., ’1 f s * u. ’ 0 ~-'- " I. I v ‘ go uta-n ‘\ O\ Michigan Stat: University This is to certify that the thesis entitled On subordination,sampling theorem and "Past and Future" of some classes of second-order Processes presented by Mohsen Pourahmadi S.A. has been accepted towards fulfillment of the requirements for Ph.D. degreein Statistics & Probability V'WJL/LQ’ Major professor natal/571m / / 0-7 639 _- _ J _ I OVERDUE Hugs: 25¢ per day per 1t- #‘“\\\‘ L W!“ Eflflmfi LIQRARV MTEQJALS: gnu!” Place in book‘nturn to move In f - diam flu circulation records 1 OI SUBORDIM “par? CU‘SLNCS .Ir L o u? L a.” ll "‘ ‘ l "'l I All 393‘ '1' Q\ ‘ fl . Fab. W" n v . ..'§ '4'- ‘ . . Mango, .'.-..> in ”PM“ *ulf‘fi'm; ''(.‘:r {A _ 00C *0; Department 0? 3;.2... (. ’3": 4 ~_ 4 0N SUBORDINATION. SAMPLING THEOREM AND "PAST AND FUTURE" OF SOME CLASSES 0F SECOND-ORDER PROCESSES 33' Mohsen Pourahmadi S.A. 2.1 ‘SSfif hlru a :1" C “ A DISSERTATION ““QU' 7:‘ Submitted to ‘ 'i‘ g} , ‘_ Michi an State University 7 $4 “-in partial ful illment of the requirements » *‘ for the degree of W_ DOCTOR OF PHILOSOPHY , 'I‘;_Department of Statistics and Probability k. I i 1980 («I/o w 7 ABSTRACT 0N SUBORDINATION, SAMPLING THEOREM AND "PAST AND FUTURE" OF SOME CLASSES 0F SECOND-ORDER PROCESSES By Mohsen Pourahmadi S.A. In this thesis, three independent problems (subordination, sampling theorem and "Past and Future") concerning harmonizable and stationary processes are studied. Chapter I contains some well-known results about such processes along with a necessary and sufficientconditions for strong subordination of q-variate stationary processes which are stationarily cross- correlated. The problem of finding analytic conditions for subordination of harmonizable and periodically correlated sequences is studied in Chapter II. Sufficient conditions for subordination of harmonizable sequences and a simple counter-example showing that these conditions are not necessary are given. In the case of periodically correlated sequences, which is a subclass of harmonizable sequences, necessary and sufficientconditions for Subordination, mutual subordination of such processes in terms of their associated multi—variate stationary sequences are derived. In Chapter III, the problem of admittance of sampling theorem of a q-variate stationary process and its relation with the admittance of sampling theorem of its components is considered. It is shown that if the components of a q-variate process (not necessarily stationary) admits a sampling theorem with the same sample spacing h > 0, then the process itself admits a sampling theorem with the same h. A sampling theorem for q-variate stationary process, under a periodicity condition on the range of the spectral measure of the process, is proved in the spirit of Lloyd's work. This sampling theorem is used to show that if a q-variate stationary process admits a sampling theorem, then each of its components will do so. In Section 5, by using Abreu's theorem, the well-known sampling theorems for harmonizable processes is proved in an easier way with more explicit coefficients for the sampling series. In Chapter IV, Helson—Sarason Theorem on ”Past and Future” is generalized from the disk algebra to a Dirichlet algebra setting by using function-algebraic method. Advantages of our method as compared to Ohno-Yabuta's method [32] on the same problem is discussed. This theorem is used to answer a question of M. Rosenblatt on the strong mixing of multi-parameter Gaussian stationary processes. To my pa rents ACKNOWLEDGEMENTS I would like to thank Professor H. Salehi for the guidance of this thesis and his constant help and encouragements during my graduate study here at Michigan State University. Also, I would like to thank Professors V. Mandrekar for his comments and critical reading of this thesis, S. Axler for his critical reading of Chapter IV and teaching me much of the mathematics which were needed to carry out this research and D. Gilliland for serving on my guidance committee. Thanks are also due to C. Hanna and L. Colon for their excellent typing of my manuscript. Finally, I am grateful to the National Science Foundation and the Department of Statistics and Probability, for financial support during my stay at Michigan State University. TABLE OF CONTENTS Chapter Page I NOTATIONS AND PRELIMINARIES .................... 1 II SUBORDINATION OF HARMONIZABLE SEQUENCES ........ 14 2.1 Introduction .............................. 14 2.2 Subordination of periodically correlated sequences ................................. 15 2.3 Subordination of harmonizable sequences... 17 III SAMPLING THEOREM FOR q-VARIATE STATIONARY AND UNIVARIATE HARMONIZABLE PROCESSES .............. 26 3 I Introduction .............................. 26 3.2 Preliminaries ............................. 28 3.3 Projection on L2 F S ................... .. 34 3.4 A sampling theorem ........................ 40 3.5 Sampling theorem for harmonizable processes ....................... . ......... 44 IV HELSON-SARASON THEOREM FOR DIRICHLET ALGEBRAS AND STRONG MIXING 0F MULIT-PARAMETER GAUSSIAN STATIONARY PROCESSES ............. . ..... . ....... 49 4.1 Introduction ............................. 49 4.2 Notations and Preliminaries ............... 51 4.3 Main results... ......................... 53 4.4 Strong mixing of multi- -parameter Gaussian stationary processes ............... ....... APPENDIX.. ............. . ................................... 54 BIBLIOGRPHY ................................................ 67 INTRODUCTION The concept of subordination was introduced, studied and used in prediction of univariate stationary sequences by A.N. Kolmogorov [18]. Analytic necessary and sufficient conditions for subordination of such processes were derived in [18]. Analogous analytic conditions for the subordination of q-variate and infinite-dimensional stationary sequences were derived by M. Rosenberg, Yu. Rosanov and others in [39], [41], [38], [23] and [22]. In [12], the notion and analytic characterization of subordination of stationary sequences have been used for optimal filtering of stationary signals. The problem of finding such an analytic characterization for the subordination of harmonizable sequences which are harmonizably cross-correlated in studied in Chapter II. The study is carried out in such a way that when specialized to stationary sequences, the results will reduce to the corresponding results of Kolmogorov [18]. In Section 2, necessary and sufficient conditions for sub- ordination, mutual subordination and necessary condition for strong subordination of periodically correlated sequences in terms of their associated multi-variate stationary sequences are derived. It is well-known that the class of periodically correlated sequences is a natural extension of stationary sequences but a subclass of har- monizable sequences. Because of the 1-1 correspondence between periodically correlated sequences and q-variate stationary sequences V vi and the existence of a shift operator for the latter sequences, we have been able to find necessary and sufficient conditions for sub- ordination of the former sequences. Sufficient conditions for the subordination of harmonizable sequences and a simple counter-example showing that these conditionsare not necessary is given in Section 3. It seems that, the fact that these conditions are not necessary can be atributed to the failure of existence of a shift operator for the harmonizable sequences. In Chapter III, the problem of sampling theorem for q-variate stationary and univariate harmonizable processes is considered. Sections 2 and 3 contain some well-known results as well as some new results which play a crucial role in the proof of our sampling theorem for q-variate stationary processes. In Section 4, a sampling theorem for a q-variate stationary process, similar to that of Lloyd's [21], is proved under the condition that the range of the spectral measure of the process considered as a linear operator- valued function from Cq to Cq is periodic. Then, this sampling theorem is used to prove that if a q-variate stationary process admits a sampling theorem,then each of its components will do so. In Section 5, by using Abreu's theorem [1], we prove in an easier way as compared to [36] and [20], sampling theorems known for har- monizable process, with the advantage that in our proof the coefficients in the sampling series for such processes are exactly the coefficients of the sampling series of its associated stationary process. The