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1 Math-Net.Ru All Russian mathematical portal P. Masani, N. Wiener, On Bivariate Stationary Processes and the Factorization of Matrix-Valued Functions, Teor. Veroyatnost. i Primenen., 1959, Volume 4, Issue 3, Use of the all-russian mathematical portal Math-Net.Ru implies that you have read and agreed to these terms of use Download details: IP: March 9, 2018, 04:56:41

2 ТЕОРИЯ ВЕРОЯТНОСТЕЙ Том IV ИЕЕПРИМЕНЕНИЯ Выпуск ON BIVARIATE STATIONARY PROCESSES AND THE FACTORIZATION OF MATRIX-VALUED FUNCTIONS By P. MAS AN I and N. WIENER* I. Introduction. Rosanov [8, Thm. 11] has shown that a (/-variate weakly stationary stochastic process (S. P.) is regular, if and only if its spectral distribution F is absolutely continuous and F' = ФЧ 1 *, where 4* L!! + -L! + is defined as the class of qxq matrix-valued function on the circle C = = [\z\ = 1] whose entries are in L 2, and whose п-th Fourier coefficient vanishes for я<0. Our terminology and notation differ from RosanoV'S cf. Sec. 5). In Sec. 2 we shall therefore re-derive his result from our previous work [10, Part I]. Rosanov's theorem reduces the study of the spectral criteria for regulariy of stationary processes to that of the factorization of matrix-valued functions. In [10, Part I, 7.13] we have shown that such factorization is possible when log det F' 6 L v Indeed, under some further assumptions we have given an algorithm for the explicit determination of the optimal factor 4*, [10, Par II, Sees. 6,8], [4]. The condition log det F' Li corresponds to the case in which the process has full rank. As Rosanov has shown and as is implicit in our previous work, det F' 0 a. e. for a regular process of degenerate rank. We are thus confronted with the following matrix-function-theoretic problem Factorization Problem. Given a qxq non-negative hermitian matrix-valued function F' in the class L x on the circle** C = [ z = l] such that detf' =0 а. е., to find the conditions under which F' = where Ф ьг In this paper we shall solve this problem for q = 2. We shall show (Thm. 4.1) that the necessary and sufficient condition for the factorizability of such a function F' = [F'ij] is that for / = 1 or 2, log Fu Ii, and for ; Ф /, F'JIIF'U be the quotient of the radial limits of two functions in a Hardy class # 5 on the disk [\z \ < 1], 0<o<oo. We are thus led to the study of quotients of such holomorphic functions (Sec. 3), which are generalizations of the beschranktartige functions of R. Nevanlinna [6] for which 8 -= oo. Our proof rests on a factorization lemma for such functions * Sincere thanks are due from Masani to Harvard University and the Massachusetts Institute of Technology for visiting appointments during , which made this collaboration possible. ** i. e. with entries in the class L\ on C.

