The question of completeness of the spectral domain of harmonizable processes has been open for some years. An example is given of a harmonizable process whose spectral domain is not complete. This shows that a recent result which claims the completeness of all such spectral domains is false

Miamee, A. G.

Salehi, H.

English

NASA Technical Reports Server

c 
AN EXAMPLE OF A EfARMONIZABLE PROCESS 
WHOSE SPECTRAL DOMAIN IS 
NOT COMPLETE 
A. G. Miamee* 
Department of Mathematics 
Hampton University 
Hampton, VA 23668 
H. Salehi  . 
Department of S t a t i s t i c s  and 
Michigan S t a t e  University 
E. Lansing, 1.11 48824 
Probabi l i ty  
ABSTRACT 
The question of completeness of t he  s p e c t r a l  domain of harmonizable 
processes has  been open f o r  some years. 
ab l e  process whose s p e c t r a l  domain is not complete, 
a recent  r e s u l t  of M. M. Rao which claims the  completeness of a l l  such,spectral  
domains is false. 
We give an example of a harmoniz- 
This shows tha t  
(Y1SA-CR-184653) A8 EXAM€LB CI A N8 9- 1 4 800 
B A B H C P I Z A B L E  PRGCESS ISHOSE S€€C'(IEALL DOBAII 
1s 109 C C B F L I I E  (Haapton Inat-) 7 p 
CSCL 128 Unclab 
G3/64 01850E5 
AMs Subject C las s i f i ca t ion  (1980) 60, 40. 
Key Words and Phrases: Harmonizable Processes, Bimeasures, Spectral  Domain, 
*Supported by NASA Grant NAG-1-768 
Completeness 
https://ntrs.nasa.gov/search.jsp?R=19890005429 2020-03-20T05:01:24+00:00Z
b . '  
I L 
I 1 
I 
1. INTRODUCTION. The completeness'of t h e  s p e c t r a l  domain of s t a t iona ry  
T h i s - r e s u l t  has played an important r o l e  in t h e  
The completeness of 
I processes is w e l l  known. 
development of pred ic t ion  theory of these  processes. 
the  spec t ra l  domain of o the r  c l a s s e s  of processes is equal ly  important, This 
completeness quest ion f o r  t h e  class of harmonizable processes has been open '. 
f o r  many years.  It w a s  always f e l t  by t h e  authors  working i n  t h e  f i e l d  that 
L 
. 
by going from t h e  s t a t i o n a r y  processes .. t o  harmonizable ones the  completeness 
may be l o s t ,  see f o r  example t h e  work of Cambanis [l] and Cramet C23, 
r ecen t ly  M.M. Rao in C71 has claimed t h a t  the s p e c t r a l  domain of any harmoniz- 
However, 
> 7. 
. .  
ab le  process is complete. 
D.K. Chang and M.M. Rao. Doubt about t he  completeness of L (F) w a s  r a i sed  by 
one of us in a recent  review which appeared i n  1987.in Zbl, Math. (cf .  
This  w a s  a l s o  mentioned in' t h e  art icle C3l by 
2 
Zbl. Y616.60009). 
- .- - - .  - 
The main purpose of t h i s  note  is t o  g ive  an example of a.harmoniz-able 
Process whose spectral  domain is not complete. This w i l l  ahow that t h e  
proofs per ta in ing  t o  t h e  completeness property,giveza ia C71 and C31 are in ' 
e r ro r .  This w i l l  a l s o  put t o  rest any speculat ion about t h i s  important question. 
In t h i s  s ec t ion  w e  g ive  soqle preliminary r e s u l t s  concern= 
ing hannonizable processes, t h e i r  s p e c t r a l  bimeasures, and i n t eg ra t ion  with respect 
