final report - probability.ca

is continuous, since
ˆ
|F (h)| ≤
H
|x| µ (dx) |h| , h ∈ H.
It follows from the Riesz representation theorem there exists m ∈ H such that
ˆ
〈m, h〉 = 〈x, h〉 µ (dx) , h ∈ H.
We call m the mean of the probability measure µ. Furthermore, we shall write
ˆ
m = xµ (dx) .
If we further assume that
ˆ
H
H
H
|x| 2 µ (dx) < ∞,
then we may consider a bilinear form G : H × H → R defined as
ˆ
G (h, k) = 〈h, x − m〉 〈k, x − m〉 µ (dx) , h, k ∈ H.
H
Furthermore, another application of the Riesz representation theorem guarantees the existence of a unique linear
bounded operator Q ∈ L (H) such that
ˆ
〈Qh, k〉 = 〈h, x − m〉 〈k, x − m〉 µ (dx) , h, k ∈ H.
The operator Q is called the covariance of the probability measure µ.
H
Proposition A.8. Let µ be a probability measure on (H, B (H)) with mean m and covariance operator Q. Then
Q ∈ L + 1 (H), i.e., Q is symmetric, positive, and of trace class.
Proof. That Q is symmetric and positive is clear from its construction. To prove that it is of trace class, observe
ˆ
〈Qe k , e k 〉 = |〈x − m, e k 〉| 2 µ (dx) , k ∈ N.
By the monotone convergence theorem and an application of Parseval’s identity, we have
traceQ =
∞∑
ˆ
k=1
H
H
∣
∣〈x − m, e k 〉 2∣ ˆ
∣ µ (dx) =
H
|x − m| 2 µ (dx) < ∞.
A.3 Gaussian Measures
For a ∈ H and Q ∈ L + 1 (H), we call the measure µ := N a,Q on (H, B (H)) a Gaussian measure µ with mean a and
covariance operator Q. Its Fourier transform is given by
̂N a,Q (h) = exp
{i 〈a, h〉 − 1 }
2 〈Qh, h〉 , h ∈ H.
The Gaussian measure N a,Q is said to be non-degenerate if Ker (Q) = {x ∈ H : Qx = 0} = {0}.
A.3.1
Existence and Uniqueness
We can show that for arbitrary a ∈ H and Q ∈ L + 1 (H), there exists a unique Gaussian measure µ = N a,Q
in (H, B (H)). Observe that since Q ∈ L + 1 (H), there exists a complete orthonormal basis {e k} k≥1
on H and a
sequence of non-negative numbers {λ k } k≥1
such that
Qe k = λe k , k ∈ N.
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