3 On Bivariate Stationary Processes and the Factorization 323 (3.10). Inched as will appear from the sequel and will be shown elsewhere (5], there is a far reaching connection between the theory of such functions and the prediction theory of bivariate stationary processes. Using our condition for factorizability we arrive at an explicit spectral criterion for the regularity of such processes (Thm. 4.2). For <7>2, Problem 1.1 presents difficulties and is still open. 2. Regularity and factorizability. We shall adhere to the terminology and notation used by us in [10, Part I], prefixing all references to this paper by I. Let (f )-oo be a <7-variate, weakly stationary S. P. with shift operator U. Here each f = (f n )Ui Ж\ where Ж\ is the Cartesian product of a complex Hilbert space Ж and U is a unitary operator on Ж such that f n = U n f 0 for l</<<7. It follows that the Gram matrix (f m, f n ) = [(fl?n)] = T m -n depends only on m n. (Г л )?оо is called the covariance sequence of the process. Let M ( n = (f )2 «Л, M (f л )2 1 < i < q, - oo < n < сю. The first is th* (cbsei) subspace of Ж spanned by the elements f for k < n\ the second is the corresponding subspace of Ж\ defined similarly except that linear combinations are taken with qxq matrix coefficients and ithe сьзигз with respect to the induced topology in Ж\ cf. 1,5.6. The orthogonal projection (<р М я ) of a vector <p = (?)Li G^Sf 7 on the subspace M n is defined as the vector whose i-\h component is the projection V clos. YM'X cf. 1,5.8 (b), 5.9. Let M_ TO = П М я. We call the S. P. поп - deterministic if М л ф M rt+1, and regular if ih I M- n ) > 0 as я-^оо. In the non-deterministic case sr«= (f«i М я. х ) =^ 0. A' s o = where Q = (g 0> g 0 ). We call Q the innovation matrix of the S. P., and following Zasuhin define the rank p of the latter to be the rank of Q. Obviously 1 < p < # for a non-deterministic process; otherwise p = 0. For reaiy reference we shall state here the following known results, I, 6.13, 7.7: 2.1. Lemma. Each of the following conditions is equivalent to the regularity of the S. P. (fj^: (a) (fj-oo is a one-sided moving average, i. e. (b) M_co= {0}. oo»«= 2 (? я,? ) = Ь тп К, КФО; (1)

4 324 P. Masani and N. Wiener 2.2. Lemma. The moving average process 2.1 (1) has an absolutely continuous spectral distribution F such that CO F' = ФФ* а. е., where Ф = ^ А Л "КК^0 б L 2 +, the sign у denoting the поп-negative hermitian square root. We are now ready to prove Rosanov's Theorem. 11, [8]; 2.3. Theorem. A q-ple S. P. (f )-co is regular, if and only if its spectral distribution F is absolutely continuous, and F' = ФФ* а. е., where ФбЬЦ 4 ", and no row of Ф vanishes a. c. Proof. From 2.1 and 2.2 it follows at once that if (У^оо is regular, then F is absolutely continuous and F' = ФФ* а. е., where W6L2*. Also no row of Ф can vanish a. e. For, the vanishing of the i ih row would entail that Fu, = 0=Fii, which would mean that our S. P. has less than q (nonzero) components. To prove the converse, suppose that F is absolutely continuous and F' = ФФ* а. е., where Ф = ^ В/ / е Lj +. (1) In (1) the same i ih row of all cannot be zero, since this would mean that the I th row of Ф is zero. Now take any orthonormal S. P. (<p )-oo, i.'e. one such that (cp OT, cpj = b m nl, and define 00 T.