In probability theory, a pregaussian class or pregaussian set of functions is a set of functions, square integrable with respect to some probability measure, such that there exists a certain Gaussian process, indexed by this set, satisfying the conditions below.
For a probability space (S, Σ, P), denote by
L
P
2
(
S
)
a set of square integrable with respect to P functions
f
:
S
→
R
, that is
∫
f
2
d
P
<
∞
Consider a set
F
⊂
L
P
2
(
S
)
. There exists a Gaussian process
G
P
, indexed by
F
, with mean 0 and covariance
Cov
(
G
P
(
f
)
,
G
P
(
g
)
)
=
E
G
P
(
f
)
G
P
(
g
)
=
∫
f
g
d
P
−
∫
f
d
P
∫
g
d
P
for
f
,
g
∈
F
Such a process exists because the given covariance is positive definite. This covariance defines a semi-inner product as well as a pseudometric on
L
P
2
(
S
)
given by
ϱ
P
(
f
,
g
)
=
(
E
(
G
P
(
f
)
−
G
P
(
g
)
)
2
)
1
/
2
Definition A class
F
⊂
L
P
2
(
S
)
is called pregaussian if for each
ω
∈
S
,
the function
f
↦
G
P
(
f
)
(
ω
)
on
F
is bounded,
ϱ
P
-uniformly continuous, and prelinear.
The
G
P
process is a generalization of the brownian bridge. Consider
S
=
[
0
,
1
]
,
with P being the uniform measure. In this case, the
G
P
process indexed by the indicator functions
I
[
0
,
x
]
, for
x
∈
[
0
,
1
]
,
is in fact the standard brownian bridge B(x). This set of the indicator functions is pregaussian, moreover, it is the Donsker class.