problem of strong mixing of multi-parameter Gaussian vii stationary sequences was first studied by M. Rosenblatt [43]. In [43], some sufficient conditions for strong mixing of such process along the work of Kolmogorov and Rosanov [19] weregiven. The problem of strong mixing of such processes is yet open. In Chapter IV, first we generalize Helson-Sarason theorem on ”Past and Future" from the disk algebra to a Dirichlet algebra setting and then specialize this theorem to the torus to obtain a necessary and sufficient condition for strong mixing of multi-parameter Gaussian stationary sequences. The Appendix explains some of the terminalogies related to a Dirichlet algebra used in Chapter IV. CHAPTER I NOTATIONS AND PRELIMINARIES Let (n.3,P) be a probability space. H = L2(D,B,P) denotes the Hilbert space of all complex-valued random variables on D with zero expectation and finite variance. The inner product in H is defined by W,y) fx(w) )yr'1r( dw) , x,y e H. In the following we introduce some basic terminologies and concepts in the spirit of the work of N. Wiener and P. Masani [50], P. Masani [23. These are used in the study of q-variate stationary processes. For q 1 1 , Hq denotes the Cartesian product of H with itself q-times, i.e. the set of all column vectors X = (x1,x2,...,xq)T with x1 6 H,for i = 1,2,...q. As usual we endow the space Hq with a Gramian structure: For X and Y in Hq their Gramian (X,Y) is defined to be the qxq matrix (X, )= [(x ,yj)]q 1,J= -1 One can easily verify that, (1.1) (X.X) :0. (X.X) = 0 <=> X = 0; (i E ) "21 i < ) AX, BY = AX,Y£B:, k=1kki=1H k=1£=1kk for any X,Xk, YREHq and any qxq matrices Ak,B£. We say that X is orthogonal to Y in Hq if (X,Y) = 0. It is well-known that Hq is a Hilbert space under the following inner product, (1.2) ((X,Y)) = trace (X,Y) = E (x‘j,yj). j=1 A closed subset H of Hq is called a subspace of Hq if it is a manifold, i.e. AX + BY 6 H whenever X,Y 6 H and A,B are 9 x 9 matrices. It is known [50] that H is a subspace of Hq if and only if there exists a subspace H of H such that H = Hq. Thus, we obtain a structure for Hq which differes from but also closely resembles that of a Hilbert space, and which we shall call Hilbertian [23]. For any x e H, its orthogonal projection on a subspace H of l,...,xq)T E Hq, its H is denoted by (x|H). Given a vector X = (x projection on a subspace H = Hq is the vector (XIH) whose i-th component is (xilH) for each i = l,...,q. 1.1 Definition: A sequence xn,n 6 Z (Xn,n E Z) of elements of H(Hq) is called a univariate (q-varjate) stochastic seguence. For convenience we may abbreviate xn,n 6 Z (Xn,n e Z) by xn(Xn) or simply by x(X). Also, throughout we use small x,y,... to denote univariate and capital X,Y,... for q-variate sequences. For random variables {xj} in H we denote by o{Xj; J Ed} is: the subspace spanned by xj. for all j in the indexed set J. Similarly for random vectors {Xj} in Hq, o{Xj3 j 6 J} is the subspace of jed Hq spanned by all Xj. J E J with matrix coefficient. 1.2 Convention: Since the class of univariate stochastic sequences is a subclass of q-variate stochastic sequences (q = 1), here, we only introduce notations and known results about q-variate sequences. The 3 corresponding notations and results for univariate sequences can be obtained by obvious specialization. 1.3 Definition: To every q-variate sequence X we associate the present and past subspaces H(X,n), n E 2, defined by, (La Ram)=dq;kin)gfl, and the terminal subspace H(X), defined by, (1.4) H(X) = H(x,m) = o{Xk; all k}. Also, we define H(X,n), n 6 Z, and H(X) by, (1.5) MM) = “”11" k :n, 11191194, (1.6) W) = H(Xa) = olxl; an k. 1:19}- It is easy to check that, (1.7) H(X,n) = Hq(X,n), n e z. In simultaneous treatment of two q-variate stochastic sequences, the concept of subordination plays an important role. Here, we define subordination and some related notions for two q-variate sequences X and Y. 1.4 Definition: Suppose X and Y are q-variate stochastic sequences. We say that, (i) Y is subordinate to X if and only if H(Y) c H(X). (ii) Y is strongly subordinate to X if and only if H(Y,n) cH(X,n), n E Z. (iii) Y and X are mutually subordinate or equivalent if and only if H(Y) = H(X). In the following, we assume that our q-variate stochastic sequence is stationary in the sense defined below. 1.5 Definition: A q-variate stochastic sequence X is said to be stationary if the covariance function R(m,n) = (Xm,Xn) depends on m-n alone. 1.6 Definition: Two q-variate stationary sequences X and Y are said to be stationarily cross-correlated if the qxq Gram matrix (Xm,Yn) depends on m-n alone. We note that a q—variate stationary sequence can be considered as a set of q univariate stationary and stationarily cross-correlated sequences. To introduce the known results about spectral analysis of q-variate stationary sequences, and for later use, we need the following concepts. Let B be a c-algebra of subsets of a space 9. M is said to be a gxg matrix—valued signed measure on (9.8) if for each A 6 B, M(A) is a qxq matrix, with finite complex entries and M(A) = E M(Ak)’ whenever A1,A2,... is a sequence of disjoint sets in B k.13hose union is A. 1.7 Definition: A qxq matrix-valued signed measure M is called a gxg matrix-valued measure if M(A) is a nonnegative hermitian matrix for each A E B. 1.8 Definition: Let 0 = (qij) be a matrix-valued function on Q and u a nonnegative real-valued measure on B. (i) We say that o is B-measurable if each function ¢ij is B-measurable. (ii) L1,u is the class of all o such that each Qij is intergrable with respect to u. (iii) For o E L . we define [a du = (fa..du). 1.9 Definition: We say that the qxq matrix—valued signed measure M is absolutely continuous (a.c.) with respect to (w.r.t.) a o-finite nonnegative real-valued measure u on (9,8) if the entries of M, dM i.e. Mij's are a.c. w.r.t. u. We write Mu = g! = ’-—il) for the Radon-Nikodym derivative of M w.r.t. u. Now, using Definition 1.8 (iii) we define integrals of the form f¢(A)M(dX)w(A), where M is any matrix-valued signed measure and o n and v are suitable functions, by (1.8) fo(>.)M(d>.) in): mi) %(A)Y(A)u(d)\) where u is some nonnegative real-valued o-finite measure on (9,3) such that M is a.c. w.r.t. u. It can be shown that the definition of the integral does note depend on the choice of u. Nhen M is a qxq matrix-valued measure it is customary to choose u to be 1M = trace M. In this case, we denote 3%“ = M; by MI. In the following, we take 9 = [0,2n) = T, B the o-algebra of Borel subsets of T = [0,2n) and as usual identify T with the unit circle {2 E t; lzl = 1} in the complex-plane. It is known that (cf.[18, Theorem 1], [38, page 14]) if Xn and Yn are q-variate stationary and stationarily cross-correlated sequences, then there exists a unitary operator u on the subspace oixl, y;; all . . i i i i n = = , 1 §_1 §_q}c:H onto 1tself such that uxn xn+1 and uyn yn+1, 1 §_i §_q. This operator u has a spectral resolution; (1.9) u = 4e-nE(dX) where E is a projection-valued measure over (T,B). The operator U may be extended to a unitary operator on H onto H in many ways, we denote this extension again by u. The inflation of u denoted by U is defined by, (1.10) 0(X) = (Ux1,...UXq), x = (mile Hq. By taking the inflation E of E analogously, we can define the following qxq matrix-valued signed measures. 1.10 Definition: With each pair of stationary and stationarily cross-correlated sequences Xn and Yn we associate the qxq matrix- valued cross-measure MXY’ not necessarilly hermitian-valued, and qxq cross-spectral distribution F defined by, XY (1.11) MXY(A) = (E(A)X0, E(A)Y0), A E B, (1.12) FXY(A) = ZnMXY(0,A] , 1 E T. * It 15 clear that MYX(A) = MXY(A)’ A E B, and (1.13) va<") = (xn.v0)=4e""*MXY(dx) = érie‘inidFXY(A). "T where these integrals are defined as in (1.8) with o (A) = e'inAI and Y(A) = I and I is the qxq identity matrix. In the special case, when X = Y, from (1.11) it is obvious that M(-) = MXX(-) is a qxq matrix-valued measure (cf, Definition 1.7). 