= 2 B **»-*- k=-0 Then (f n )-oo is a g-ple weakly stationary S. P., which is a one-sided moving average and therefore (by 2.1) regular. Hence by 2.2 (in which we take К = I) its spectral distribution F is absolutely continuous and CO F' = ФФ*, where Ф = 2 В/' 8 6 L 2 +. (2) By (1) and (2) Ф = Ф a. e. and therefore F' = F' a. e. Since F, F are absolutely continuous, and F(1)=0=F(1) we have F = F. Thus (У-ос, (?l)-oo have the same spectral distribution and therefore the same covariance sequence. If with an obvious notation we let N (f 0 1 М_ л ) = lim V A?f_*. (7 0 1 M_ ) = lim V A?f_ ft, N->oo ^ Л ; ->оо ~ we may take A* = A*, of e. g. [10, Part II, Sec. 2]. Therefore (f 0 M_ ) = (f 0 M- n )l' N

5 On Bivariate Stationary Processes and the Factorization 325 whence the regularity of (fj^ follows from that of (fj^oo (Q. E. D.) 2.4. Corollary. If (fj-co is regular and has rank p<</, then detf'=0 a. e. Proof. Let the regular S. P. (fj"*, be given by 2.1 (1), say. Then by 2.2 det F' = det4f 2. Now det 4* is the radial limit a. e. of a function in the Hardy class H 2 / q on the disk D + =[ z <l]. Hence by I, 2.6 log j det W б L x or det 4* = 0 a. e. By I, 7.10 the first alternative would imply that p = q. Нэпсе det Ч 1 = 0 a. e. and so det F' = 0 a. e. (Q. E. D.) 3. Boundary values of functions in the Nevanlinna class. We shall now discuss the radial limits of quotients of complex valued functions f l9, which are in the Hardy class #s on the disk D + =[ zj<l], cf. I, 2.1. Such limits will be encountered in our solution of Problem 1.1 for the сазе q = 2 in Sec. 4. In the sequel it will be understood that С is the circle [ z - 1] and that 0<8<oo Definition, (a) We shall say that a function f is in the Nevanlinna class Nb on D +, if and only if there exist functions f u 6#s on D + such that f a ф 0, and for all z 6 D + for which (z) ф 0, we have f (z) = f x (z)/ (z). (b) A function / will be said to be in the class L% + or Qs + on C, if and only if f is the radial limit a. e. of a function in the class Нь or N* on respectively. The following lemma is a simple consequence of well-known properties of Hardy-class functions, cf. e. g. I, Triviality, (a) с L 5 ; for 1 <8<oo, L b is the class of all functions in Ьь on С whose n-th Fourier coefficients vanish for n < 0. (b) f6q + on C, if and only if there exist functions f l9 dlb + on С such that ^=0 а. е., and for all CGC for which ( )= 0, we have f(c) = = fi(c)/f,(c). (c) If f6l? + or Q? +, then log / 6Li or / = 0 a. e. (d) f Ll\ gtlp implies that f. g L B +, where 1/e = 1/8+ 1/8'; analogous results hold with # a, JV 8, Qs + replacing L? +. Let cp>0 on C, cp^ls, log<p Li, and define g + by g + (z) = ехр ± j log cp (/) {P (z, /) + iq (z, /)} d6, z 6 D +, (3.3) о where P is the Poisson kernal and Q the conjugate kernal. Then as is well-known (cf. e. g. the arguments used in proving I, 2.8), on D +9 and if g is its radial limit, then g =? a. e. on C. Thus log I g" j б L ly and (3.3) yields + (2) = exp 1 j log I g (/) I {P (z, /) + iq (z, e*)) d9, z D +. (3.4) Definition, (a) We shall say that a function g + in Нь on D + is optimal *, if and only if the relation (3.4) holds be ween g and its radial limit g. (b) Likewise, we shall call a function g* Ls + optimal, if and only * Beurling [1] calls such functions outer function

