be a second-order and mean-value-zero vector-valued process, i.e., for t E

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1 CONFERENCE REPORT 617 DSCUSSON OF TWO PROCEDURES FOR EXPANDNG A VECTOR-VALUED STOCHASTC PROCESS N AN ORTHONORMAL WAY by R. GUTkRREZ and M. J. VALDERRAMA 1. ntroducton Snce K. Karhunen [l] and M. Lo&e [2] establshed a general method for representng a second-order process as a denumerable seres of orthogonal and random terms, ths representaton, called the Karhunen-Lo&e expanson, has been the subject of extensve theoretcal and practcal treatment. Sometmes t s of nterest to consder smultaneously several processes as one vectoral process wth several components. Such a stuaton s the one n whch the recepton of several sgnals by a receptor s studed [S]. Therefore, a vector-valued process s defned as an one-parameter famly of vectors whose components are one-dmensonal stochastc processes wth the same parameter space. The am of the present paper s to study an orthonormal expanson for a vector-valued process wth a fnte number of components, such as s usually consdered n practcal experments. For that, we consder twodmensonal vectoral processes, the generalzaton beng trval. Furthermore, we wll follow two alternatve approaches, whch we wll call global expanson and term-by-term expanson; both are obtanable from the classcal Karhunen- Loeve theorem (see [6]). 2. Vector-Valued Processes Let be a second-order and mean-value-zero vector-valued process,.e., for t E Department of Statstcs and Operatonal Research, Unversty of Granada, Granada, Span.

2 618 [a, bl, E[ W ) ] =E[X(t) m] =E[[X,(t)[2]+E R. BRU AND J. VT6RA We say that x s a quadratc mean (q.m.) contnuous process f ~~~{E[X,(t+h)-X,(1)/2]+E[lX~(t+h)-X,(t)/2]) =O. ts components are correlated stochastc processes f E[X,(G X,(t) ] =+ 0 for t E [a,b]. Otherwse they are called uncorrelated processes. The covarance matrx s expressed as R,(t,s) = E[X(t)X(s) ] = E[X,(t) Xl(S) _ E[X,(t) X,(s) E[X&)X,(s). E[X,WX,o When X,(t) and X,(t) are uncorrelated processes, then R,,(t, s) = R,Xt, s) = 0 and R.( t, s) s a dagonal matrx. 3. Global Expanson Let x be a twodmensonal second-order vector-valued process wth mean value zero, and consder the set H,(x) whose elements are all the fnte lnear combnatons and q.m. lmts of such summatons. Let us defne the nner operaton =E[Y,z,]+E[Y,z,], E H,(x).

3 CONFERENCE REPORT 619 Then H,(x) s a Hlbert space; f x s a q.m. contnuous vectoral process, t can be easly proved that H,(x) s separable. Under all these assumptons, there s an orthonormal and denumerable bass n H,(x) such that x can be represented as follows: X(t) = x,(t) x,(t) = n~ou wn= 2 fj,(t) zt,, n=o z,z o,(t) =E[X(t) Z,] (1) where a,( t ) are contnuous and lnearly ndependent functons. From (1) the covarance matrx can be wrtten as R,(k 4 = E f ~n(tbm(s) E[Z;Z;] E[Z;Z;] EZ2Zl 1 E[Z2Z2 l (2) n=o m=o n m n m Suppose the followng addtonal assumpton: Then (2) s expressed as a dagonal matrx: R&s) = f o,(t) o,(s) n-0-2 / E GJ&) U (S) n=o 0

4 620 R. BRU AND J. VT6RA Lkewse, let us suppose that {u,,(t), n E N} s an orthonormal system n L2[a, b],.e. / % ( a t ) u,( t ) dt = a,,. (5) Operatng and ntegratng n (4) we have / u brx(t, s)u*(s) ds = L% ( t 1 o So, under the hypotheses (3) and (5), un( t ) are the egenfunctons matrx ntegral equaton of the (7) where (A) s a real, dagonal, and postve matrx. Recaptulatng the above study, we can establsh the followng result: THEOREM 1. Let x be a vector-valued process under the ubove assumptons, and { $,( t ), n E N} an orthonormu 1 egenfuncton system of the equaton (7), assocated to the matrx egenvalues {(A,), n E N}. Then x can be represented as (lmt n mean), (8) where { B,, n E N} s a sequence of orthonormal vectoral varables 1 m&(t) dt d 2x, (1 B,qz [2(A,)]-1 2/bmX(t)dt= 1 * (9) a J?$h(t) dt,-- 2x2, / The seres (8) converges unformly n [a, b].