1.11 ngipjtjpp; The qxq nonnegative hermitian matrix-valued function F defined by, (1.14) F(1) = 21M(0,A] , A E T, is called the spectral distribution of the stationary sequence X. 1.12 Definition: By the §pectral representation of the stationary sequence Xn and its covariance R(n) we mean (1.15) xn = 4e'i"*E(d1)xo = {e‘i"*g(d1) (1.15) R(n) = (xn,x0) = {e‘inAM(dX) = égye‘i"*dr(1), where in (1.15), 5(A) = E(A)X0, A e B, is an Hq-valued countably additive, orthogonally scattered (c.a.o.s.) measure, so-called be- cause of its decisive property, A,B e B and A,B disjoint implies €(A) 1 5(8). The last integral in (1.15) is difined as (Ie'mshqu . T j=1 With the definition of integral for matrix-valued functions as in (1.8), we define the L2 class of such functions with respect to amatrix-valued masure M associated to a q-variate stationary sequence by, (1.17) L2,F = L2,M = {a ; Io(A)M(dA)o *(1) exists}. We put the following natural norm on L2 F; (1.18) ”illr = [trace (1(1)M(d1)1*(1)1’2. T It is known that L2 F is complete under this norm (cf. [40], E38,page 30]). We can introduce in L2 F a matricial and scalar-valued inner products by, (1.19) (1,1))F = (¢,v)M = {D(A)M(dA)Y (1), ¢,v 6 L2 F, (1.20) ((453))F = ((o,v))M = trace (¢,v)F. Thus, the norm introduced in (1.18) can then be written as (1.21) 11111,. = E((¢.¢))FJ% The following theorem of [40] connects L2 F and H(X) : Hq. The integral appearing in the theorem is defined in [40]. 1.13 Tpgppgm. For a q-variate stationary sequence Xn’ the correspondence o + fo(X)E(dA)XO is an isomorphism on the space H(X) gLHq. T The following theorem is an extension of Kolmogorov's Theorems 8, 9 and 10 DB] in a form which is given in EB]. Actually, this theorem gives analytical necessary and sufficient conditions for subordination and mutual subordination in terms of the spectral measures of two q-variate stationary and stationarily cross-correlated sequences. 1.14 Theorem. Suppose Xn and Yn are stationary and stationarily cross-correlated sequences, then (i) Yn is subordinate to Xn if and only if there exists a o E L such that 2,FXX (1.22) dFYY(X) = 1(1)drxx(1)¢*(1), (1.23) dFYx(>.) = ¢(A)dF X). XX( In the sense that for any A E B, FYY(A) = £1(1)drxx(1)1*(1) and FYX(A) = £¢(X)dFXX(A). (ii) Let Yn be subordinate to Xn and o as in (i). Then Xn and Yn are mutually subordinate if and only if, dex * _ dFXX F ) (1.24) rank{¢(x) 3;F;;(1)¢ (A)} - rank{3;:;;(x)} a.e(r XX . 10 Condition forstrongsubordination of stationary and stationarily cross-correlated sequences is not available in the literature. In the following, by using Theorem 1.14, we give necessary and sufficient conditions for strong subordination of such sequences. For F the gectral distribution of a q-variate stationary -inXI sequence we define H2 F = oie ; n1: 0} in L2,F‘ In the special case, when dF(A) = Idx, H2,F is the usual matricial Hardy class of functions denoted by Hz. 1.15 Theorem. Suppose Xn and Yn are stationary and stationarily cross-correlated sequences, then Yn is strongly subordinated to Xn if and only if there exists a function e 6 H2 F such that ’ X (1.25) dFYY(X) = 1(1)dFXX(1:1(1)* (1.26) dFYX(X) = o(x)dFXX(A). Proof. Suppose Yn is strongly subordinate to Xn’ then Y e H(X,0) 0 and since H(X,0) and H2 F are isomorphic, there exists a function ’ XX o 6 H2 F such that Yn = f éi" o(A)E(dX)XO, for every n (c.f. Theorem ’ XX T 1.13). Thus for all integers m and n we have; 1 -i(m-n)X = _.l_ -i(m-n)X * 2F 4 e dFYY(X) (Ym,Yn) — 2" 4 e ¢(A)dFXX(A)¢(A) , §F¥ él(m‘“)*dFYX(1) = (vm,xn) = %F 4 él(m'")*1(1)drxx(1). Which implies (1.25) and (1.26) respectively. 11 Conversely, suppose that there exists a function a 6 H2 F ’ XX satisfying (1.25) and (1.26). Then by Theorem 1.14 (1') Yn is subordinate to X". Define Zn =fe '1"A¢ fi(dX)X0, then it is easy to check that, dFZZ(X) = dFYY(A) and dFXZ(X) = dFXY(A). Thus by Lemm 8.1 [38, page 35], it follows that; Yn = 2n = 48-1nA¢(A)E(dA)X0, for all integers n, which shows that Yn E H(X,n) for all n thus Yn is strongly subordinate to Xn by (1.7) and Definition 1.4 (ii). Q.E.D. Using Theorem 1.15, we note Ehat if FX XX is a.c. w.r.t. the d Lebesgue measure dX and FXX = HIKE = 11* a.e. (d1), where w 6 H2, then (1.25) implies that FYY(A) = oww*o* =(owX¢W)* a.e. (dA), i.e. if Yn is strongly subordinate to Xn and Xn is purely nondeterministic (cf.[23], Theorem 9.7), then knowledge of a and v facilitates the task of finding an optimal factor of FYY' It is known that this type of analytic factorization plays a major role in prediction theory of stationary sequences, (cf. [23], section 13). Next, we define a class of H-valued stochastic sequences, which are a natural generalization of univariate stationary sequences and closely related to q-variate stationary sequences. 1.16 Definition. A stochastic sequence xn is said to be periodically correlated of period q if the function R(m,n) = ) = R(m+n,n) (xm+n’xm is periodic in m of period q (we note that when q = 1 the sequence 12 is stationary). Since R(m,n) is periodic in m of period q, one Znikm) q . ~ q can write R(m,n) = X Rk(n)exp( k=1 For convenience we extend the definition of these functions Rk(n). k = 1,2...,q, to all integers by Rk(n) = Rk+q(")‘ It is shown in 001 that each R (n) has the representation k (1.27) R n) = %fe"'"*ark(1), "T where each Fk(-) is a complex-valued measure on T. Let F(-) be the qxq matrix-valued measure, given on intervals by A1+2nj X2+2nj)]q-1 T: q . k—O, A1_ A2. (1.28) F(11.121= [Fk_J-( J _ It is proved in Balthat F(-) is a matrix-valued measure. It is also shown that 1 -i(m+n)).1+im).2 (1.29) R(m,n) = 4—fge dF(>.1,A2). 11' where the spectral measure F(-,-) is given by q-1 (1.30) F(A,B) = X f dF (1). k=~q+1 Anus-33$) " In other words the spectral measure F(-,-) is concentrated on 2q -1 straight line segments 11 - 12 = g%£, k = -q + l,...,q - 1, contained inside the square T2, and the measures Fk(-) give the mass of F(-,-) on these lines according to (1.30). 13 With any H-valued sequence xn we associate the Hq- valued sequence Xn whose i-th coordinate x; is given by an+i’ i = 0,1,2,...q -1. This correspondence establishes a one-to-one linear transformation from the H-valued sequences onto the Hq-valued sequences and we have, _ T Xn - (an, an+1""an+q-1) 1.17 Lemma. xn is periodically correlated with period q if and only if Xn is a q-variate stationary sequence. By Definition 1.12, this associated q-variate stationary sequence has a spectral measure F which is a qxq nonnegative definite matrix- valued measure such that R(n) = (X , X ) = l— Ie'1nxdF(1). n 0 211T The following theorem which gives the relation between this measure F and the measure F given in (1.28) can be found in [10] and [27]. 1.18 Theorem. With the notations as above, we have; F(A)= £qu*(x)dF(x)u(1), A 63. Where U is a unitary matrix-valued function whose (j,k) - th entry is given by qJ5 expEzn1.:+ikA]. CHAPTER II SUBORDINATION 0F HARMONIZABLE SEQUENCES 2.1 Introduction: The concept of subordination was introduced, studied and used in prediction of univariate stationary sequences by A.N. Kolmogorov ENE. Conditions for subordination in terms of the spectral measures of the sequences were derived in [18]. Analogous conditions for the subordination of q-variate stationary sequences were derived by M. Rosenberg BQAIh Yu. A. Rosanov [38] and P. Masani [23] and for infinite-dimensional stationary sequences by V. Mandrekar and H. Salehi [22]. In [41] and [22] the notion of subordination have been used to gain some new insight into some problems in analysis. T.N. Siraya [4m gives conditions for subordination and strong subordination (cf. Definition 1.4) of second-order (not necessarily stationary) processes in terms of their covariances and corresponding reproducing kernel Hilbert spaces. In [49] conditions for subordination and strong subordination of one second-order process to another such process with orthogonal increments, in terms of the structural measure of the latter has been derived. In [12], the notion and analytic characterization of subordination of stationary sequences have been used for optimal filtering of stationary signals. In [3 I it is shown that under some general conditions the output of a linear system is a harmonizable stochastic process. 14 15 In this chapter we give analytic conditions for subordination of periodically correlated and harmonizable sequences in the spirit of Kolmogorov [18,section 4], see also Theorems 1.14 and 1.15. In section 2, necessary and sufficient conditions for subordination, mutual subordination and necessary conditions for strong subordination of periodically correlated sequences in terms of their associated multi-variate stationary sequences (cf. Leann 1.17) is studied. Sufficient conditions for subordination of harmonizable sequences and a counter-example showing that these conditions are not necessary along with the problem of linear transformation of har- monizable sequences is discussed in section 3. 