6 326 P. Masani and N. Wiener if the relation (3.4) holds between g and its holomorphic extension g + on D +. Obviously if g + is an optimal function in Нь, then g + (0) = exp i. j log I g (/) I d6 > 0. (3.6> 0 ig + (г)i = exp 1 j logu(e ib )\-P(z, e*)db>0, z6d +. (3.7> 0 On the other hand we know, cf. I, 2.6 (c), that for an arbitrary function f + in Нь with radial limit f, /,(2) <expi- f log /( e /9) P( Z, e*)db, 2 D +. (3.7> Hence our choice of the term optimal by (3.4). 0 to describe the functions governed' 3.8. Lemma (Uniqueness). // f, g are optimal functions in Ll + on С and \f\ = \g\ a. е., then f = g a. e. Proof. This is just a restatement of the uniqueness theorem I, 2.9. For letting f +> g + be the extensions of f, g to D + and ср = \ f = g а. e* on C, we have by 3.1 (b), 3.5 (b) and 3.6 / + ея 0 on D +, f(r/) ->cp(/) а. е., as r->l-0, / + (0) = expijlog<p( e 'Ve ( 0 and the same statement for g +. Hence by I, 2.9 / + * on] D +, and thereforef = g a. e. on 3.9. Lemma. Every (поп-zero) function f in L? + on C, 0<&-<oo admits the unique factorization f= g ф, ay/iere g^ls* /s optimal, ф!2" a/id Ф = 1 я- од C. Proof. Since /^0, it follows by 3.2 (c) that log f Ii. Define g + by (3.3) taking <p = f. Then g + 6#8 and if g is its radial limit, we have- \g\ = \f \ a. e. on C. (1> Hence g satisfies (3.4) i. e. is optimal, and so by (3.7) g + has no zerose on* D +. Letting f + be the holomorphic extension of f to D +, it follows that и = 8 + тл*)> ( 2 > where ф + = f + /g + is holomorphic on D. By (1), (3.7), (3.7') we see that \U ( z ) I < I ( 2 ) I o n D+ SO THAT Ф+ Яоо. Now define ф as the radial limit of ф +. Then фб/^4" and by (1) ф = 1 a. e. The existence of the desired factorization now follows on putting z = re lb This factorization is unique. For if * Beurling [1] calls g, ф the owter and inner factors of f. in (2) and letting r->i 0. / = ф is another such factorization^

7 On Bivariate Stationary Processes and the Factorization 327 then since ф = 1 = ф а. е., it follows that \g\ = \f \ = \g\ a. e. Hence by 3.8 g = g and therefore ф = ф. (Q. E. D.) Lemma. Every function f in Ql + on C, 0<8<oo such that I f I = 1 a. e. is in Q 0^ and admits a factorization f == фхф 2, where ф х> ф 2 е^ + and ф 4 = 1 = ф а а. е. Proof. By 3.2 (b) /-fi/ а. е., where f lf 6L? + and } 2 ф0 a. e. Since / = 1 а. е., therefore / 2 0 a. e. Let then f. = g-ф, be the unique factorizations of f i as in 3.9, /= 1, 2. Then since ф,. = 1, а. е., it follows that \g t \ = ) fx I = I = g 2, a. e. Hence by 3.8 gi = g 2 a. e. Therefore h Я2Ф2 Ф2 Since ф ь ф а 61», this shows of course that fcq«. (Q. E. D.) The last factorization, unlike that in 3.9 is not unique. It is not disturbed when we multiply both ф ъ ф 2 by a function фб!^ such that ф = 1 a. e. Supplementary conditions required to secure uniqueness will be given in [5]. 4. Factorizability. We are now ready to prove the following theorem, in the stochastic applications of which the function F is to be interpreted as the derivative of the spectral distribution Factorization Theorem. Let F = be a 2x2 (поп-zero) non-negatice, hermitian matrix-ialued function which is in Li on the circle С and such that detf = 0 a. e. Then (A) F - ФФ* а. е., where Ф^ 4 " if and only if (B) for /= 1 or, 2 l o g a n d forj= i yfaeq 0^, 0<B<oo. Proof. Let (A) hold, and let Ф = [<p J ]. Then By (1) and (2), fii4?nl 2 + l?i 2 2 a. е. (1) 2=? 2 i 2 +? a. e. (2) i = <p 21 <p u ^12 =7i2 a. e. (3) <Pi2?a = <Pn<P22 a. e. (4) log /и > 2 log I <p u, 21og cp 12, log 1 > 2 log J <p 21 j, 21og?22 i (5 * We now break up the discussion into that of three cases.

