Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
.. Department of Statistics~ University of North Carolina~ Chapel Hill~ North CaroZ':;.a 27514. This research was supported by the Air Force Office of Scientific Research under Grant AFOSR-68-1415 and by the Office of Naval Research under Grant N00014-67A-0321-0006. TH~EQU1 VALEiJCE ORS INGULARITYOF STOCHASTI C' PROCESSES Aim OF THE HEASURES THEY INDUCE ON L 2 by Stamatis Cambanis * Depa1>tment of Statistics University of North Carolina at Chapel Hill Institute of Statistics Mimeo Series No. 759 July~ 1971 THE EQUIVALENCE OR SIHGULARITY OF STOCHASTIC PROCESSES AND OF THE l·jEASURES THEY INDUCE 011 L2 Stamatis Cambanis* ABSTRACT It is 'shown that the proposition: "Two stochastic processes are equi- valent or singular if and only if the measures they induce on an appropriate L 2 space are equivalent or singular respectively", is not true in general, and sufficient conditions are given for its validity. For a wide class of square integrable martingales it is shown that this proposition is valid and a number of results are obtained which generalize known results for the Wiener process. >+ Department of Statistics~ University of North Carolina~ Chapel HiU~ North Carolina 27514. This research was supported by the Air Force Office of Scientific Research under Grant AFOSR-68-1415 and by the Office of Naval Research under Grant N00014-67A-0321-0006. 1. INTRODUCTION Problems in detection theory and in classification theory or pattern recognition reduce to studying the equivalence or singularity of the probability measures induced on the probability space by the stochastic processes under consideration; and in computing the Radon-Nikodym derivative in the former case. These processes induce probability measures on appropriate L 2 spaces, and the original detection or discrimination problem is often treated on this Hilbert space where a powerful structure is available (see for instance [16, 17, 1]). It is the purpose of this paper to study the validity of the proposition stated in the abstract. It is shown that this proposition is not true in general (Proposition I and Theorem 4); specifically, two stochastic processes may be singular and yet their induced measures on 4). L 2 be equivalent (Theorem Under appropriate continuity conditions it is shown that the proposition is valid (Theorem 2). The second order stochastic processes for which the proposition is shown to be valid are those that are continuous in probability or smooth, a notion introduced in [4]. Weakly, as well as mean square, continuous processes are smooth and a further class of smooth processes is given in Section 3. It is shown there that all square integrable martingales satisfying certain conditions are smooth stochastic processes (Theorem 5). Also their linear span, their reproducing kernel Hilbert space, and the range of the square root of their covariance operator are characterized (Theorem 6); these spaces playa significant role, especially in the Gaussiom case. These results generalize well known facts for the "Uener process and, in the case of the reproducing kernel Hilbert space, for mean square continuous processes with orthogonal increments. 2 Finally, the relationship between the reproducing kernel Hilbert space of a stochastic process and the range of the square root of its covariance operator is established in the most general case (Theorem 3), generalizing a known result for mean square continuous processes defined on compact interva1s. This relationship is of primary importance in the Gaussian case. 2. THE RELATIOI~SHIP BEHJEEi',1 THE EQUIVALEiKE OR SINGULARITY OF STOCHASTIC PROCESSES AHU OF THE HEASURES THEY Ii~DUCE OIJ L • 2 Throughout this section the following nota,tion is used. are two probability measures on stochastic process {~(t,w), and covariance functions r (t,s), 1 R1(t,S) tET} respectively; R, DEFINITION 1. P and 1 with respect to which the measurable is second order with mean, autocorrelation mO(t)=O, generated by the random variables the real line (n,F) Po rO(t,s), RO(t,s) F is the and m (t)=m(t). 