2.2 Subordination of Periodically Correlated Sequences: Throughout this section we assume that xn and yn are periodically correlated sequences with period q and that they are periodically cross- correlated i.e. the function ny(n,k) = (xn+k’ yn) is periodic in n of period q. 2.2.1 Remark. If xn and yn are periodically cross-correlated with x period q, then [y:], n 6 Z, is a two-dimensional periodically correlated sequence. Thus by [10], ny(-,-) has an spectral representation similar to the spectral representation of the covariance of xn (cf. 1.29). 2.2.2 Lenna. If xn and yn are periodically cross-correlated with period q and xn’Yn are their associated q-variate stationary sequences. Then Xn and Yn are stationarily cross-correlated. 16 Proof. It is easy to check that, for all integers m,n; q-1 (Xm’Yn) ‘ [(x(m-n)q+i’ yJ)]1',j=o which depends on m-n alone. Q.E.D. For xn a periodically correlated sequence and Xn its assocaited q-variate stationary sequence we have for all integers n, (1) H(X,n)=c{ ;m:n.0:i:q-1} xmq+i oixk; kinq +q - 1} =H(x; nq+q -1). Thus, letting n + m, we get the following important equality, (2) H(X) = H(x). In the following theorem we give necessary and sufficient conditions for subordination and mutual subordination of periodically correlated sequences in terms of their associated q-variate stationary sequences. Necessary and sufficient conditions in terms of matricial spectral measures for subordination and mutual subordination of periodically correlated sequences can be obtained by using Theorems 1.14 and 1.18. 22.3 Theorem. Suppose xn and yn are periodically cross-correlated sequences of period q and X", Yn their associated q-variate stationary sequences, then 17 (i) yn is subordinate to xn if and only if Yn is subordinate to Xn‘ (ii) yn and xn are mutually subordinate if and only if Yn and Xn are mutually subordinate. Proof: (i) and (ii) are obvious because of Lemma 2.2.2 and relation (2). Q.E.D. 2.2.4 Remppgz If yn is strongly subordinate to X", then by relation (1), Yn is also strongly subordinate to Xn‘ But, the converse is not necessarily true. For an example, let an be a periodically correlated sequence of period q = 2 with an i o{€k; k 5_n -1}. Define x11 and yn by X2n = g2n-1’ x2n+1 = 52n’ y2n = E2n and y2n+1 = €2n-1’ then it is clear that Yn is strongly subordinate to X", but H(y, 2n) 3 H(x,2n) i.e. yn is not strongly subordinate to X . n 2.3 Subordination of Harmonizable Sequences: In this section we study the problem of subordination of harmonizable sequences and its relation with linear transformation of such sequences. First we develop a few concepts which are essential in this study. 2.3.1 Definition: A stochastic sequence xn is said to be harmonizable 1r xn =4e'1m‘n(d>.) “d -‘imA +inA (1) R(m,n) = fiffe 1 2U(dl T2 dA 1’ 2): 18 where n is a countably additive H-valued measure (not necessarily orthogonally scattered) on T and for A,B 6 B, u(A,B) = (n(A), n(B)) extends to a complex-valued measure of bounded variation on T2. u is called the spectral measure of the sequence. 2.3.2 figmapk. Comparison of (1) and (1.29) reveals that the class of periodically correlated sequences is a special subclass of harmonizable sequences. 2.3.3 The Hilbert Spgge 12(gg). For p, v measurable functions on T, m o v will denote the tensor product of o and w i.e. (P 9 Y) (A1112) = m(11)w(12), for 11, 12 e T. Let S be the class of all step functions on T, it is clear that S is a linear space and for all e, v E S, the double integrals Ié e o i du = I; ¢(Al)§(12)u(dxl,d12) is defined in the obvious way (u T is a measure satisfying (1)). Two step functions o and V will be considered identical if, I] (<1 -1) «1 (op—Juan = 0. T2 If we define for o, v e S, <¢1Y> = ff o e T du, then(S, <-,.>) 2 T is an inner product space. In fact, it is obvious that has the ordinary bilinear and conjugate symmetric properties and further <¢, ¢> 3 0(this follows from property of u).and <¢, ¢> = 0 only when If p o m du = 0 i.e. when p is identical with 0. Also, it follows from 3 0, that we have the Cauchy-Schwartz inequality i.e. |<¢,y>|2 f <¢,¢> . Let A2(du) be the completion of (S, <-,->) so that it is a Hilbert space with an inner product denoted again by <.,.>. 19 Elements in A2(du) may no longer be functions on T. A typical element in A2(du) can be realized as a Cauchy sequence of step functions. However, we treat elements in A2(du) as ”formal" functions on T and use the improper but suggestive notation I; m e E du for the inner T product with p, Y e A2(du). Of course, [g p o D du = l;m jg on e indp, where on and T Yn are Cauchy sequences of step functions from S converging to o and v, respectively, in the norm of A2(du). Let A(du) = {all measurable functions m on T; [flu o $[dlul < w 2 T and fjleld|u| < m}, where Iul denotes the total variation measure 2 of u and the double integrals are in the sense of Lebesgue. We say that the function e in A(du) represents an element in A2(du) if there exists a o'eA2(du) such that for all N e S, = jgcp(>.1)v(12)u(dxl, dxz). T We note that if such m' exists, it is unique, since S is dense in A2(du). Then, we denote a' by a and write 9 e A2(du). With this convention and Theorem 1.1 of C 4],A(du) is a dense subset of 2 . . A (du) and 1f cpl, cpz e A(du) Wlth fg lcp1(>11) (p2(A2)| lul(d}11,d}12) < co, . T then < $1, $2 > = If o1 (A1) o2 (A2)u(d11.dx ). 2 T2 where the double integral is in the sense of Lebesgue. For n, as in the Definition 2.4.1, we define H(n) = o{n(A); A e B} in H. 20 It is shown in [5 I that {en(1) = e-inA; n e Z} forms a basis in A2(du), H(h) = H(x) and further that there exists an isomorphism between 112(1111) and H(n) defined by cp+f\2)’ (2') ny(AaB) ' If CP(A25 dex(A1:A2)c A B Proof. Suppose there exists o e A(dex) such that, yn =1 é‘mw) 1. (d1) . n e z. T Since yn is harmonizable, it has its own spectral representation, i.e. there exists an H-valued measure (cf. Definition 2.3.1) g(-) such that yn = IT 6mm). Thus, for all integers m and n we have; [gem o endey = (xm,yn) = If em 0 en r dex T [gem o endFyy = (ym,yn) = [gem 9 en cp a (p dex , T which implies (2). 23 Conversely, suppose that there exists a o e A(dex) such that (2) holds. We define 2n = f éinlm (A) n (d1), then it is easy to check T that, (xm,yn) = (xm.Zn) (ym.yn) = (Zm.Zn). Thus, by a slight extension of Lemma 8.1 [38, page 35], we get -inX . . . yn = €_ e o (X) n (d1), 1.e. yn 1s obta1nable from xn by means of a linear transformation. Q.E.D. 2.3.11 Theorem. Suppose x and yn are harmonizable and harmonizably n cross-correlated. If there exists a function a e A(dex) such that dFyy = o e m dex dey 3 (P dFXX’ then yn is subordinate to X". Proof of this theorem is an easy consequence of Theorem 2.3.9 and Remark 2.3.8. 2.3.12 A counter example. Here we give an example which shows that, unlike the stationary and periodically correlated sequences, the conditions of Theorem 2.3.11 are not necessary for subordination of harmonizable sequences. rfifi 24 Let a be a random variable on some probability space with E g = 0 and E lgl2 = 1. Let f,g e L1(T,dX) where f is not identically zero. Define the following stochastic sequences . . . 2n . xn = f(n): and y = g(n)a, where f(n) =-l; f e’1nAf(A) d1. n 2110 It is easy to check that xn and yn are harmonizable and harmonizably cross-correlated with dFXX = f e‘f dm, dFyy = 9 5‘5 dm 2 and de = f o 5 dm, where m is Lebesgue measure on T . y For any choice of such functions f and 9 we have H(y) §_H(x) i.e. yn is subordinate to X". But, in the following, we show that it is possible to choose f and g in such a way that none of the relations in (2) (or (2')) can hold. Suppose, there exists a a e A(dex) such that conditions in Theorem 2.3.11 are satisfied, then, for A = B we have; 2 _ 12 <3) 0‘90)de - Womb. , A a a. For A = [0,H] choose 9 e L1(T,dX) such that Ig(A)dX # 0. Then with f = x[n,2n] we have i o(A)f(A)dA = 0, whiéh contradicts (3). 2.3.13 Bgmgpk. Theorem 2.3.11 can also be proved by using Theorem 1 of [48] and Lemma 2.3.5. 2.3.14 339355. In Definition 2.3.1, if u is a measure which is concentrated on the main diagonal of the equare T2, then the cor- responding process xn is stationary. In this case, we can think of u as a nonnegative measure on T, then it is easy to see that A(du) (as defined in 2.3.3) is the same as the space of measurable functions on T which are square integrable with respect to u i.e. J W—‘—————7 A(du) = L2(du). Thus Theorems 2.3.9 and 2.3.11 specialized to the case when xn and yn are stationary and stationarilly cross- correlated will reduce to Theorem 8.1 of [38, page 36] and sufficient part of Theorem 9 of [18], respectively. 