8 328 P. Masani and N. Wiener Case 1. Suppose that log <pn GLi. Then by (5) log f u ^ ^ v Thus <p n, /ii>0a. e. Hence by (3), (4), (1).?21 hi fnv '?ii " J Фи a. e. + Since <p 21> <pn6l 2, we have i//n6q +. Thus (B) holds with / = 1 and 8 = 2. Case 2. Let log <p 22 6:Li. Then as before we can show that log 2 Li + and /W 2 = 912/922 б Q 2, so that (B) holds with / = 2 and S = 2. + Case 3. Suppose that log <p /7 1( L x for /=1, 2. Then since 9^6 L 2, it follows (cf. 3.2 (c)) that cp /; == 0, а. е., i = 1,2. Hence by (3) 1 = 0=f l 2 а. е., and by (4) <Pi2?2i = a. e. Since cp 12, <p 2 i L 2, the last relation gives the following alternatives: (i) Ф12 =?2i == 0. a. e. (ii) <p 12 = 0 а. е., log I cp2i i Z. 2 (Ш) logl^l^, <p 2 i = 0. a. e. The first is untenable, since it entails that Ф = О a. e. If (ii), then by (5) log /226L1, and f 12//22 = 0 Qa 4 ". К (Hi), then similarly logf n Li and f«/fn = 0 QS +. We have thus derived (B) in all cases. Next let (B) hold, and suppose for definiteness that /=1, so that Then, cf. I, 2.9, log f u el l f and 1 //n6qr, 0<8^oo. (6) /11 = I 9i 2, where cp^lr. (7) We again have to consider two cases: Case 1. Suppose that / 22 = 0 a. e. Then F being non-negative hermitian, we have f 12 = /21 = 0 a. e. Hence F = hi o] о 0 = ФФ\ Ф = <Pl 0 о 0 Thus (A). Case 2. Suppose that 2 > 0 on a set A of positive measure. Since by (6) fn>0 a. e. it follows that 1 2 = fnf22>0 a. e. on A. (8) Thus the function / 2l /f u in Qs + is non-zero; hence by 3.2 (c) log i/fnl But by (8) logfe = log f«i/fnl*+logfn.

9 On Bivariate Stationary Processes and the Factorization 32 9 Hence log2 L v Therefore, in analogy with (7) By (7) and (7') we can write (7') where "1 f "?i0" 0 <p 2 _ 1 0 <p 2 /21 _ /21 9i ф1ф2 /ll ф2 (9) Now since i/f U 6Q8 + and cpi/'^gq^, it follows by 3.2 (d) that ^GQ 0+ where e = 25/(2 + 8). Also from (8), (7), (7'), ф = 1 a. e. Hence by 3.10 we have the factorization Thus Ф = Ф1Ф2 a - e -> where ^6L^, ^ = 1 а. е., /= 1, 2. 1 f >1 Ф2" Л 1. ф a. e. and therefore by (9) F = ФФ*, where ф =?2<Ь 0 Thus (A). (Q. Е. D.) Combining the last theorem with our first, 2.3, and with our earlier result 1,7.12, #nd noting that in 2.3 no row of ф can vanish, we may conclude as follows. 4.2 Theorem. The bivariate process (fj-oo is regular, if and only if it has an absolutely continuous spectral distribution F == [Fy] such that either (i) log det Ft/a. or (ii) detf' = 0 a. e.\ for /=1 or 2, log Fa б/*; and for \Ф1, In case (i) the S. P. has rank 2, and in case (ii) rank 1.* 5. On the definitions of a ^-variate process. We shall now discuss the relationship of the definition of a 4-variate weakly stationary S. P., given in Sec. 2, which is due to Zasuhin [11], with that employed by Rosanov in [8]. Zasuhin's definition is a direct generalization of Kolmogorov's [3]: a <7-variate, discrete parameter, weakly stationary S. P. is a stationary sequence in S% q, Ж being a Hilbert space, and stationarity being defined by the Gram-matricial relation (f m +A»f/i+*) = (fm»f/i)- In application^ will consist of course of L 2 -random functions over a probability space. Thus the earlier definitions of * We can easily show that incase (ii) log F-^Li for / = 1,2. 5 Теория вероятностей, 3

10 330 P. Masani and N. Wiener Khinehine, Cramer and Wiener are subsumed. Gram matrices enter naturally into the theory, since in applications the components of the g-variate process are stationarily cross-correlated. For this reason orthogonality has to be defined in terms of such matrices rather than by inner products, and linear combinations have to be taken with matrix rather than complex coefficients. This procedure has been adopted by Doob [2, Ch. 12, Sec. 7] by us in [10], cf. especially Part I, Sees. 5, 6 and by Rosenblatt [9}. With it have been proved all the results announced by Zasuhin, and