Table Of Content-1 --
1 POSITIVE DEFINITE SEQUENCES
Denote by Z the set (... -1.0,1,...)of the integers.
Definition 1.1. A sequence (an) n s Z of complex
numbers is called ~ositive definite if~ for each
system of complex numbers zl,...,z N we have
N
- an_mZnZ - m ~0
n,m-1
Let (F,B,p) be a probability space (i.e. a set F, a
Borel-field B of subsets of F and a real valued, positive,
~-additive, normed set function p from B to the reals)
and X n :F-@C (n ~Z) a sequence of random variables
(i.e. measurable functions from the set F to the set
C of complex numbers).
Definition 1.2. The sequence (Xn) n G Z is called stationary
iff
EX n = O, ~X n < ~ (n ~Z)
and
(n,m,k~ Z) .
In addition we say that
(n G z)
r n ~ coV(Xn,X o)
is the covariance s~qRenc ~ generated by the stationary
sequence (Xn) nE Z "
Theorem 1.1. The covariance sequence (rn)n~ Z has the
following properties
1. rn ~ ~-n (neZ)
2. (rn)nE Z is positive definite .
-2-
Proof: 1. is trivial, 2. follows from
N N
r n_mZnZ- m - ~ EXn-XmZn~m
1=m,Lr n,m=l
N
2
_'It ~ 20
- ,
k=l JL, ,IL.
2 HERGLOTZ" THEOR~
Theorem 2.1. A sequence (an) n (cid:12)9 Z of complex numbers is
positive definite iff there exists a finite
Borel measure m on the unit interval 0,1with
1
a n - ~ e2Wintm(dt) (n = Z)
O
Given ~an)ne Zjm is uniquely determined.
Proof: Let (an) n e Z be a positive definite sequence
of complex numbers. Define
I N
(1) hN(t):- ~ ~ an_me 2 i(m-n)t (N=1,2,...
n,m=1 0~t~l) (cid:12)9
Since (an)me Z is positive definite, we have (choose
z n - e -~int (a- I,...,~)) ~(t) '~ o.
We continue by defi~ing a sequence of Borel measures
m~ (N - 1,2,...). The value of ~N on a measurable set
G ~ C0,11 is
~(G) - f ~(t)dt (cid:12)9
G
Then mN(0,1) - a o . ~ow it is well known that the
-2-
Proof: 1. is trivial, 2. follows from
N N
r n_mZnZ- m - ~ EXn-XmZn~m
1=m,Lr n,m=l
N
2
_'It ~ 20
- ,
k=l JL, ,IL.
2 HERGLOTZ" THEOR~
Theorem 2.1. A sequence (an) n (cid:12)9 Z of complex numbers is
positive definite iff there exists a finite
Borel measure m on the unit interval 0,1with
1
a n - ~ e2Wintm(dt) (n = Z)
O
Given ~an)ne Zjm is uniquely determined.
Proof: Let (an) n e Z be a positive definite sequence
of complex numbers. Define
I N
(1) hN(t):- ~ ~ an_me 2 i(m-n)t (N=1,2,...
n,m=1 0~t~l) (cid:12)9
Since (an)me Z is positive definite, we have (choose
z n - e -~int (a- I,...,~)) ~(t) '~ o.
We continue by defi~ing a sequence of Borel measures
m~ (N - 1,2,...). The value of ~N on a measurable set
G ~ C0,11 is
~(G) - f ~(t)dt (cid:12)9
G
Then mN(0,1) - a o . ~ow it is well known that the
-3-
Borel measures on 0,1 are linear functionals on the
space C(0,I~) of continuous, complex valued functions-
on 0,1 . Using this picture the topology induced
by the weak convergence of measures is often called the
w ~ - topology in the dual space of C(0,1). By the
w - compactness of the unit sphere we can choose a
subsequence (mNk)k=1'2'''" of (m N)N=I,2,... , which
converges weakly to a certain measure m :
mNk--* m (weakly) .