2.3.15 Bgmgpk. We note that for stationary sequences, the property that yn is obtainable from xn is equivalent to the subordination of yn to xn [38, Theorem 8.1] and Theorem 1.14. But, this is not the case for harmonizable sequences, as counter example 2.3.12 shows. 2.3.16 An Open Problem. It is conjectured that the assertions of Theorems 2.3.9 and 2.3.11 are true even when p e A2(dex) instead of belonging to A(dex). Although this can be estiablished formally, we have been unable to prove it rigorously. It seems that a rigorous proof of these assertions in this new setting hinges on giving a proper meaning to the relation (2) in Theorem 2.3.9. L... . ‘ CHAPTER III SAMPLING THEOREM FOR q-VARIATE STATIONARY AND UNIVARIATE HARMONIZABLE PROCESSES 3.1 Introduction. It is well-known that a stationary stochastic process x(t) e H, t e R, has the sampling series (D ' _1 x(t) = Z x(nh) §lfl_flfl:TiE;flflT n=-m nh (t-nh if the spectral measure u of x(t) is supported by the interval h'1 h'1 . . (--§ ,~§ ). Th1s so called ”sampl1ng theorem” dates back to Cauchy and is of considerable importance in communication and information theory [117 and [29]. Such processes with bounded spectra are called ”band limited”. This sampling series, which converges in mean-square and also almost surely, enablesa band-limited process to be exactly reconstructed from its sample {x(nh); n e Z}. Of course, a process need not be band-limited to admit an error- free reconstruction from its samples. S.P. Lloyd [21] gave a necessary and sufficient oonditionon the spectral measure for a stationary process to admit such a reconstruction. More precisely, a process x(t) e H (not necessarily stationary) can in principle be exactly reconstructed from its samples {x(nh); n e Z} if H(x) = Hs(x), where Hs(x) = o{x(nh); n e Z} in H. 3.1.1 Definition. For a fixed h > 0, we say that the process x(t), t e R, admits a sampling theorem if H(x) = Hs(x). Lloyd uses the terminology that “x is linearly determined by its samples" when H(x) = H (x). We will refer to the fixed positive h as s 26 Ft: "p|IIIIIaIII---------------------------------------------q 27 "sample spacing" and to the set {tn} = {nhg n e 2} as "sample times". In [21], Lloyd proved the following remarkable resuts for a stationary stochastic process. 3.1.2 Theorem. Let x(t), t e R, be a stationary process with spectral measure u. then (i) x admits a sampling theorem if and only if u has a support A '1 i.e.{A + nh'l; n e Z} are such that the translates of A by nh i mutually disjoint. (ii) If the measure n has an open support A whose translates {A + nh'l; n e Z} are mutually disjoint, then we have i a x(t) = l.i.m. ) (1 - ifil) x(nh)K(t-nh), t e R, N+oo =-N where K(t) = h f Enixt dA, t e R, and l.i.m. stands for limit in mean A square. T (iii) If the A from (ii) is a finite union of intervals, or, more generally if sup [tK(t)| < m, then t N x(t) .m. X x(nh)K(t - nh), t e R. = 1.1 N + n=—N For more information on sampling theorems and its applications in different fields, as well as a complete bibliography of this subject, [17] may be consulted. The extension of sampling theorem for multi- parameter stationary processes have been studied by Parzen [23], Miyakawa [28] and others [17]. For q-variate stationary processes no sampling theorem is available in the literature. Due to the importance of such processes in application, it is important to have theorems similar to 3.1.2 for q-variate stationary 28 processes. Also, it is important to know whether anything is gained by studying sampling theorem and sampling series for q-variate stationary processes. It is useful to know whether there is any connection between admittance of a sampling theorem for a q-variate process and its components. If it is so, then it is desirable to know something about the rate of convergence of the q-variate sampling series and its relation with the rate of convergence of its individual component's sampling series. In sections 2 and 3, using the ideas of Lloyd, we develop the necessary machinery which is needed to prove a sampling theorem for a q~variate stationary stochastic process. Also, we show that if the com- ponents of a q-variate process (not necessarily stationary) admits a sampling theorem with the same sample spacing h > 0, then the process it- self admits a sampling theorem with the same h. In section 4 we prove a sampling theorem for q-variate stationary process and use this sampling theorem to show that if a q-variate stationary process admits a sampling theorem then each of its components will admit a sampling theorem. In section 5, by using Theorem 3.1.2 and Abreu's Theorem [ 1] we obtain a sampling theorem and a sampling series with explicit coefficients for harmonizable stochastic processes. 3.2 Preliminaries. In the study of sampling theorem for q-variate stationary stochastic processes the notion of absolute continuity of a matrix-valued sgined measure (defined in Chapter I) with respect to another such measure plays an important role. The problem of defining a "proper" notion of absolute continuity for such measures was first posed by P. Masani [23]. Later J. B. Robertson and M. Rosenberg [37] dealt with this question and obtained a satisfactory solution to it. F'V'T **** 29 Here, we will briefly review some of their results and some other concepts which are needed for the proof of our sampling theorem. Throughout this Chapter 9 = R and B is the o-algebra of Borel subsets of R. The customary definition of absolute continuity for matrix-valued signed measures does not guarantee the existence of a Radon-Nikodym derivative. 3.2.1 Definition. Let M1 and H2 be q x q matrix-valued signed measures on (0,3) respectively, let u be any o-finite nonnegative real-valued measure on (9,8) such that M1 and M2 are a.c. w.r.t u. We say that M2 is strongly absolutely continuous (s.a.c) w.r.t. M1 1f, N(M'1,u(>\))c M(Ml2911()‘)) 3.8. (L1), where for each matrix M, N(M) = (X; MX = 0}. It can be shown that this definition is indpendent of u. Hence, we supress the dependence of Mi,“ and Mé,u on u i.e. we only write M; for M; u’ i = 1,2. The following theorem is proved in [37]. 3.2.2 Theorem (Robertson-Rosenberg). Let M1 and M2 be q x q matrix- valued signed measures on (9.8) then, (i) M2 is s.a.c. w.r.t M1 if and only if there exists a measurable q x q matrix-valued function e on 9 such that for all A e B M2(A) = {a dMl' (ii) Let a and u be measurable q x q matrix-valued functions on n. Then for each A e B, It dM1 = [v dM1 if and only if ed = id a.e. (u). A A 30 where J is the orthogonal projection matrix-valued function onto the range of Mi and u is any o-finite nonnegative real-valued measure on (9,8) w.r.t. which M1 is a.c.. Thus, if M2 is s.a.c. w.r.t. M1, then by Theorem 3.2.2 (1) there exists a measurable matrix-valued function a such that for each A e B, M2(A) = it dMl' o is called the Radon-Nikodym derivative of M2 w.r.t M1 and will be denoted by EM; . To make this notation more clear 1 and for later use we need to introduce the concept of generalized inverse of matrices due to R. Penrose [34]. 3.2.3 Theorem (Penrose). Let A be any q x q matrix, then there exists a unigue q x q matrix X such that, A = AXA, x = XAX, (Ax)* = Ax and (XA)* = XA. 3.2.4 Definiton. The matrix X in Theorem 3.2.3 is called the generalized inverse of A, and will be denoted by A‘. It can be shown that the generalized inverse of a matrix A has the following important properties: AA = PR(A) = PN(A*).L, ”A = P12W) = PM(A)i Where R(A) stands for the range of the matrix A considered as an operator from ¢q to ¢q and P denotes orthogonal projection. From Theorem 3.2.3 and Definiton 3.2.4, if M2 is s.a.c. w.r.t. dM M we define the Radon-Nikodym derivative -—31 = dM .dM' by I dM1 2 1 31 22% (A) = Mé (A).Mi' (A) a.e. (u), where p is any nonnegative measure such tat M1 is a.c. w.r.t. u. Next, we introduce some basic notions about continuous time q-variate stationary processes. We note that all definitions and results of Chapter I are still valid for continuous time stationary processes, if n is replaced by t and the region of integration by R = (-w, w), [38. Chapter I]. To be consistent with the literature on sampling theorem, through- out this chapter, we replace é'it by é2n1At contrary to our standard notation of earlier chapters. Let X(t), t e R, be a q-variate mean continuous stationary stochastic process with the spectral distribution, q x q matrix-valued function, F defined on n. Then, X(t) has the spectral representation X(t) = ? ‘5"‘At E (dA) X(0), (c.f. Definition 1.12). By Theorem 1.13, + éZTTIAt under the map X(t) I, t e R. where I is the q x q identity matrix, H(X) is isometric to L2 F. For fixed h > 0, by the samples of the procg§s X(t) , we mean the collection {X(nh); n e Z} of random vectors. The samples (x(nh); n e Z} span a closed subspace of H(X). We denote this subspace by Hg(X). The random vectors in H;(X) are those determined linearly by the samples with matrix coefficients. 