1 a-algebra of subsets of {x.(t,w), tET}, and n T is any interval on open or closed, bounded or unbounded. N is the class of measu~es the a-algebra of Lebesgue measurable subsets of the Lebesgue measure, v ~ v T) on (T,B(T)) (B(T) is which are equivalent to Leb, (i.e. mutually absolutely continuous) and satisfy fT r.(t,t) dv(t) < + ~ ,i=O,1. 00 (1) N is a nonempty class of measures, as it is demonstrated by the following construction. Define g E L (T,B(T),Leb), 1 v on g(t) > 0 (Tf> (T)) by a.e. [Leb] [dv/d Leb](t) = f(t)g(t), on T, and where 3 = { f(t) Clearly v€N v€N, For every x(·,w) € L (T,B(T),v) 2 v (H =L (V) , C(H v . 2 », nV , subsets of v and also is finite. the measurability of x(t,w) and = L 2 (V) i=O,1. Also the map where a.s. [Pi]' v B(H) is the (1) imply that T: (n,n-+ a-algebra generated by the open defined by T w = x(·,w) (2) is measurable [15], and thus the probability measures induce probability measures v ~i P., i=O,l, ~ on en, F) defined by on (3) for all v B€B(H ). It follows by (1) and (3) that E [ i II ull 2V ] = H . 2 Ilx(·,w)11 V dP.(w) n H ~ J Hence [13] the mean elements and the covariance operators of the probability measures v ~i' i=O,l, are defined as the elements u v i v € H and as the bounded, linear, nonnegative, self-adjoint. trace @lass, operators which satisfy v E.[<u-u., v> ~ 1 HV v <u-u , w> ] = i HV 4 v v, weH • for all It is easily seen that integral operator with kernel Ri(t,s) and that [15]. {x(t,w), teT} if the stochastic process v ].Ii then the induced measure s~ ~ is an It is also shown in [15] that is Gaussian with respect to P. ~ is Gaussian. A remark should be made about the need to consider the family of measures N. What is really needed is that the probability measures Pi induce probability measures interval T, ].Ii on some space of square integrable functions on the and that ].Ii have covariance operators. vEN, for this induction to be possible is that square integrable functions on that if x(t,w) x(t,w) \leN V H • is Gaussian with respect to necessary condition. choice of T is in which case the space of In particular, it is shown in [15], Pi' \lEN then is also a If the Lebesgue measure satisfies (1), then the natural is clearly \I = Leb. This is the case if, for instance, is mean square continuous with respect to both is a compact interval. For example, if A sufficient condition x(t,w) Po and PI and T However the Lebesgue measure is not always in is wide sense stationary with respect to Po N. and PI and T is an unbounded interval, then the Lebesgue measure is not in N, but every finite measure equivalent to the Lebesgue measure belongs to N. Before we consider the relationship between the pairs of measures we prove that the equivalence and of the measures and ].I~ does not depend on (~) or singularity (~) \I€N. THEOREM'l. If anyone of the following relations is satisfied for some \I€i'J: then it is satisfied for all \lEN. For the proof of Theorem 1 we need the following 5 LEr~~ 1 [4, Theorem 2J. functions in LZ(V)' vEN, {f~(t)}~=l Let be a complete set of "" {n k (w)}k=l and the random variables be defined by (4) a. e. [P.]. 1 Then the subspace random variables v {fk}k in Hi(X,{f~}k'V) of v {nk(w)}k' L (Q,F,P ), 2 i does not depend on L (V); it is denoted by 2 which is spanned by the vEN and on the complete set H(x,smooth). PROOF OF THEOREM 1. For vEN and Hv = LZ (v) define Then T GV is a a-algebra of subsets of Gv (defined by (2», c F. Q and because of the measurability of depend only on the values of the measures the completed and prove that for all G~1 = G~1 on 11 i We will denote by a-algebras with respect to the- measure p., i=O,l, 1 v,AEN , i=O,1. (5) is the smallest If algebra of subsets of v It is clear from (3) that the measures a- V H with respect to which the linear functionals v }"" v v {Fk(u) = <u,f v k > v k=l are measurable, and since nk(w) = (Fko T)(w), it H follows that GV = FV(n), where FV(n) is the a-albebra of subsets of generated by FV(n) c F. Because of the measurability of x(t,w) and (4) , Thus it suffices to prove