t o  bimeasures. For mor information t he  reader  is re fered  t o  t h e  articles of 
2. PRELIMINARIES. 
Rozanov C83, N i e m i  C61 and Chaw h Rao 133 
2 ;*- 
0 
L e t  L (P) be t h e  space of a l l  complex random va r i ab le s  X on a Probabi l i ty  
2 space (n, X, P) f o r  which Ex10 and E l X l  
Suppose that t h e  s t o c h a s t i c  process {Xt : t c R = real numbers) c Lo (P) admits 
<. =, E s tands  f o r  expectat ion operator ,  
2 
a harmonic representa t ion  
P 
$ # . . I  
(1) xt = jR eiteZ(dO), t . c  B, 
where Z(*) is a u-additive Lt (P) - valued measure. Le t  F(A,B) - E Z ( A ) m , '  
be the  bimeasure induced by Z(*). One can see t h a t  F ( a ,  A) and F(A, e )  are 
I L 
2 
scalar measures f o r  eaih- A 6 - Z  and that P(* , *)  is of bounde-d 'semi-variation. 
(See f o r  example [8]). Any process {X,, t 6 R) with a representat ion as in 
(1) is ca l led  weakly harmonizable, i t  is ca l l ed  s t rongly harmonizable if I n  
addi t ion  F ( * , * )  is of bounded va r i a t ion .  
2 
concentrated on the  diagonal X - 8 of R . 
s p e c t r a l  measure of t he  process. 
bounded va r i a t ion  i t  can be extended t o  a Radon measure on R2 and hence f o r  
It is ca l led  s t a t iona ry  i f  P ( * , * )  IS 
In a l l  these cases F is ca l l ed  the  
I n  case t h a t  our bimeasure F(*,*) is of 
any two bounded measureable f and g t he  i n t e g r a l  
jR(R f(e)g(i) '  FW, dh) 
can be defined as a Lebesque i n t e g r a l  and the  covariance of t h e  corresponding 
s t rongly  hanaonizable process {Xt }has t h e  representat ion 
F(de, dX), t , S E  R it0 - i s x  - (2) Extxs =/-ie 
When our process {Xt} is weakly harmonizable t h e  s p e c t r a l  measure F ( * , * )  
However is of bounded semi-variation and t he  representat ion (2) still holds, 
In  t h i s  case the  i n t e g r a l  m u s t  be given a new i n t e r p r e t a t i o n  in the  following 
sense due t o  Morse and Transue [SI. The Morse-Transue in t eg ra t ion  with respect  
t o  bimeasures and r e l a t ed  problems are t r ea t ed  i n  d e t a i l  i n  the comprehengive 
a r t i c l e  of D.K. Chang and M.M. b o  C33. The following d e f i n i t i o n s a n d  Theorem. 
are taken from C31, 
DEFINITION. 
bimeasure on Xl x Z2. I f  t he  funct ion fi: $ 2 1 3 '  is Xi measurable, then 
( f l .  f2) is s a i d  t o  be s t r i c t l y  F-integrable, provided the  following two 
conditions hold: (a) f l  I s  F(*B B) - in tegrable  f o r  each B E t2, and f 2  is 
F(A,*)  - in tegrable  f o r  each A L C1, that 
L e t  (ni, Xi) I - 1,2  be measurable spaces and F(*,*) be a 
: A+/i f 2 ( X ) .  F(A, dX) and 
(b) f is 
each N c .Zl, and t h a t  
- Integrable ,  f o r  each 14 E C2,  and f I s  FN - In tegrable ,  for 
1 2 2 
I 
f 
3 
(3) ’ fp) +le) = J f 2 ( h )  F;(dA). 
The common value i n  (3) is denoted by ki, f l ( e )  f20) F W ,  dh) e 
DEFINITION, Let {Xt , t E R) be a weekly harmonizable process, with 
spec t r a l  measure F ( *  , * )  . (F( e ,  0 )  is 
sets of R). Then following E31 and 
denoted by L @),is defined t o  be 
0 
2 
2 L (F) = i f :  R J C  
now a biueasure on BxB,’B be;ing the  Bore1 ’ 