5 CONFERENCE REPORT 621 Proof. s easy to prove that B, are orthonormal varables. To obtan (8) let us observe that 2 x(t) - $ (W.)) 2~n,(~)% n=o But R 11( t, s) and R,( t, s) are contnuous functons because they are the covarances of two one-dmensonal q.m. contnuous processes [3]. Therefore, Mercer s theorem [4] can be appled to both functons, and the second member of (10) vanshes when N --, co unformly. W 4. Term-by-Term Expanson Let x be a second-order and q.m. contnuous twedmensonal vector-valued process wth mean value zero, whose components are correlated processes (for example, two messages sent smultaneously). t can be assumed that X Xt ) and X,( t ) are processes of the same type, and they generate dentcal Hlbert spaces, so that both can be represented n terms of the same orthonormal and complete bass { Z,, n E N}: wth u;(t) = E[X,(@,], = 1,2. Then the covarance matrx s

6 622 R. BRU AND J. VTORA Furthennore, suppose that (13) Then (14) a ~ ( an d t h e vectors t ) ) h f. f h. 2 are t e elgen unctlons 0 t e equaton ( an (t) lbr At, s ) 4> ( s ) ds = ;\.4>( t ), a te[a,bj. (15) THEOREM 2. Let x be a vector-valued stochastc process under the above assumptons, and an orthonormal vectoral egenfuncton system of the equaton (15) assocated to the egenvalues {;\.n, n EN}. Then x can be represented as unformly n [a, b], (16) where {b n, n E N) s a sequence of orthonormal varables q.m. 1 Jb-- btl =!\ 4>n(t)' X(t) dt. V;\.tl a

7 CONFERENCE REPORT 623 Proof. Smlar to that of Theorem 1, because + R,,(w) - f q+3)12 n=o and by Mercer s theorem the proof s concluded. 5. Dscusson The two procedures descrbed above assume that the components of the vector-valued process are one-dmensonal stochastc processes. By means of the global expanson, an ntegral equaton wth matrx egenvalues and scalar egenfunctons s obtaned, whereas the term-by-term expanson drves to recprocal ssues. For (8) to be managed n a smple way s necessary to mpose the restrcton (3), whch reduces the covarance to a dagonal matrx. Otherwse, the term-by-term expanson also ncludes a restrcton, whch s that the Hlbert spaces assocated to the components of the vectoral process must be dentcal (n q.m.). However, ths second procedure s a generalzaton of the frst one when X and X2 are not correlated. n fact, let us denote by b and b,f the projectons of b,, on the Hlbert spaces generated by X, and X2 respectvely, and let us set BL = ba/llb\l, = 1,2; then, under the above descrbed hypotheses on the expanson (16) becomes as n (8). From the arguments just descrbed, the two procedures each are subject to a pror lmtaton for ther expanson, but the algorthm expanded for the term-by-term representaton s more general and drect than the other one. REFERENCES K. Karhunen, ber lneare Methoden n der Wahrschenlchketsrechnuug, Ann. Acad. Sc. Fenn. 37 (1947). M. L&we, Fonctons albatores de second order, C. R. Acad. Sc. Pars 220 (1945). M. Lobe, Probablty Theory, Sprnger-Verlag, F. Resz and B. Nagy, Lepns D Analyse Fonctonnelle, Gauther-Vllars, H. L. Van Trees, Detecton, Estmaton and Modulaton Theory, Wley, E. Wong and K. B. Hajek, Stochastc Processes n Engneerng Systems, Sprnger- Verlag, 1985.