RANDOM FIELDS AND THEIR GEOMETRY

42 2. Gaussian fields
finite set B ⊂ A0 (the number of whose elements will in general depend on
ε) such that
� � � � �
′ ′
(s, t) ∈ T × T : max τ(s, s ), τ(t, t ) ≤ ε
(s ′ ,t ′ )∈B
covers Aη for some η = η(ε) > 0, with η(ε) → 0 as ε → 0. . That is,
whenever (s, t) ∈ Aη there is a (s ′ , t ′ ) ∈ B with τ(s, s ′ ), τ(t, t ′ ) ≤ ε. Note
that
|ft − fs| ≤ |fs − fs ′| + |fs ′ − ft ′| + |ft ′ − ft|.
Since (s ′ , t ′ ) ∈ B ⊂ A0, we have f(s ′ ) = f(t ′ ) a.s. Thus
sup |ft − fs| ≤ 2 sup |ft − fs|,
d(s,t)≤η(ε)
τ(s,t)≤ε
and the τ-continuity of f implies its d-continuity. ✷
The astute reader will have noted that in the statement of Lemma 2.1.1
the parameter space T was quietly assumed to be compact, and that this additional
assumption was needed in the proof. Indeed, from now on we shall
assume that this is always the case, and shall rely on it heavily. Fortunately,
however, it is not a serious problem. As far as continuity is concerned, if
T is σ-compact 4 then a.s. continuity on its compact subsets immediately
implies a.s. continuity over T itself. We shall not go beyond σ-compact
spaces in this book. The same is not true for boundedness, nor should it
be 5 . However, we shall see that, at least on compact T , boundedness and
continuity are equivalent problems.
Now recall the Brownian noise processes from Section 1.3. There we saw
that the same process could be continuous, or discontinuous, depending on
how we specified its parameter set. We shall now see that the difference
between the parameter sets had nothing to do with their geometrical properties,
but rather their “size” as measured in terms of d, via the tool of
metric entropy.
Definition 2.1.2 Let f be a centered Gaussian field on T , and d the canonical
metric (2.1.1). Assume that T is d-compact, and write
(2.1.2)
Bd(t, ε) ∆ = {s ∈ T : d(s, t) ≤ ε}
for the d ball centered on t ∈ T and of radius ε. Let N(T, d : ε) ≡ N(ε)
denote the smallest number of such balls that cover T , and set
(2.1.3)
H(T, d : ε) ≡ H(ε) = ln (N(ε)) .
4 T is σ-compact if it can be represented as the countable union of compact sets.
5 Think of simple Brownian motion on R+. While bounded on every finite interval, it
is a consequence of the law of the iterated logarithm that it is unbounded on R+.