L73 t h e  s p e c t r a l  domain of i X t ,  t q R}, . 
I. 
Cf , f )  is s t r i c t l y  F- integrab1.e). I 
For any s tochas t i c  measure 2 onB,L (2) stands f o r  t h e  set  of a l l  funct ions I s  1 
f : R + C  f o r  which the  i n t e g r a l  f ( e >  Z(d0) e x i s t s  ( i n  the  Donford and 
Schwartz sense C 4 3 ) .  
/R 
The following theorem proved by Chang and Rao i n  C31 (cf, PPI 34-44)iS ’ 
used t o  show t h a t  our exynple constructed in the  next . sec t ion”has  the  desired property. 
THEOREM. Let{Xt)be a weakly harmonizable process with s tochas t i c  measure 
L (Z) and f o r  1 2 as i n  (1) and s p e c t r a l  measure F ( * , * )  as i n  ( 2 ) .  
any two funct ions f and g i n  t h i s  space w e  have IR fR f ( e )  g(X) F(de, dh) = Then L2(F) - 
f ( e >  z (de) I, z A) 
( B e w a r e  t ha t  the  i n t e g r a l  i n  the  l e f t  hand s i d e  is i n  the sense of Morse - 
Transue). I n  t h i s  case L (F) can be considered as an inner  product space 2 
3.  EXAMPLE. Before w e  proceed t o  present %ur example of a harmonizable 
process whose s p e c t r a l  domain is not complete w e  need t o  prove a Lemma which 
is  e s s e n t i a l  f o r  es tab l i sh ing  our counter example. 
For any s t a t iona ry  s tochas t i c  process {Yns n E Z = in tegers ) ,  i t s  time 
I 
domain is t he  subspace H (i-) = S P  (Yk : k c  21, its past  subspace i s  the  
Y 
{Y, : k n) ,  and i t s  remote p a s t  is the  subspace H (--) = 
Y subspace H (n) = -- 
nHy(n). n 
Y 
The s t a t iona ry  sequence {Yn, n E Z} is ca l led  nondeterministic if 
. 
1 
4 
Xn 4 Hy(n-1) f o r  some and hence every n. It is called purely nondeterministic 
if Hy(-m) = 0. 
LEMMA. Let {Y,, n E 2) be a stationary stochastic process which is non- 
deterministic but is not purely nondeterministic, Then there exists a nonzero 
vector V in the remote past of H (-)of theprocess which cannot be expressed as 
a series of the form. . Y 
m 
= "kY,k ,. 
k=O 
PROOF. Suppose not, i.e., suppose any nonzero vector  in,^ (-a) has a 
series representation as in ( 4 ) .  Take a nonzero vector V in H (-,a). Then 
Y 
Y 
and hence 
where U is the usual shift operator on H (+-),associated with our stationary 
Process . Y On the other hand since Ut+% itself is in H (--) we can write 
Y 
m C - A Y,k. 
y! k=O at 
This contradicts our nondeterministic assumption. Hence the proof of *. 
the Lemma is complete. 
Now we are ready to give our counter example mentioned before. 
Take a stationary stochastic process {Y : n e 2) c Lo(f) which is non- 2 
deterministic but not purely nondeterministic. -Take any sequence A 
positive numbers which is summable, i.e. Z X < m. Define.a stochastic measure 
Z on Bore1 subsets of R which is Concentrated on the positive integerg I by 
n 
i E 2,of I' 
m 
i I- 1 
5 
Consider t he  harmonizable process {X t e R) c Lo(P) 2 given by Xt -d  eiteZ(d8) t '  
and its spec t r a l  bimeasure which is induced by Z ,  i .e . ,  
F(A,B)'= E Z(A) ZBT. 
2 W e  claim tha t  t he  corresponding spec t r a l  domain L (P) i n  t h i s  case is not complete. 