3.2.5 Definition. We say that the q-variate stochastic process X(t) admits a sampling theorem if H(X) = H;(X). Now, we prove the following important but simple theorem. 3.2.6 Theorem. If the components of X(t) i.e. xi(t), 1 f i 5 q admits 32 a sampling theorem with the same h, then X(t) admits a sampling theorem with the same h. Eppgf: From Definitions 3.2.5 and 1.3 it follows that X(t) admits a sampling theorem if and only if H(X) = HS(X), where HS(X) = oixi(nh); 1 f i IA q, n e Z} in H. From this observation and the fact H(x‘) = Hs(x1), 1 f i f q, it follows that H(X) = o{H(xi); 1 f i f q} = 0{H5(X1), 1 f 1 IA q} = HS(X) i.e. X(t) admits a sampling theorem. Q.E.D. We note that this theorem holds for any second-order q-variate process. The converse of this theorem is not that easy. In the case of q-variate stationary processes we get that as a corollary of our main theorem. We denote by L2 F s the image in L2 F of fig(x) under the isomorphism. According to this isomorphism to X(nh) e Hg(X) corresponds éZNlnhA -2nlnhA I 6 L2 F s’ n e Z. Since for each n e Z, e I is periodic with period h'1 in A, it is tempting to characterize L2 F s as equivalent classes of all matrix-valued functions in L2 F which are periodic with period h'1 . But, this is not true in general. Next, we put enough conditions on F which gaurantees that L2,F,s is the equivalent classes of matrix-valued functions which are periodic with period 11'1 . 3.2.7 Assumption. Throughout this chapter we assume that the spectral distribution F is such that R(F'(A)) is periodic in A a.e. (r) with period h.1 (i.e. R(F'(A)) = R(F'(X+nh-1)) if A, A + nh"1 e support of 1) where r = trace F and F' = gg-a.e. (1). 33 It is obvious that when F' is of full-rank or F' has constant range or F' has periodic entries, on the support of 1, then Assumption 3.2.7 is satisfied. Now, we show that under Assumption 3.2.7, L2 F s can be identified as equivalent classes of functions in L2 F which are periodic with period h'1 . 3.2.8 Lemma. Under Assumption 3.2.7, L2 F 5 consists of equivalent classes of matrix-valued functions in L2 F which are periodic with period h‘1 . Proof. First we note that L2,F,s = a1e2"'"“* I; n e Z} in L2,F . Thus, for o 8 L2 F 5 there exists a sequence on of matrix-valued 1 functions which are periodic with period h- such that on + o in L2 F or what is the same on/E“+ o /F7 in L2 11’ This implies that there exists a subsequence on such that 1 On. 1417+ 4 14:7 a.e. (T). 1 Thus, a F' + o F' a.e. (1) , "i which implies that therefore, (1) o + o a.e. (I) on R(F'). Now, we show that o, as a function in L2 F, is periodic with period h'1 . From (1), we have for almost all A, 34 (2) on. (A) + ¢(A) on R(F'(A)) l (3). a (1) = on (A + nh'l) . ¢(A + nh‘l) on R(F'(A + nh'1)) = R(F'(A)) n1 1 (by Assumption 3.2.7). Thus, (2) and (3) implies that for almost all A 1(1 + nh‘l) = a(1) on R(F'(A)). Thus, L2 F s is contained in the collection of all equivalent classes of matrix-valued functions in L2 F which are periodic with period h'l. Next, suppose that 0 t b e L2 F is periodic with period h‘1 such that, (4) j é2"i""*dF(A)o(A) = o , for all integers n. By periodicity of a, (4) is the same as h'1 . e j ez"‘"“*( I dF(A+mh‘1))o(A) = o, for all n, which implies o m=-m that o e 0 in L2 F. This contradiction proves that L2 F 5 contains all equivalent classes of matrix-valued functions in L2 F which are periodic with period h'l. Q.E.D. 3.3 Projection on L2 F S’ For the proof of our main result, Theorem 3.4.1, we need to have an explicit form for the operator P projecting L2 F onto L2 F S(Lemma 3.3.5). In this section we find such a form for P along the line of Lloyd's Lemma [21]. Let Bb denote the family of all bounded sets in B. For A e B and given a e L2 F we define the following countably additive b and o-finite set functions on 8b: 35 M(A) = E f dF (A) and M¢(A) f¢n(A) an(A), I n=-m A o(A + nh-l), n e Z, A e R are where Fn(A) = F(A + nh‘l) and ¢n(A) 1. translates of F and b. We note that M¢ is equal to M, when a the q x q constant matrix. These functions are determined by their values for sets in 8b. Let A e 3b have diameter less than h'l, so 1 that its translates {An = A - nh- ; n e Z} are mutually disjoint. Then, M¢(A) = f a dF, and the countable-additivity and o-finiteness of UA [111 M¢ follows from this and the fact that each set in 8b can be written 1 as finite union of Borel sets with diameter less than h' . Due to this latter fact, without loss of generality, we assume throughout this chapter that A e B has diameter less than h'l. 3.3.1 Bgmppg, Here, we note that although H and M¢ are not (necessarily) defined on the o-algebra B, neverthless, the assertions of Theorem 3.2.2, concerning s.a.c. and Radon-Nikodym derivative and its uniqueness, are still valid when M1, M2 and B are replaced by M,M¢ and 8b, respectively. This can be proved by applying Theorem 3.2.2 to each bounded Borel set and the o-algebra of its Borel subsets. 3.3.2 Lgppg. M¢ is s.a.c. w.r.t. M. Eppgf, We must find a o-finite nonnegative measure H such that M¢ << p and M << u and then show that: N(M'(A))<;.N(M; (A)) a.e. (u). Let u = X 1 r , where T = rF (IF = trace F ), then it n n n n is clear that Fn << Tn << p . Thus, we can define 36 M(A) = A(§F8)dp, M¢(A) = {(gonFa)du, which implies M' = {F6 a.c. (p) and M’ = In F' a.e. (u). n p a n n Let X e N(M'), then (XF6)X = O which implies: Since FT'1 is nonnegative definite [40,Lemma 2.3], we get * x* F; x = o for every n. But, x* Fax==(/f; X) . (/f; x) = 0, which implies that /F; X = 0, for every n. Thus, M; x = (XanFa) x = { (o/F;) 7F; x = o i.e. x e N(M$). Q.E.D. [I n By Lemma 3.3.2, Theorem 3.2.2 and Remark 3.3.1 the Radon-Nikodym dM derivative HMQ-exists. So we can define the operator P on L2 F into 9 the space of matrix-valued functions by, dM (M(A) = 21112“) a.e. (.1), <1) e 12 F. It is clear that P is matricial linear, also since for each fixed o e L2,F’ A and integer k, M¢(A) = Z f¢n an = f o dF : n A UA n n dM 1) f o dF = M¢(A + kh' , it follows that ani' can be chosen to be NAn+k periodic with period h.1 , this fact plays a key role in the proof of boundedness of P. 37 dM To show that ENE is in L2 F, it is enough to prove that P is norm bounded in L2 F. For this, we need to prove the following matricial Cauchy-Schawrtz inequality for matricial inner product in L2 F' 3.3.3 Lemma. For ¢,v e L2 F with matricial inner product °° * ($11)?)F: f iI’dF ‘i’ 9 -oo we have, (o,v)F (nap); (u,o)F 5 (o.o)F . Proof. For every q x q constant matrix A we have, c.f. (1.1), (i + At, a + Ap)F 3 0 'k * or, (¢,®)F + A (u,w)F A + A (w.¢)F + (t,b)F A > 0 . For choice of A = - (oaw)F (W9W); and using the defining properties of the generalized inverse of matrices, (c.f. Theorem 3.2.3), we get the result. Q.E.D. 3.3.3 Lemma. P is a contraction on L2,F into L2,F,S’ 38 Proof. For o e L2 F we have; .. dM dM .. dM * (1'1 ,2 - __q>_ ___4>_ - e - “PM'F T [mdM dF (dM ) T [0°(dM ) d—M— dF - -1 h dM dH 1 z] (m2 (A + nh1))* mi (A + nh'l) dF (A + nh'l) n 0 = If ('dM (m T (A) Z(“En“) = If dM dM d” O n 0 -1 * -1 h dM dM h if A dM —-9: - if dM dM dM dn‘ dM = dM d” 0 0 h']. T f dM dM dn 0 ¢ dH In this chain of equalities we have used the fact that 3M3 can be chosen to be periodic with period h'l. Since M (A) = f o(A) dF(A) I with diameter of A less than ¢ UA [111 -1 h , by letting bl = ongn and v = IXUAn we get (o1,v)F = H¢(A), n * (r,v)F = f dF = M(A) and (o1,o1)F = 6A ¢(A) dF(A) o (A) = N(A). in _ * Thus, from Lemma 3.3.3 we get M¢(A) M (A) M¢(A) f N(A), therefore; -1 h .2 * .2 IIPMIF 5 Tfo dN = 1 Z I Andi;1 on = llbliF. Which shows that P is a contraction on L2 F into L2 F . But, since di 3M2 can be chosen to be periodic with period h"1 , it follows that the range of P is inside L2 F S . Q.E.D. In the following, a bounded matricial linear operator P on L2 F is said to be a projection if P2 = P. In this case P is the 39 identity operator on its range. 3.3.5 Lgpmg. The operator P is a projection onto L2,F,S . 