A R K I V F O R NIATEMATIK Band 8 nr 1
1.68 10 1 Communicated 5 June 1968 by H~Ax~ C ~ - ~ and ULV GRE~A~rDEJt
On the Hellinger integrals and interpolation of q-variate stationary stochastic processes
B y HABIB SALEHI
I n t r o d u c t i o n
Let (X~)~¢@ be a q-variate continuous parameter, mean continuous, weakly stationary stochastic process (SP) with the spectral distribution measure F de- fined on B the Borel family of subsets of the real line; cf. [1]. I t is known [10] t h a t for matrix-valued measures M and ~V the Hellinger integral (M, N) = S-~
(dMdN*/dF)
(*= conjugate) m a y be defined in such a way that
H~.F
the space of all matrix- valued measures M for which (M,M)F= S~-~ (dMdM*/d-~)
exist becomes a Hil- bert space under the inner product T(M, N) 2 (~ = trace). The significance of these integrals when M and ~V are complex-valued measures and lv is a non-negative real-valued measure has been pointed out by H. Cramgr [2, p. 487] and U.Grenander [3, p. 207; 4, p. 195] in relation to unvariate SP's. The importance of Hellinger integrals with regard to the theory of interpolation of a q-variate weakly stationary SP with discrete time has been discussed by the author in [11]. In this paper we propose to use the Hellinger integrals and obtain similar results concerning the interpolability of a q-variate continuous parameter, mean con- tinous, weakly stationary SP. The question of interpolability of a univariate SP with continuous time has been looked at by K. Karhunen [6]. Our results extend his work in a natural way.
Let K be a n y bounded measurable subset of the real line.
K'
will denote the complement of K in the set of the real numbers. ~ K and 7hE" will denote the (closed) subspaees spanned by Xt,t6K
and Xt, t E K ' respectively, i.e., 7~K=~{Xt,
tEK}
and ~ K ' = ~ { X t ,tEK'}. 7H¢@
will denote ~{Xt, t real} and finally S ~ will denote 7H~ N 7n~., where 7n~:, denotes the orthogonal complement of 7~K' in a fixed Hilbert space ~/~ containing the SP (Xt)-~.Definition 1.
We say t h a t (a) K is interpolable with respect to (w.r.t.) ( X t ) ~if ~={0}.
(b) (Xt)_~ is interpolable if each bounded measurable subset K of the real line is interpolable w.r.t. (Xt)_~.
For each X 6 ~ ¢ ~ ,
(X, Xt)
is a continuous function on ( - c ~ , c~). Moreover, (X, Xt)= 0 iff t ilK'. Thus the following definition makes sense.H. SALEm,
On the Hellinger integrals De/inition 2.
For each X e }lK, we letPx(2) = f ?** e-'~'(X, :L)dt
=
The properties of
P x
are given in the next lemma.Lemma 1. (a)
The entries o/ the matrix-valued /unction P x are integrable
w.r.t.Lebesgue measure. Hence ]or each B 6 B, the integral ~BPz(I)d2 exists.
(b)
I/ /or each Be B we de/iue
Mpx(B) = fBPx(2)d2,
then MPx E H~. r.Proof.
(a) Let X G ~ and tF be in /~.F such thatV~F=X,
where F is the isomorphism on L2.p onto ~ [9, pp. 279-98]. Then(X, Xt) = (VW, Ve-aq
= 1 (W, e-"a)r(1) Also by definition of
Px
Px(2) -- e-aqX, Xt)d~.
(2)
B y (1) and (2) it follows that for each B E B
(3)
Thus (a) follows from (3).
(b) Since b y (a) for each BfiB,
SsPx(~)d2
exists, therefore Me~ is a matrix- valued measure on B. B y the definition of M~x, (3) and [10, Theorem 2] (b) follows.(Q.E.D.)
Thus the following definition makes sense.
Definition 3.