Verif icat ion.  By bur..Eemma there  e x i s t s  a nonzero vector i n  H (--) 'which does 
not have a series representat ion as i n  ( 4 ) .  Take one such vector V, Since V 
is c lea r ly  i n  H (0) there  exists a sequence C a Y 
2 combination of Y k ' s ;  k < 0 which converges t o  V i n  L (P). 
0 
Y 
of f i n i t e  l inear  n Y k -k ='n 
We can write - 
where the.nonzero functions f a r e  defined on pos i t ive  in tegers  with f (k) = 
n a By our Theorem i n  sec t ion  2 we have k' 
n n 
I I f n  - f m  It, = I1 vn - v m t t  
Now s ince  v converges t o  v and hence is Cauchy so is 
par t i cu la r  sequence f of functions i n  L (F) does not 
f i n  L (P). Because otherwise another appl icat ion of 
2 shows tha t  f is i n  L (Z) and 
n 
2 
n 
2 
1 
However t h i s  fn*  
converge t o  any element 
t he  Theorem i n  sec t ion  
Thus ye see tha tV  a l so  converges t o  fdz,  So 
JR OD 
n 
f dZ == ? f(i) Z ({ i ) )  = C f ( i ) h i  Ymi, 
i=o i=o 
which contradicts  our choice ofV,  
REMARK 11 Our example shows tha t  the  main r e s u l t  of C71 claiming the  
completeness of the  spec t r a l  domain of any mul t ivar ia te  weakly harmonizable 
process X is f a l s e  even f o r  a univar ia te  s t rongly harmonizable process, t 
. 2. W e  f e e l  t ha t  the  e r ro r  i n  C71 occurs i n  l i n e s  8 and 9 of the  
secopd column of page 4612 ,  where the  existence of a "cer ta in  project ion onto 
. 
a subspace" is asser ted  and a reference t o  page 33 bf 194 is.made t o  
support it. In  view of the r e s u l t s  established i n  t h i s  note the  r e s u l t s  i n  
- 
4 6 . 
2 
t he  later p a r t  of C33 which are based on t h e  completeness of L (P) deserves 
re inves t iga t ion .  
REFERENCES . .  
1. Cambanis, S. (1975). The Measurabili ty of a' Stochas t ic  Process of ,  Second 
Order and Its Linear Space, Proc.'Amer. Math. SOC, 47, 467-475. 
Proceeding of Second Berkeley Symp. Math. S t a t .  Prob., 329-339. 
t c  
2. Cramer, H. (1952). A Contribution t o  t h e  Theorylof S tocha i t i c  Processes,  *. 
3. Chang, D.K. and h o ,  M.M. (1986). BirPeasures and Noastotioxiary Processes,  
Real and Stochas t ic  Analysis, Wiley Ser,  Probab, Math., Sta t , .P rob .  Math, 
S t a t . ,  7-118, ed i t ed  by M.M. bo. 
Dunford , N. and Shuartz , J.T. (1958) . Liheat  Operators, P a r t  I: General 
Theory, Wiley-Interscience, New York. 
4. 
5 .  Morse, H. and Transue, W. (1956), C-Bimeasrues and Their I n t e g r a l  Extensions, 
b * M a t h .  64, 480-504. - -  
6. N i e m i ,  H. (1975). Stochas t ic  Processes as Fourier  Transformations of 
S tochas t ic  Heasures,Ann. Acad. Sci .  Fenn. AI Math >sD 1-47. 
Rao, M.M. (1984). The Spec t r a l  Domain of Multivariate HarPlonizable Processes,  
Proc. Nat. Acad. Sci.  '(U.S.A,) 8 l ,  4611-4612. 
Prob. Appl. 4, 271-287. 
7. 
8. bzanov,  Yu. A. (1959). Spectral Analysis of Abstract Functions, Theor. 
9. Rozanov, Yu. A. (1967). Sta t ionary  Random Processes,  ttoldan-Day, Sea 
- 
Francisco Cal i forn ia .  
. .  


An example of a harmonizable process whose spectral domain is not complete

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