RANDOM FIELDS AND THEIR GEOMETRY

40 2. Gaussian fields
Their diversity shows the power of the abstract approach, in that all can
be treated via the general theory without further probabilistic arguments.
The reader who is not interested in the general Gaussian theory, and cares
mainly about the geometry of fields on R N , need only read Sections 2.2.1
and 2.2.2 on continuity and differentiability in this scenario.
The next two important results are the Borell-TIS inequality and Slepian’s
inequality (and its newer relatives) in Sections 2.3 and 2.4 respectively. The
Borell-TIS inequality gives a universal bound for the tail probability
P{sup f(t) ≥ u},
t∈T
u > 0, for any centered, continuous Gaussian field. As such, it is a truly basic
tool of Gaussian processes, somewhat akin to Chebychev’s inequality in
Statistics or maximal inequalities in Martingale Theory. Slepian’s inequality
and its relatives are just as important and basic, and allow one to use
relationships between covariance functions of Gaussian fields to compare
the tail probabilities and expectations of their suprema.
The final major result of this Chapter is encapsulated in Theorem 2.5.1,
which gives an expansion for a Gaussian field in terms of deterministic
eigenfunctions with independent N(0, 1) coefficients. A special case of this
expansion is the Karhunen-Loève expansion of Section 2.5.1, with which
many readers will already be familiar. Together with the spectral representations
of Section 1.4, they make up what are probably the most important
tools in the Gaussian modeller’s box of tricks. However, these expansions
are also an extremely important theoretical tool, whose development has
far reaching consequences.
2.1 Boundedness and continuity
The aim of this Section is to develop a useful sufficient condition for a
centered Gaussian field on a parameter space T to be almost surely bounded
and/or continuous; i.e. to determine conditions for which
P{sup |f(t)| < ∞} = 1 or P{lim |f(t) − f(s)| = 0, ∀t ∈ T } = 1.
t∈T
s→t
Of course, in order to talk about continuity – i.e. for the notation s → t
above to have some meaning – it is necessary that T have some topology, so
we assume that (T, τ) is a metric space, and that continuity is in terms of
the τ-topology. Our first step is to show that τ is irrelevant to the question
of continuity 2 . This is rather useful, since we shall also soon show that
2 However, τ will come back into the picture when we talk about moduli of continuity
later in this Section.