59.21. POSITIVE DEFINITE FUNCTIONS, BOCHNER’S THEOREM 1937

Then this is a random variable whose covariance matrix is just Σi j = ( fi, f j)H and whosecharacteristic equation is e−

12 t∗Σt so this verifies that

(W ( f1) ,W ( f2) , · · · ,W ( fn))

is normally distributed with covariance Σ. If you have two of them, W (g) ,W (h) , thenE (W (h)W (g)) = (h,g)H . This follows from what was just shown that (W ( f ) ,W (g)) isnormally distributed and so the covariance will be(

| f |2 ( f ,g)( f ,g) |g|2

)=

 E(

W ( f )2)

E (W ( f )W (g))

E (W ( f )W (g)) E(

W (g)2) 

Finally consider the claim about independence. Any finite subset of {W (ei)} is gener-alized normal with the covariance matrix being a diagonal. Therefore, writing in terms ofthe distribution measures, this diagonal matrix allows the iterated integrals to be split apartand it follows that

E

(exp

(i

m

∑k=1

tkW (ek)

))=

m

∏k=1

exp(itkW (ek))

and so this follows from Proposition 59.11.1. Note that in this case, the covariance matrixwill not have zero determinant.

59.21 Positive Definite Functions, Bochner’s TheoremFirst here is a nice little lemma about matrices.

Lemma 59.21.1 Suppose M is an n×n matrix. Suppose also that

α∗Mα = 0

for all α ∈ Cn. Then M = 0.

Proof: Suppose λ is an eigenvalue for M and let α be an associated eigenvector.

0 = α∗Mα = α

∗λα = λα

∗α = λ |α|2

and so all the eigenvalues of M equal zero. By Schur’s theorem there is a unitary matrix Usuch that

M =U

 0 ∗1. . .

0 0

U∗ (59.21.51)

where the matrix in the middle has zeros down the main diagonal and zeros below the maindiagonal. Thus

M∗ =U

 0 0. . .

∗2 0

U∗