We define the operator TK on ~K into H2.F as follows: for each X E ~1
The important properties of T k are given in the following theorem.
2
~kRKIV FOR MATEMKTIK. B d 8 nr 1 Theorem 1. (a) Let X 6 ~ and 1F 6 L~. F such that l z ~ = X, where V is the isomorphism on L2.~ onto 71L¢ [9, pp. 297-98]. Then ]or each B 6 ~, Me~(B)=
(b) TK is an isometry on ~ into H2.p. I n /act /or all X and Y in ?IK (X, Y ) = (T~:X, T K Y)~,.
(e) The range o/ T~: is a closed subspace o/ the Hilbert space H2.F.
Proo/. (a) follows from the proof of L e m m a 1.
(b) L e t X and Y be in 7/K, and let (I) and LF be in L2.Fsuch t h a t VqD=X a n d VuF = Y. Then b y (a) and [10, Theorem 1]
2st(Tz, Tr)F = (¢, ~F)F. (1)
Also b y [9, p. 297]
2 n ( x , y ) = (op, ~F)F. (2)
F r o m (1) and (2), (b) follows.
(c) Since ~K is a (closed) subspace and since b y (b) T~ is an isometry on
~K into H2.F, therefore range of T~ is a closed subspaee of H2.p. (Q.E.D.) I t is convenient at this point to introduce the following definition.
De]inition 4. (a) A q × q matrix-valued function P on ( - oo, oo) is called time- limited if
(i) The entries of P are integrable w.r.t. Lebesgue measure.
(ii) P ( t ) = f ~_ ¢~ e-~tG(t)dt,
where G is a q × q matrix-valued function whose entries have bounded supports and are square-integrable w.r.t. Lebesgue measure.
(b) C will denote the class of all time-limited q × q matrix-valued functions on ( - oo, oo).
(c) for each P E E the matrix-valued measure Mp is defined on B as follows;
for each B 6 B
Me(B) = f~ P(1)dl.
We note that if X 6 7 ~ K and Px(1)f]_~oe-t~(X, Xt)dt , then b y L e m m a 1 (a),
2xe£.
L e m m a 2. Let X 6 7t~ 0 7tL. Then
H. sAt.~.m, On the Hellinger integrals
Proo/. I t is clear t h a t ~ x t ] ~ z = 7 / K n z . Hence T K X = T ~ : n z X = T z X . (Q.E.E.) Making use of this lemma, Tx's m a y be put together to introduce a well- defined operator with a bigger domain. This is done in the following theorem.
Theorem 2. Let 7 / = U ~ x , where K is a bounded measurable subset o / ( - oo, oo).
De]ine the o,perator T on 7,l by
T X = T ~ : X , i/ X 671~:.
Then
(a) ~ is a linear manqolg in ~ , i.e., X, Y e ~ and A, B matrices ~ A X + BYE71.
(b)
T is a single-valued linear o~ra$or on ~, i.e., q X, Y E74 and A, B are nuarices, thenA ( A X + B Y ) = A T X + B T Y .
(c) T is an isametry on 71 into H~.~. In /act /or X, Yfi71 (X, Y)= (TX, TY)z,.
(d) The range o] T con4i~ o] all matrix-valued measures M1, /or which the Hellinger integrals S~_~ (dM~dM*/d]) exit where P e E, E is as in de/inition 4(b) and Me is related to P as in de/inition 4 (c).
Proo]. (a) follows from the fact t h a t ~ x U ~ r _ ~ ~EuL.
(b) and (c) are consequences of Lemma 1 and Theorem 1.
(d) L e t X E ~ . Then X E ~E for some K. I t then follows from the definition of T t h a t
T X = T x X - - MPx, (1)
where for each B f B , Mpx(B)=~nPx(2)d2 and Px(2)=~-o~e-a~(X, Xt)dt. Since b y Theorem 1 (a) the entries of P x are integrable w.r.t. Lebesgue measure, hence P x £ 1 : . From (1) and (c) it follows t h a t ( X , X ) = (Mpx, M~x)~ and henc~
(Mex, Mex) is Helllnger integrable w . r . t . F .
Conversely let Mp be a matrix-valued measure such t h a t for each B fi]~
~(B) = fBe(,~)aa,
where P 6 l: and f _ ~ (dMpdM*/d]) exists. Then by [10, Theorem 1 (c)], ¢ = (dMe/dl~) (dF/dl~)- £/~. F, where /~ is a n y ~-finite non-negative real-valued meas- ure w.r.t, which M e and P are a.c. {(dF/d/~)- denotes the generalized inverse of dF/d/u; cf. [8]}. If X 6 ~ such t h a t V O = X , where V is as in Theorem 1, then b y [9, p. 297] and [10, Theorem 2]
4
ARKIV Ffil~ MA.TEMATIKo Bd 8 nr 1 1 e_i~t) F
( x , x~) = ~
(o,
if:.
= 2---~ (dMe/dt~) (d-~/dl~)- (dF/d#) eatdl ~
= 2~ e'~qdM~/d#) d#
Since P E £ then
where the entries of G have bounded supports and are square-integrable w.r.t.
Lebesgue measure. I t then follows that
G(t) = ~ f_ ~ e'atP(2) dl.
(2)B y (1) and (2) we conclude t h a t
(X, Xt) = G(t) a.e.
Therefore the entries of (X, Xt) have bounded supports and hence their supports are contained in [ - e , e] for some e > 0 . Since X is in 77/o~ it follows t h a t XG~/c_~.~ ] and therefore X E ~ l = t ) K ~ . I t is clear that M~,x=Mp and the re- sult follows. (Q.E.D.)
We are now ready to give a characterization for the interpolability of a SP.
Theorem 3. ( X t ) ~ is interpolable i]] /or any time-limited matrix-valued ]unction P /or which Mp is not a null-point in He, F, (Me, Me) is not HeUinger w.r.t. ~.
Proo[. ( ¢ ) I f K is a n y bounded measurable subset of ( - ~ o , ¢0), it is a consequence of Lemma 1 {b) and Theorem 2 (d) that ~/K = {0}. Hence b y de- finition 1 (a) K is interpolable w.r.t. (Xt)_~o~. Since K is a r b i t r a r y it foUows t h a t ~/= U K = {0} so that b y definition 1 (b), (Xt)_~o¢ is interpolable.
( ~ ) I t follows t h a t ~ = {0}. Hence b y Theorem 2 (d) range of T = {0}. The result follows from Theorem 2 (c). (Q.E.D.)
Remark 1. Since U ~ / t .... ~-- U K~/K = ~ and since b y [7, Theorem 10] P is a time-limited matrix-valued function in the form P(2)=S~_~e-ttaG(t)dt if the en- tries of P(2) are integrable as well as square-integrable w.r.t. Lebesgue measure and P(z)=o('l~l), where P(z) is the unique analytic extension of P(2), we im- mediately obtain the following theorem, which generalizes the corresponding
H. SALEHI, On, |h~
Hellinger integrals
T h e o r e m 4. (Xt)_~=o is interpolable i// /or any analytic matrix-valued /unction P ( z ) o~ the /orm P(z)=o(e "lzl) ~uch that the entries o/ P(~) are integrable as well as 8quare-inte~rable w.r.t. Lebesgue measure i/ Mp is not a nulLpoint in tt2. F, then (Mp, Mp) is not Hellinger integrable w.r.t. F.
A C K N O W L E D G E M E N T
This research was p a r t i a l l y s u p p o r t e d b y National Science F o u n d a t i o n GP-7535 a n d GP- 8614.
M{ch{gan 8tc~e University, Eea~t Lan~ing, .Michigan 48823, U~A
R E F E R E N C E S
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T r y c k t d e n 20 december 1968 Uppsala 1968. Almqvist & Wiksells Boktryckerl AB