F~(n) = F~(n) , i=O,l. Because of the symmetry, it is enough to show Q (6) and this for 6 i=O. for every A j, A v) = HO(x,{fk}k' HO(x,{f~}k' Since by Lemma 1 'V n j E HO(x, {fk}k' v) A), it follows that and thus for some constants {ajk}k' 00 I a jk n~ (w) k=l where the convergence is in n.A(w) which implies that A =- FO(n) L (Q,F,P ). Hence along some subsequence 2 O is J F~(n) - measurable for all v FO(n). Note that (5) implies that every set F = where A.DC., ~ ~ and Clearly zero. and thus j, C. ~ F c.EF. G V is a subset of a can be taken disjoint from and in this case A is of the form in G p. set of measure ~ A. (if not replace by C .'VA.) ~ ~ ~ This will be used in the following. ~ We now complete the proof of the theorem by proving the following proper ties v v ].10 Pl/Gv « P \) OIG V if and only i f v ]..11 J. lJ O Pl/Gv Pi/Gv where v v A Proof of (7). FEG V A ]..11 J. ].10· , P l/Gv i f and only if v V,AEN , ]..11 « A A i f and only i f and only if Since POIGv J. Pl/Gv « POIGv- i f and only i f denotes the restriction of show that for any ]..11 'V ]..10 i f and only i f ]..11 « ]..11 'V ]..10' v ].10 P. ~ Pl/GIi. « J. (7) POIGA (9) POIGv Pl/GA to GV • i f and only i f J. POIGA (10) Clearly (7) and (8) A ].11 « Assume F = T-l(B) and (9) and (10) imply that v ].11 «].10 for some A and hence, ]..10 For notational convenience we will put v Then PO(F) =0 v \) if ].11 J. ].10 Pi/Gv be such that and let v BE0(H ). (8) = Pv.• PO(F) ~ = O. and (3) imply 7 v = 0, ~O(B) v P~« Conversely assume Then, by (3), v (3), ~l(B) PO(F) = 0, . Iies 1mp = O. PO(F) Since PO(D ) = 0, O = 0, ) and by (3, Pl(F) P~ B€o(H and let where = O. Thus Pvl « Pov • v v ~~(B) be such that ) v = T-1 (B)€G. F Hence Pl(F) =0 = O. and by v ~l« which implies Proof of (8). v v P1 «p 0 =0 ~l(B) hence ~O· Because of the sYmmetry it suffices to prove that A A PI «PO. A F€G, we have which implies Pl(F) $ Pl(AOUD O) A A proves that Pl « PO. = AOUC O' F = O. Pl(DO) and thus v v P l « Po Assume A v =0 PO(F) = 0, F€G AO€G, where Now Pl(AO)+Pl(D O) $ and let i.e. be such that C~DO€G implies Pl(F) = 0, v Pl(AO) and =0 which Then t here exi sts B~-l3(Hv) ~ suc h t hat v v v ~ and by (3), Pl(F) = 1 = PO(F c) ~l(B) = 1 = ~O(B). If F = T-l(B), then F€G v v which proves that P l « PO. Proof of (9). v Assume v Conversely assume v ~l L PI ~O· v F€Gv , Pl(F) = 1 = PO(F~). Since some v BO' Bl€J(H ). It follows by (3) that -1 T -1 (B nB ) = T 1 O v ~l(Bl~BO) (Bl)nT v L v Po that ioplies Pl(F) A Ai€G , Ci€F Po (AO) and since c = F = T-l(B l ), ~~(Bl) = 1 0 and by (3), which implies v v ~l L ~O = ~~(BO). v ~i(BlnBO) = for But O. Hence • Because of the symmetry it suffices to show that A A PI L PO. = 1 = PO(F'~. where we have (B ) = FnF O v = 1 = ~O(BO~Bl)' Proof of (10). PI -1 such that Oc F = T-l(B ) O Then there exists PO. L and v v PO. Then there exists F€G such v F, FC€G we have F = AlUC l , Fe = AOUCO' Assume Since Pi(C i ) AlnAO =~, v PI = 0, L i=O,l. A A PI L PO. It follows that Pl(A l ) = 1 = 0 We now consider the relationship between the equivalence or singularity of the pairs of measures (PO,P ) l and it suffices to do this for a fixed v€N. script indicating the dependence of H, v ' v (~O' ~l)' v€N. In view of Theorem 1 Thus in the following the super~i and Si on v€N will be omitted. 8 The proof of the following two propositions is contained in the proof of Theorem 1. PROPOSITION 1. (1) PI « Po (2) PI (3) III .L 'V implies Po implies Il O implies ]ll « III 'V Il O Il O PI .L Po In general the inverses of (1), (2) and (3) of Proposition 1 are not true. This is not surprising because of the following. algebra of subsets of defined by (2). the measures Since Il T is measurable, Thus i f F is larger than PI on 'V Po Theorem 4. G. G be the which is the inverse image of B (H) Q depend only i Let GcF under a- T, and it is clear from (3) that .. on the values of the measures G it may happen tha t Pl.L Po Pion G. on.F and ye t That this is not an hypothetical situation, it is proven in Thus Proposition 1 cannot be improved unless restrictive assump- An appropriate assumption