3599:. By Lemma 3.3.4 it is enough to show that P is (equivalent to) the identity operator on L2,F,S . Since any a e L2,F,S is equivalent to some t e L2,F,S which is periodic with period h‘1,(c.i. Lemma 3.2.8), thus by definition of M¢, Lemma 3.3.2, Theorem 3.2.2 (1) and Remark 3.3.1: 1 . dM M¢(A) = E A on(A) an(A) = T o dn = g afii- dM. Hence, by Theorem 3.2.2 (ii) we get; where J(A) is the orthogonal projection matrix onto the range of M'(A) dM a.e. (u). Since 3M2 5 L2 F’ (c.f. Lemma 3.3.4) and R(F')g; R(M'), it dM ’ follows that, HM$'= a a.e. (F). Thus, for o e L2,F,S we have Po = o a.e. (F). Since range of P is contained in L2,F,S it follows that P is the projection onto L2,F,S° dM Next, we find a version of HMQ' which will play a major role in the proof of our main theorem. For each n, let Fn denotes the Lebesgue-Stieltjes matrix-valued measure induced by the functions Fn(A) = F(A + nh'l), A e R, n e Z. Each of these measures may be decomposed, by Crdmer-Lebesgue theorem [37], into a TF-continuous part and a TF-singular part; Fn(A) = A fn(A)dr(A) + FnU-U) Sn) , A 88,, n e Z, 40 where fn, a q x q nonnegative definite matrix-valued function, is the Radon-Nikodym derivative of the rF-continuous part of Fn with respect to if, and the tF-singular part of Fn is supported on the Sn i.e. 1(Sn)= 0 (F(Sn) = 0). Let S = p5", then 1(5) = 0 and Fn(A) = A fn(A)dT(A) + Fn(A H S), A e B, n e Z. thus, the measures M and Mo will have the form, (A))dr(A) + M(A n s) M¢(A) = A (E ¢n(A)fn(A))dT(A) + M¢(A n 5). Hence, we arrive at the following important result. dM - 3.3.6 Lemma. (Po)(A) = 5M9 (A) = (Z ¢n(A)fn(A)) (E fn(A)) on R\S. n n which is a.e.(r). We note that this version of the projection is no longer formally periodic, but it plays a major role in the proof of Theorem 3.4.1. 3.4 A SamplinggTheorem. From Definition 3.2.5, it easily follows that the statement that, for all values of t e R not of the form nh, the random vector X(t) can be obtained by linear combination of the sample random vectors {X(nh); n e Z} with matrix coefficients. In this section we find necessary and sufficient conditons on the support of the spectral measure F or equivalently the trace measure of F so that the process admits a sampling theorem. By a support of a measure I we mean any set A e 3 whose complement has T measure zero i.e. t(R \ A) = 0. Here is our main theorem which is stated and proved in the spirit of Theorem 1 of [21]. 41 3.4.1 Theorem. Under Assumption 3.2.7 the following properties of a q-variate stationary stochastic process X(t) continuous in mean are equivalent. (1) Each random vector X(t), t c R, of the process is determined linearly by the samples {X(nh); n e Z}. (ii) For some irrational number a, X(ah) is determined linearly by the samples. (iii) There exists a support A of the trace measure I of the spectral distribution of the process whose translates {A + nh'l, n e Z} are mutually disjoint. Pppgfi. That (1) implies (ii) is clear. We show that (ii) implies (iii) and then (iii) implies (i). Suppoese X(gh) is determined linearly by the samples i.e. ZWIAEhI X(gh) e H; (x), then e which is the isomorph of X(gh) in L2,F belongs to L2,F,S so is equal to its projection on L2,F,S . Thus, by Lemmas 3.3.5 and 3.3.6 we have; 1 )5“ I)f 1-1 éZflTAEh I 3 P éZNTnEh I = [2(é2N1(A + "h n n 3 M -h 3 L..l O) m f"\ a V Which implies, - o - o + ' eZWIAgh (Z fn) = (Z e2fl1 0 there exists n-mo F e H” and a positive integer n such that |Arg(Fh2Z1'n)( < e and |log|F|| < e a.e. (m) on X. Proof. Lim on = lim in i (ll-21.n 31¢Fum = 0, if and only if for every n-wo n...» FEH e > 0 there exists a positive integer n such that; inf; Hl-zl‘” 51 ¢ Fljm < c . FeH This holds if and only if, there exists an F e H00 such that nl-zl‘" 31¢ rum < 6. And this in turn holds if and only if, lArg(Fh2Z1'n)| < e and |log|F|| < e a.e. (m) on X (In this proof 5 may not be the same throughout). Next, we quote the following result from [25]. Let Hp denote the closure in Lp(dm) of the set of polynomials in Z and Lp the closure in Lp(dm) of the set of polynomials in Z and ‘Z . For 1 g p f m, we put P P ‘k I = {f e H ; [f2 dm = 0, k = O,1,2,,,,}. 4.3.2 Lemma [25]. If 1 f p g m, then Hp Hp o Ip Lp Lp e Np 55 where 0 denotes the algebraic direct sum and Np denotes the closure of ID + 1p in Lp(dm). Here is our main theorem, whose proof is essentially the same as that of Theorem 6 in [31]. For the sake of completeness and comparison we present its proof in detail. 4.3.3 Theorem. Lim p = 0 if and only if, for every positive e <«% n n+oo there exist real functions r, s e Lm(dm) with lirllco < 5.th00 < e such that w = TPIZ E+CS, where P is a function in H” so that P 1 A3 in L2(dm) for some n and Cs denotes the conjugate function of s. Proof. Assume lim p = 0, then by Theorem 4.3.1,for each 0 < e <-% n—rco n there exists a positive integer n and F e H°o such that lLoglFil < e and |Arg(Fh2Z1'n)i < 5 Let s be the real function bounded by e such that; (5) s + Arg(Fh221‘") = 0. From here on we proceed as the proof of Theorem 6 of [31]. We put, (6) s = thzl‘" eCSI'S . Then by (5) S 3 0. From Theorem 10 of [7 ], we conclude that éCS+iseH1 is outer. By proposition 4 of [31] and the fact that 2 1-n 2 |Arg(Fh Z )1 < e < %-, we may write Fh = ZmB, where B e H1, dem f 0 and 0 f m f n-1. Therefore (7) s = Bz'k eCS+ls > o and so 56 (8) zks = B eC5+AS c 111/2 where k = n-m-l. Furthermore, by Jensen's inequality, (9) flogllkSIdm = flongldm + flog|éCS+iSIdm 3 log|dem| + logif éCS+isdm| > -m . Using Theorem 2 of [9 ], it follows from (8) and (9) that there exists an outer function P in H1 and an inner function g in H” such that (10) zks = q P2. Since S = |S| and |S| = |P|2, we have from (10) that (11) qP2= zklpiz. Since P is outer, it follows that P is not zero. Thus we may divide (11) by P and obtain (12) q P = ZkP‘. By Lemma 4.3.2, we can write P = E a.Zj + a e H1 e I1 i=0 3 I where a1 belongs to 11. Now (13) ZkP = 302k + Elzk'1+ ....+ Ek_lz + 3k + §k+lZ+ 310.272 +. + Zk 011. 57 1 —k 1 1 Because ak+1Z + ak+2 +... 6 H0 and Z 61 e I c: H0, we have 9 = ak+1Z + ak+2Z2 + ... + ZkoI e H3 . By (12) and ZkP e H1 we conclude g'e H1 by (13). Hence 9 e H'1 n H5. Since A'+ A0 is weak*- dense in L”(dm), we have 9 = 0 and k _-— k -— k-l -— -— Z‘P — aOZ + all + ... + ak_1Z + ak . Thus P has the form k P = + a Z + + Z 30 1 ... ak where 0 f k 5 n-1. Therefore P e H00 and P 1 A3 in L2(dm). Indeed, Q if G e A3<: (H0)n, then G = ZnK for some K e H” and we have; k . k . (p,c) [(2 a.zJ)‘z’"T<'am= ) ajfin‘Jram .= J '=0 3 0 k . I ajdemJZn'J'lem = o, 3:0 since m is multiplicative on H” and n-1 3 k. Now by (10) and (6) we have; IP = 5 = |5| = lFllhl and since w = |h|2, w =|PI2lF|'135 = 1212 5"“ where r = -log|F|, Hr!)0° < c and (isl)co < 5 . Conversely, suppose w satisfies conditions of the theorem. 2 Let s = {Pl . “)" 0 for f e I”, we have Since zn‘lp f e (H 58 oo (14) fzn‘ls f dm = (z"'1P f, P) = o , f c I If f e I”, then it is easy to see that. Z (n-1) f is also in I . Therefore, by (14) f 7”"13 f dm = fzn'1572(“'1)f dm = o , f e I” . since S = S, (15) [2”‘15 T'dm = o , f e I“ . It follows from (14) and (15) that jzn'ls f dm = o , f e I e 1 I "-15 c L . By Lemma 4.3.2, Z Furthermore, we have (z"’1‘kP,P) = o (n-l-k 3 n, i.e., k = -1,-2,...) fzn'lsik dm = k+1-n (P,Z P) = O (k + l—n 3 n, i.e., k = 2n-1,2n...). n-I We conclude that Z S has the form n-l _ 2n-2 Z S - a0 + all + ... + a2n_22 We put, k = max{m; 0 f m f n-l, am+n-1 f 0}. Since S f 0 and S'= S, such k exists. Then ZkS c H"0 and kaS dm # 0, therefore by Theorem 2 of :9 1, zks has the factoring zks = do2 where q is inner and G is outer in H”. If we take an outer function F in H0° such that [PI = Er, then 59 (16) zkas Es'is = th , up to constant factors of modulus 1. Indeed, by Theorem 10 of [7 ], 85-15 k G2 is also outer in H”, so that the left hand side of (16) is outer in H1. is outer in H1, and Z 55 Furthermore, since F is m 2 . . . outer in H and h is outer in H , the right hand s1de of (16) 15 also outer in H1. Now by the assumption on w ._ Cs-is Cs Cs _ )quS e | = S e = [PI2 e = wer = |h|2|F| = |Fh2|. Since an outer function is determined up to a constant factor by its modulus, (16) follows. From (16), s = thz‘kq ecsiis and s 3 o, it follows that ZZ-k éCs+is) = 0. Arg(Fh q Hence [Arg(FhZZ‘k q)| = [s] 5 lislico < e and [loglFll = [rl f “rum < e. n-I-k If we put B = F q Z , then B e H” and lArg(Bh221‘")l = (Arg(fhzz‘k q)| < c , llongll = lloglfll < e - Thus the assertion follows from Theorem 4.3.1. Q.E.D. 4.3.4 Corollary. :1: pn = 0 if and only if, for every 0 < e < %3 there exists real-valued functions r, s e L”(dm) with “rum < c, “5“,, < E: and w = lplz eT‘i-CS ,where P is a polynomial in Z of arbitrary degree and Cs is the conjugate function of 5. 