 2 N −1 (si ∨ ti ) − 2 i=1 (si ∧ ti ).
Eik )) ∂ti1 . . ∂tik of f of various orders. 33) E ∂ k f (s) ∂ k f (t) ∂ti1 ∂ti1 . . ∂tik ∂ti1 ∂ti1 . . ∂tik = ∂ 2k C(s, t) . ∂si1 ∂ti1 . . ∂sik ∂tik 27 This is an immediate consequence of the fact that a sequence X of random variables n converges in L2 if, and only if, E{Xn Xm } converges to a constant as n, m → ∞. 26 1. Random fields The corresponding variances have a nice interpretation in terms of spectral moments when f is stationary. For example, if f has mean square partial derivatives of orders α + β and γ + δ for α, β, γ, δ ∈ {0, 1, 2, .
In the following Chapter we shall concentrate exclusively on Gaussian fields. Despite, and perhaps because of, this it is time to take a moment to explain both the centrality of Gaussian fields and how to best move away from them. It will become clear as you progress through this book that while appeals to the Central Limit Theorem may be a nice way to justify concentrating on the Gaussian case, the real reason for this concentration is somewhat more mundane. The relatively uncomplicated form of the multivariate Gaussian density (and hence finite-dimensional distributions of Gaussian fields) makes it a reasonably straightforward task to carry out detailed computations and allows one to obtain explicit results and precise formulae for many facets of Gaussian fields.