Hence
I I
2Tist .~_.
aril (cid:12)9 mNk~a~J = I e2WiStm(dt) (s ~ Z) .
k ~ o J o
For any fixed s and N k (cid:12)9 s it is easy to prove that
1
~ e 2Tist (dr) = i
o kNm Nk as(Nk - s)
because for Nk-S terms under the sum (I) we have
m-n = s and the remaining terms yield 0 after integration.
After all ahyiomsl~j
1
e2TiStm(dt) - a
S
and we have finished the first part of the proof.
Now let us assume that we are given a finite Borel
measure m on 0,qwith
1
a n = ~e2~intm(dt) (n~ Z) .
O
We have to show, that (an)ng Z is a positive definite
sequence. But
N N 1
an_mZnZ-m = ~ ~ e 21ri(n-m) m(dt)Zjm =
n,m=l n,m=q 0
-4-
q N
~ z e2"iktl 2 m(dt) s0 .
o k=q k
The uniqueness of m follows, since each linear functional
f(t)m(dt) on C(0,1) is uniquely determined by its
values on the generating set of functions (e2Tikt)k e Z "
Corollary 2.1. Let (rn) n m Z be a covariance sequence.
There exists one and only one measure m on 0,q
with
1
rn = I e2~intm(dt) (n eZ)
0
Definition 2.1. Let (Xn) ne Z be a stationary sequence of
random variables defined on a probability space
(F,B,p) , (rn) n E Z the generated covariance sequence.
If
1
r n = f e2~intm(dt)
0
then m is said to be the spectral measure corresponding
to z "
Theorem 2.2. Every finite Borel measure m on 0,1is
a spectral measure corresponding to a certain
stationary sequence (Xn) n e Z on a suitable
probability space.
Proof: If m(0,11) = ,O let X n m 0 (neZ). If m(~0,13) =
c > O,then ~m is a probability measure on K0,11. Let
Y be a real valued random variable with distribution
m ~d let ~ be a random variable with values 1 and -1
1
and distribution p((Z=q}) = p((Z=-I}) = ~ . Let Y and ~
be independent(this is the only necessary property of
the underlying probability space~. We now define
-5-
X n = ~Se 2~inY (ne )Z
and show that~n)n e Z is the desired stationary
sequence. Indeed, since E~ = 0 it follows that
EX n = c EZEe 2 inY = 0 (neZ)
and, in addition,
coV(Xn,Xk) = EXnX k = cEZ~i(n-k) Y
1
.e -c i(n-k) 2 - elc 2 (n-k)t m(dt)
1 o
=~ e2~i(n-k)tm(dt) (n,k e Z)
C
depends only om the difference n-k.So we conclude~
that (Xn) n a Z is stationary and m is the corresponding
spectral measure.
Let ~n)n e Z be a stationary sequence of random variables
defined on a probability space (F,B,p). In the sequel
we denote by ~ the subspace of ~ generated by (Xn) naZ.
Theorem 2.3. Let (Xn) n e Z be a stationary sequence and
m the corresponding spectral measure. There exists
2 e2Wint
an isomorphism 1 : ~ ~-- L m with l(Xn) =
(nez).
Proof: 1. If X n = X k (mod p),then e 2Tint = e 2~ikt (mod m)
according to
1 1
J lm(dt) = cov(Xn,Xn) = coV(Xn,Xk) = ~ e2~i(n-k)tm(dt),
0 0
(n,k G Z)
2. Hence the mapping
i : (~I neZ}
.._ e 2~int (nez)
X n
-6-
is well defined. This mapping can be extended in a
natural way to
N
1 , ( ;q. akXkl ~ integer, a k ~ C (k - -~,... )~, }
N-,,k
N
-'~ { .~ ak e2~ikt I N integer, a k e C (k = -N,...,N).}
k=-N
The inner product is invariant under I according to
N N N I
j,k=-N ~ ~ 0
N N
Z ka east, ~ m)tJi"2e~b
"(
(cid:12)9
k=-N N---j
SO we have established an isomorphism between a dense
subset of ~ and a dense subset of L 2, which is enough
to prove the theorem.
3 VECTOR VALUED MEASURES
Let H be a Hilbert space.