4.3.5 Example. Let X = T, A be the disc algebra on T i.e. A = {f a C(T); f(n) =f:fléinAf(A) %%-= o, n = -1,-2, ... and m normalized Lebesgue measure on T. Then A is a Dirichlet algebra on 60 T and it is well-known that the Gleason part of m is non-trivial. Wermer's embedding function, in this case, is Z = eAA. Thus, by Sarason's Lemma [46], Corollary 4.3.4 reduces to Theorem 4.1.1. 4.3.6 Example. Let x = T x T and s = {(m,n) e 22; m > O}U{(O,n)e 22; n 3 0}. Let A = A(S) be the Dirichlet algebra of continuous functions on T x T which are uniform limits of polynomials in 3(mx+ny), (m,n) c S. Let m be the normalized Lebesgue measure on T x T (torus). Then the Gleason part of m can be identified with {(0,6) e ¢2; la) < 1} which is non-trivial, the Wermer's embedding function is given by Z(eix,eiy) = eiy and P of corollary 4.3.4 is a polynomial in eiy [31]. Corollary 4.3.4 is similar to Theorem 5 of [15] and its form resembles that of Theorem 1 of [13]. Example 4.3.5 shows that when X is the unit circle, the characterization of u does not depend on e. Actually, this is the case when X is any compact Hausdorff space as is shown in 532;. Let C(Z) = {f(Z); f c C(T)}, then H” + C(Z) is closed in L”(dm) [32, Lemma 3]. Thus, by using an extension of Sarason's Lemma [46]; Corollary 4.3.4 can be restated as: 4.3.7 Theorem. Lim p = 0 if and only if w has the form n-mn 2 er(Z) + CS(Z), where Z is the Wermer's embedding function, w = |P(Z)| P is an analytic polynomial, r and s are real valued continuous functions on the unit circle T and Cs is the usual harmonic conjugate function of s. 4.4 StronggMixing of Multi:parameter Gaussian Stationary Processes. Unlike the prediction theory for stationary stochastic processes with one parameter, prediction theory for multi-parameter stationary stochastic processes is more diversified. Because there is no natural 61 distinction between "past" and ”future” in the latter case as compared to the former one. Here, for simplicity, we only consider the two- parameter or doubly stationary stochastic processes with discrete parameters. 2( Let (9,8,P) be a probability space and x e L n,B,P) m,n 2 such that [xm n(w)dP(w) = 0, (m,n) e Z . We say that {xm n} is a two- parameter stationary stochastic process if for all integers m,n,k,l we have = (xk,l’x0,0)' In this case, we call C(k,l) = (xm+k,n+l’xm,n) (xk,T,x0’0) the covariance of the process. It is easy to see that C(-,-) is a positive definite function on Z2. Thus, by Herglotz- Bochner-Weil Theorem [45, Page 19] on positive definite functions, there exists a finite non-negative measure u on Borel sets of the tours such that C(k,l) = ff ei(kX+IY)du(x,y), (k.l) e 22. a is called the spectral measure of the process. H. Helson and D. Lowdenslager [14] developed the theory for pre- dicting x0,0 by linear combination of elements xm,n with (m,n) e S, where S is a half-plane of lattice points. The fact that the proofs and some of the results of [14] are independent of the particular choice of S have been crucial hithe development of abstract Hardy spaces. Also, this fact is very useful in theory and applications of two-parameter stationary stochastic processes as will be seen in this section. Here, we adopt the following definition of half-plane of lattice 2 points. A set S of lattice points of Z is called a half-plane if; 1) (0.0) e S, 2) (m,n) e S if and only if (-m,-n) t S unless m=n=0, 3) (m,n) e S and (m',n') e S imply (m+m', n+n') c S. 62 For X = T2 and S a fixed half-plane of lattice points, it . 2 2 . is easy to show that A = A(S) = {f e C(X); f(m,n) = “lg-ff é1(mx+ny) 4H 0 O f(x,y)dxdy = 0, (m,n) t S} is a Dirichlet algebra on the torus. Let Sk = {(m,n)e 221‘3(mx*"Y)e Ag} and B(Sk) the o-algebra generated by the collection of random variables {xm n; (m,n) e Sk}. We say that the process is strongly mixing if, Sup|P(AnB) - P(A)P(B)|= a(n) s o A,B as n + m, where A and B range over 8(5) and B(Sn), respectively. BY using a remarkable result of Kolmogorov and Rosanov [19] it can be shown that a Gaussian stationary process {xm,n; (m,n) e Z2} is strongly mixing if and only if A'= {T; f c A} and A3 are asymptotically orthogonal in L2(du), that is, if and only if pn + 0 as n + m. Therefore, necessary and sufficient conditons for strong mixing of such processes is obtained by specializing Theorem 4.3.7 to the case when X is the torus and S is any half-plane of lattice points. Thus, the problem of strong mixing of two-parameter Gaussian stationary processes is solved in the spirit of [14]. A slightly different notion of strong mixing and a sufficient condition for strong mixing of such processes is given in [43]. 2 4.4.1 An Open Problem. In this special case i.e. when X = T , S a fixed half-plane and m a complex homomorphism of A(S) whose Gleason part G(m) is non-trivial, it is important to know whether there exists a complex homomorphism in G(m) such that its corresponding Wermer's . . . . iimx+nyl embedd1ng funct1on Z shifts the exponent1als e , (m,n)cS, "properly". To make this problem more clear, in Example 4.3.6, the 63 Wermer's embedding function Z(eix,eiy) = eiy corresponding to m (the normalized Lebesgue measure on the torus) shifts the desired exponentials along the y-axis, in this case from viewpoint of application to strong mixing problem, it would be more meaningfull if we could find a complex homomorphism in G(m) such that its corresponding Wermer's embedding function would shift exponentials along the x-axis or along the line y = x. APPENDIX APPENDIX Here, we explain in more detail some of the terminologies related to a Dirichlet algebra. Throughout this appendix, X will denote a compact Hausdorff space, C(X) will denote the linear space of all continuous complex- valued functions on X. It is well-known that this linear space is a Banach space (Banach algebra) under the sup norm Hf“ = sup|f(x) XeX By a measure on X we mean a finite complex measure on X. A uniform algebra on X is a complex linear subalgebra A of C(X) which satisfies: (i) A is uniformly closed; (ii) The constant functions are in A; (iii) A separates the points of X, i.e. if x and y are distinct points of X, there is an f in A with f(x) f f(y). If A is a uniform algebra on X, then a complex homomorphism of A is an algebra homomorphism from A onto the field of complex numbers. Since the uniform algebra A is closed, it is a Banach space (Banach algebra) under the sup norm, it can be shown that each complex homomorphism 4 is a bounded linear functional on that Banach space. A representing measure for o is a positive measure m on X such that <1>(f) = ff dm, f e A. x 64 65 Since 4(l) = 1, we have fdm = 1, therefore a representing measure for o is a probability méasure on X. For a uniform algebra A, we denote by M(A) the set of all complex homomorphisms of A. With each f in A we associate a complex-valued function f (called Gelfand transform of f) on M(A) by f(1) = (f) , a e M(A). If we topologize M(A) with the weakest topology which makes all these functions f continuous, then it can be shown that M(A) is a compact Hausdorff space. This space M(A) is known as the the space of complex homomorphisms of A or the maximal ideal space of A or the space of multiplicative linear functionals on A. By Riesz representation theorem, it can be shown that for each complex homomorphism of A, there exists at least one representing measure m on X. To show that this measure m is unique it is necessary to impose more restrictions on A. A uniform algebra A is called a Dirichlet algebra on X if the real parts of the functions in A are uniformly dense in the space of real continuous functions on X. It can be shown that A is a Dirichlet algebra on X if, and only if A + A' is uniformly dense in C(X), or, if, and only if, there is no non-zero real measure on X which is orthogonal to A. For a Dirichlet algebra A, it can be shown that the relation 41 ~ 42 defined by Hal - 42“ < 2 is an equivalence relation on 66 M(A). The equivalence classes for this relation is called the Gleason parts of M(A). For 4 a complex homomorphism of A with the unique representing measure m, G(m) the Gleason part of o is defined by G(m) = {P e M(A); 9 ~ 4}. For more information on this subject and proof of the state- ments made earlier the following paper of K. Hoffman may be consulted (Analytic functions and logmodular Banach algebras, Acta Math., 108 (1962), 271-317). BIBLIOGRAPHY Ch 10. 11. 12. 13. 14. BIBLIOGRAPHY L. 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