RANDOM FIELDS AND THEIR GEOMETRY

46 2. Gaussian fields
By this and (2.1.10), it follows that
(2.1.13)
�
��fπj(t) �
P − f �
πj−1(t) ≥
�
r/ √ � �
−j
2
≤ exp
�
−(r/ √ 2) −2j
2(2r−j+1 ) 2
�
= exp � −2 j /8r 2� ,
which is emminently summable. By Borel-Cantelli, and recalling that r ≥ 2,
we have that the sum in (2.1.11) converges absolutely, with probability one.
We now start the main part of the proof. Define
Mj = NjNj−1, aj = 2 3/2 r −j+1
�
ln(2j−iMj), S = �
aj.
Then Mj is the maximum number of possible pairs (πj(t), πj−1(t)) as t
varies through T and aj was chosen so as to make later formulae simplify.
Applying (2.1.12) once again, we have, for all u > 0,
(2.1.14)
and so
P � �
∃ t ∈ T : fπj(t) − fπj−1(t) > uaj ≤ Mj exp
�
P sup ft − fto
t∈T
�
≥ uS
For u > 1 this is at most
� � j−i
2 �−u 2
j>i
≤ �
Mj exp
j>i
= �
j>i
Mj
j>i
� −u 2 a 2 j
2(2r −j+1 ) 2
� −u 2 a 2 j
2(2r −j+1 ) 2
� � 2
j−i −u
2 Mj .
≤ 2 −u2 �
2 j−i+1
j>i
= 2 · 2 −u2
.
The basic relationship that, for non-negative random variables X,
E{X} =
� ∞
0
P{X ≥ u} du,
together with the observation that supt∈T (ft − fto ) ≥ 0 since to
immediately yields
∈ T ,
� �
(2.1.15)
E ≤ KS,
sup ft
t∈T
�
