on the alternating projections theorem and … · 2018. 11. 16. · bivariate stationary stochastic...

ON THE ALTERNATING PROJECTIONS THEOREM ANDBIVARIATE STATIONARY STOCHASTIC PROCESSES

BYHABIB SALEHIC)

Summary. In this paper we shall first use the theorem of von Neumann onalternating projections to obtain an algorithm for finding the projection of anelement x in a Hubert space Jtf onto the subspace spanned by JP-valued orthog-onally scattered measures £x and |2. We then specialize this algorithm to the casethat £x and |2 are the canonical measures of the components of a bivariate stationarystochastic process (SP), and thereby get an algorithm for finding the best linearpredictor in the time domain.

1. Preliminary results. In this section we state some results which are used inlater sections.

The following theorem is due to J. von Neumann (cf. [7, p. 55]).

Theorem 1.1 (Alternating Projections). Let Px, P2, and T be projectionoperators on a Hilbert space Jf onto the subspaces Mx, Ji2, and Jíx n Jí2. If Tnis the nth term of either of the sequences

Fx, F2PX, PXP2PX, P2PXP2PX, • • •,

F2, PiP%, P 00.

This at once yields the following corollary, which we need later.

Corollary 1.2. With the notation of 1.1, if P is the projection operator on 0, p(X — h, X + h)>0}. Then (a) the limit D(X, h), ash^yO, exists (finite) almost everywhere with respect to p (a.e. p).

Received by the editors June 15, 1966.P) Publication supported by Public Health Service Research Grant No. NIH-GM-13138-

01 from the National Institute of General Medical Sciences.(2) I.e., for each x in Jf, \Tnx—Px\ -* 0, as n -* 00, | | being the norm in Jf.

121

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

122 HABIB SALEHI [July

(b) Ifv is absolutely continuous (a.c.) with respect to p and if the Radon-Nikodymderivative ofv with respect to p. is dv\dp, then

(dv\dp)() = lim D(-,h), ash-^O, a.e. p..

We shall denote lim D(-, «), as « -* 0, by (Dv\Dp)() and shall call it the Besi-covitch derivative of v with respect to p.

The following is an immediate consequence of 1.3.

Corollary 1.4. Let (i) M( (/=1, 2) be a a-finite, ca., nonnegative measure onthe family 38 of Bor el subsets of the real line R.

(ii) 38i = {B: Beí%i and Mt(B)

1967] ON THE ALTERNATING PROJECTIONS THEOREM 123

Theorem 2.2. If £P is the set of all stochastic integrals ¡R 9 d£ then(a) \R # exists iff jB \9\2 dM< oo.(b) JB (a + bf) di; = a \R dÇ + b lRy d(¡, where a and b are complex numbers.(c) if=the cyclic subspace spanned by Ç, i.e.,

£P = ^(B);Be380}.

(d)(ÍR9de,¡RVd?)=¡R94>dM.(e) The correspondence -*- JB di is an isomorphism from L2(R, 38, M) onto if

such that (a) and (b) hold, and therefore

If ¿dt? = f \9\2dM.\Jb Jb

Let x be in ^f and {£(m)} be an orthonormal basis for a subspace Jt ofJ*P. Then2m \(x, f(m))|2


Corollary 2.4. Let (i) | be an o.s. Jf-valued measure over (R, 38) whose asso-ciated spectral measure M is a-finite.

(ii) e L2(R, 38, M).Then (a) for each x e Jf, the complex-valued function (f, x) defined in 2.3 is a c.a.measure over (R, 38) which is a.c. with respect to M.

(h) e L2(R, 38, M)and by 2.3(c), d(& x)\dM e L2(R, 38, M), therefore

H ■ (£■'* W«) -J>t ww(2)

= JR ft«) ̂ f (


We shall generalize equation (A) to the case in which J(x and Jt2 are spannedby such measures t-x and f2. We shall show that under certain conditions we getinstead of (A) :

-/.{/.^w%? fco**»»}«*»

where MX2, M2X are the product measures Mx x M2, M2 x Mx, and d( )/d denotesthe Radon-Nikodym derivative.

Notation 3.1. 38 will denote the family of Borel subsets of the real line F.3^ will denote a fixed Hubert space. (t (i=l, 2) will denote an o.s. ^f-valuedmeasure over (F, J1) whose associated spectral measure M¡ (j"=1, 2) is a-finite. Weset

3St = {B : Be38 & Af((F)


The next lemma whose proof is easily seen shows that (f¡, f,) is finitely additiveon My.

Lemma 3.4. (a) (|(, $,) is finitely additive on 3ii} and is a.c. with respect to Myon Mi,.

(b) (ii, èj) is the unique finitely additive extension of the measure defined in 3.2 on38iX38j toSHi,.

To proceed further, some restriction has to be imposed on £i and £2.Assumption 3.5. There exist functions


Theorem 3.9. With the notation of 3.1, if i¡ (i—I, 2) are o.s. J^-valued measuresover (R, 38) satisfying Assumption 3.5, and if (it, $,) 's me CM- measure on 38^ givenby 3.7, then for each x in ¿P

(a) d(x,ii)¡dMieL2(R,38,Mi),

d^dM~(-) ^Èff {'t)£Ll(R'a-Mi) a'e-t(Mi)'

L ÍL d-W « W «• *>*«*>} %7(?' ■>***>e L*(* *** • • • •(b) If P¡ (i=l, 2) is the projection operator onto the cyclic subspaces Jii (i=l, 2)

which is spanned by £¡ (/= 1, 2), then

rf* - 5. ii. %? w %?

128 HABIB SALEHI

By the Schwarz inequality, we have

|¿(x,¿ri),J|¿íii,£2)

[July

dMx(s,t) Mx(ds)f P^ (s)\

)n\ dMx {)\

where by (1), C=[\R \(d(x, £x)ldMx)(s)\zMx(ds)]112 *.«*>}«*>■This completes the proof of the first relation in (a) and the first two relations in (b).

To obtain the expression for P2PxX given by (b), we essentially made use of thefact that

Mn — ess. lub.tsR dM12 (;')

< 00.2,Mj

d(x, fi) (■)eL2(R,38,Mx),dMx



Now since

A/i - ess. lub.dM2 (•.«)

< oo,2,M2

¿Vl* = f


Let Jt(t) be the past-present subspace of the SP (x(t), teR)up to time t in Jt",i.e., Jf(t) =


Px2(t). Applying the results of §3 we will be able to obtain convergent infiniteseries expressions for Pxx(t) and Px2(t), and hence the best linear prediction Tx(r)is determinable.

Since (\(t), t e R) is purely nondeterministic, so are its component processes(xx(t), t e R) and (x2(i), t e R) (cf. [11, p. 79]). Accordingly £x and £2 will denotethe canonical measures of the processes (x^r), t e R) and (x2(t), t e R). AlsoJ(x(t) and J(2(t) will denote the past-present subspaces of (xx(t), t e R) and(x2(t), t e R) up to time t in Jf.

We will be able to effect our algorithm under the following assumption.Assumption 4.3. The spectral density F' = [Fy], l¿i, j¿2, of the bivariate,

purely nondeterministic SP (x(i), t e F) satisfies the condition

FÍ2¡(F'xxy>2(F22r2 eLx(R, 38, Leb)(4).

Remark 4.4. We assert that the SP has rank 2. For if its rank were less than 2,then F'XXF22 - F[2F2X = det F' = 0a.e. on F (cf. [11, p. 36]). Hence F[2l(Flx)ll2(F22)112= 1


Proof. Let Yl0)=(l¡2ir) füa e~ÜÁ (Fy(A)/&(À)&(A)) dX. We define 9ij on R2 by

Leb-A*w«ere t (/'= 1, 2) is the generating function of the component process (x¡(í), t e R),Hi^2.

(iv)


Then for each r 2:0,

Fx¡(r) = f Ci(r+s)ii(-ds)+ j j f ci(T+5)yiXr-j)d'i|^(-a'j)

-J if |J ci(T + s)yit(t-s)ds\yti(u-t)dt\ii(-du)+---.

Proof. By the third paragraph of §4 and 4.1(b),/» CO

Xi(r) = I Ci(r + s)Íi(-ds),

(1)Jlifi) =

134 HABIB SALEHI

4. L. H. Koopmans, On the coefficient of coherence for weakly stationary stochastic processes,Ann. Math. Statist. 35 (1964), 532-549.

5. P. Masani and J. Robertson, The time domain analysis of a continuous weakly stationarystochastic process, Pacific J. Math. 11 (1962), 1361-1378.

6. R. F. Matveev, On multi-dimensional regular stationary processes, Theor. ProbabilityAppl. 6 (1961), 149-165.

7. J. von Neumann, Functional operators, Vol. II, Annals of Mathematics Studies No. 21,Princeton Univ. Press, Princeton, N. J., 1950.

8. F. Riesz and B. Sz.-Nagy, Functional analysis, Ungar, New York, 1955.9. M. Rosenberg, The square-integrability of matrix-valued functions with respect to a non-

negative hermitian measure, Duke Math. J. 31 (1964), 291-298.10. H. Salehi, The Hilbert space of square-integrable matrix-valued functions with respect to

a o-finite nonnegative, hermitian measure and stochastic integrals, Research memorandum,Michigan State Univ., East Lansing, 1966.

11. -, The prediction theory of multivariate stochastic processes with continuous time,Doctoral Thesis, Indiana Univ., Bloomington, 1965.

12. N. Wiener and P. Masani, The prediction theory of multivariate stochastic processes. I,Acta Math. 98 (1957), 111-150; II, Acta Math. 99 (1958), 93-137.

Michigan State University,East Lansing, Michigan


on the alternating projections theorem and … · 2018. 11. 16. · bivariate stationary stochastic...

Documents