Definition 3.1. ~HTvalued function M, defined on
the Borel subsets of s , is said to be a
measure with crthc~onal val~es (m.o.v.) iff
1.For every sequence EI,E2,... of disjoint Borel
subsets of the unit interval
n
~M(E )k (n=1,2,...)
1=k
converges (with respect to the norm of H) and
(we write E~F = E+F iff E and F are disjoint)
it follows
-6-
is well defined. This mapping can be extended in a
natural way to
N
1 , ( ;q. akXkl ~ integer, a k ~ C (k - -~,... )~, }
N-,,k
N
-'~ { .~ ak e2~ikt I N integer, a k e C (k = -N,...,N).}
k=-N
The inner product is invariant under I according to
N N N I
j,k=-N ~ ~ 0
N N
Z ka east, ~ m)tJi"2e~b
"(
(cid:12)9
k=-N N---j
SO we have established an isomorphism between a dense
subset of ~ and a dense subset of L 2, which is enough
to prove the theorem.
3 VECTOR VALUED MEASURES
Let H be a Hilbert space.
Definition 3.1. ~HTvalued function M, defined on
the Borel subsets of s , is said to be a
measure with crthc~onal val~es (m.o.v.) iff
1.For every sequence EI,E2,... of disjoint Borel
subsets of the unit interval
n
~M(E )k (n=1,2,...)
1=k
converges (with respect to the norm of H) and
(we write E~F = E+F iff E and F are disjoint)
it follows
-7-
2. (M(EI),M(E2)) = 0 whenever EI~E 2 = ~ (cid:12)9
We continue with several remarks.
a) If M is a m.o.v, and q(E) = I~M(E)U 2 ( E Borel in(O,1 )
then q is a measure in the usual sense. This is
a consequence of the formula
q(nEk) . iU(~.Ek) | 2 = i~U(~_,k)l I 2 . z~a(~), 2
= z q(~:)
b) We may define an integral with respect to M as follows:
N
1. Let f - ~cbl . (E k (k=l,...,N) disjoint and Borel,
k=l = =k
c k (k=l,...,N) complex numbers,
I E the indicator function of E)
be a simple function. Put
)td(m)tCf )k~(mkO1j~--~k
(cid:12)9 Then
=
1 N
(2) no j fCt)uCdt)il 2 . ~lCkl21uC%)U 2
N 1
. ,~lCki2q(Ek) = ~ If(t)i 2 q(dt)
1 o
2. Let f (cid:12)9 . There exists a sequence (fn)n=l,2,...
consisting of simple functions with fn *-" f (n --*~ )
2
(convergence with respect to Lq norm). We put
I I
(3) J f(t)M(dt) = llm fn(tlMCdt)
0 ~ ~ 0
and, according to formula (2), we have
-8-
1 1
~ ; fn(t)M(dt) - f fm(t)M(dt)I 2
o o
= ~ i fn(t) _ fm(t) ~ 2 q(dt) .
o
_2
Since (fn)n=l,2,... is a f~udamental sequence in L~ ,
1
( fn(t)~dt))n=1,2,..__ is a fundamental sequence in H
o
and(5) makes sense. The same argument ensures us that
the integral (3) does not depend on the choice of
(fn)n=l'2''''" 1
f'-~I
c) The mapping f(t)M(dt) from Lq 2 into H is linear
o
and isometric.
1 1
o o
1 1 1
o o o
f) Let H I and H 2 be Hilbert spaces and 11 an isomorphism,
11: H 1 --r 2 . If M I is a m.o.v, in H I then M2(E) =
ll(M(E)) is a m.o.v, in H 2 with
1 1
11( f(t)M1(dt)) = J f(t) a(dt) (cid:12)9
o o
To prove this assertion,note that M 1 and M 2 will
correspond to the same Borel measure q, because the
latter depends only on the (ll-invariant) norm. Furthe~-
more for simple functions the asserted equation is trivial.
Now here are a few examples:
1. Let H = ~ , m a measure defined on t~ ~orel field B
in 0,1 , and Mo(E ) = 1E. Orthogenality is trivial;
also we have
M(~Ek) = 1 ~E k = Z 1Ek = ~(M(Ek) and