many more. According to Rosanov's definition, however, a ^-variate, discrete parameter, weakly stationary S. P. is a pair (Jt, (х л )-оо) where Л is a ^-dimensional unitary space, and the x are isometric mappings on Л into a Hilbert space Ж, such that the inner product (х л+ (a), x n (а')), а, а' С A is independent of n. This definition is shown to be equivalent to Zasuhin's. But its adoption gives the impression that the difficulties of the subject arealgebraic, and that the remedy lies in a coordinate-free formulation of the theory. Actually, in all stochastic work the given family of random functions does have a privileged status; and it is not clear what is to be gained from a formulation which obliterates this fact. Indeed, all that Rosanov's mappings seern to accomplish is to replace the matrix notation by the more cumbersome language of quadratic forms or functionals. The work done so far has shown that the subject is confronted with analytical problems concerning matrix - valued functions akin to those encountered elsewhere, e. g. by Potapov [7]. In the Zasuhin presentation the results of the g-variate theory retain the simple form they have when <7=1; only the symbols have to be interpreted as matrices or vectors. This is a great advantage from the operational, heuristic and esthetic standpoints. We therefore see no good reasons for departing from Zasuhin's formulation, as far as ^-variate, discrete parameter processes are concerned. But we admit the possibility that Rosanov's definition may prove useful in the abstract generalizations of the subject. The Institute of Science, Песту пила в редакцию Bombey (India) Massachusetts Institute Technology, Cambridge, Mass. (U. S. A.) REFERENCES [I] A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Mathematica, 81 (1949), [2] JLL. Doob. Stochastic Processes, New York, [3] А. Н.Колмогоров, Стационарные последовательности з гильбертовом пространстве, Бюлл. МГУ, 2, вып. 6 (1941), [4] P. Masani, Sur la prevision lineaire d'un processus vectoriel densite spectrale npn toornee, C. R. Acad. Sci. Paris, 246, (1958), Ш. [5] P. Masani, The generating function of a bivariate stationary process (to appear). [6] R. N e v a n 1 i n n a, Eindeutige analytische Funktionen (Zweite Auflage), Berlin, [7] В. П.Потапов, Мультипликативная структура /-нерастягивающдх матрицфункций, Труды Моск. Матем. Общ., 4 (1955),

11 О двумерных стационарных процессах в разложении матршно-значных функций 331 [8] Ю. А. Р о з а н о в, Спектральная теория многомерных стационарных случайных процессов с дискретным временем, УМН, XXIII, 12, (1957), [9] M.Rosenblatt. The multidimensional prediction problem, Proc. Nat. Acad. Sci., U.S.A., 43 (1957), [10] N.Wiener and P.Masani, The prediction theory of multivariate stochastic processes, Part I, Acta Math., 98 (1957), ; Part II, Acta Math., 99 (1958), [11] В.Засухин, К теории многомерных стационарных случайных процессов, ДАН СССР, 33 (1941), 435. О ДВУМЕРНЫХ СТАЦИОНАРНЫХ ПРОЦЕССАХ И РАЗЛОЖЕНИИ МАТРИЧНО-ЗНАЧНЫХ ФУНКЦИЙ П. МАЗАНИ (ИНДИЯ) а Я. ВИНЕР (США) (Резюме) Из теоремы Розанова. И [8] следует, что спектральный критерий регулярности <?-мерного слабо стационарного процесса относится к разложению q X Я неотрицательной эрмитовой матричной функции F' L\ на окружности [ z \ 1] в произведение ТТ*, где W имеет односторонний ряд Фурье. В [10] (часть I) мы показали, что такое разложение возможно при условии lgdetfgli. В настоящей работе задача решается для вырожденного случая, т. е. когда det F' = 0, но только для q = 2. Мы показываем, что F' = [F^-] разложима тогда и только тогда, когда \gf i (< L\ (i = 1,2), и F^/F^ (I Ф j) есть частное радиального предела двух функций в классе Харди Н ь на круге [ z \ < 1], 0 < В < оо. Так как терминология и обозначения Розанова отличаются от наших, его теорема была выведена заново.