would be that tions are made. larger than G, specifically that F is not essentially Gi = Fi , where the subscript i completion with respect to the measure P., i=O,L 1. denotes The implications of this assumption are stated in Proposition 2 and it is shown in Theorem 2 that a large class of processes satisfies this assumption. PROPOS ITION 2. If Gi = Fi , if and only i f then i=O,l, (1) PI « (2) PI Po if and only i f III (3) PI .L Po if and only if III .L 11 0 'V Po DEFINITION 2. processes: III « 'V Il 11 O 0 Define the following classes of real, measurable stochastic 9 Sl is the class of continuous in probability stochastic processes {x(t,w), t€T} such that S2 Leb« v and x(',w) € a-finite measure L (v) 2 v on (T,8(T» a.s. is the class of continuous in probability, second order stochastic processes . S3 for which there exists a {x(t,w), t€T} • is the class of smooth, second order stochastic processes {x(t,w), t€T}, defined in [4, Theorem 4]. S is the union of Sr S2 and p S3' It is shown in [4] that the weakly continuous, and therefore the mean square continuous proces~are smooth. Further classes of smooth second order processes are given in Section 3. THEOREM 2. {x(t,w), t€T} If with respect to both probabilities Po and PI' belongs to the class $, then (1), (2) and (3) of Proposition 2 are valid. PROOF. For x(t,w) x(t,w) take any complete set in as in (4). in v L (v) 2 Denote by Sl a measure v on to be any measure in (T,8(T» N. Let is given. For {fk(t)}~=l be 00 and the random variables F(n) the {nk(w)}k=l a-algebra of subsets of be defined n generated by (11) This is shown in [3,11] for the class for the class S3 $1; in [11] for the class S2; in the same way as (6) or as in [11, Theorem 3] • shown in the proof of Theorem 1 (between (5) and (6» follows by (11) that G = F i i that G = F(n). and thus Proposition 2 applies. 0 and It is It 10 Before we proceed we need the result stated in Theorem 3. L (n,F,P ) O 2 be the subspace of tET} and let covariance RKHS(R ) O RO(t,S). spanned by the random variables be the reproducing form RKHS(R ) O f(t) = EO[~ ~ernel It is well known [14] that isomorphic with corresponding elements and that x(t,w) for all the integral type operator from t€T HO(x) and HO(x) and some to L (V) 2 ~ HO(x) {x(t,w), Hilbert space of the and RKHS(RO) RO(·,t) consists of all real valued functions x(t)] Let E HO(x). are respectively, f on T of the Let also with kernel A be x(t,w). Since by (1), A is a Hilbert-Schmidt operator. THEOREH 3. fE range i.e. (1) If fERKHS(R ) O then f€ range (S~). Conversely, if (S~) then f is equal a.e.[Leb] on T to a function in RKHS(RO); every equivalence class in range (2) range (A) (S~) contains a function in = range (S~). The relationship between the reproducing kernel Hilbert space of a stochastic process and the range of the square root of its covariance operator, established in Theorem 3.1, plays a significant role for Gaussian processes as it is demonstrated in Theorem 4. Theorem 3 generalizes a well known result for zero mean, mean square continuous stochastic processes defined on a closed and bounded interval (see for example [8]). Theorem 3 that the linear manifold .range the sense explained in the theorem. Note that it follows from ~S~~) is invariant of vEN, in 11 For the proof of Theorem 3 we need the following LEf1MA 2[4, Theorem 3]. eigenvalues and the corresponding eignefunctions of SO' 00 {~k(t)}k=l let be the versions of the eigenfunctions which are defined for all Ak~k(t) RO(t,s)~k(s)dv(s), and let the by ~k(w) = IT x(t'W)~k(t) dv(t) = IT be defined EO[~k~j] = AkO kj , and for all by {~k(w)}~=l ~kEHO(x), random variables a.e. [Po]· tET Then t€T 00 x(t,w) = l: k=l !jJk(t) ~k(w) + w(t,w) (12) where the equality as well as the convergence of the series are in L (G,F,P )' 2 O and PROOF OF THEOREr:J 3. Note that 00 I ~~k range(sJ) = {f = 00 in L (v), 2 k=l (l.i) Let fERKHS(R O). I a 2 k < + oo} (13) k=l A k Then for some ~€HO(x) and all tET, we obtain by (12), 00 00 f(t) = 00 L k=1 00 Hence L k=l 2 ak<+oo l: ak~k(t) k=1 2 00 a k = L EO[~ A k=l k for all ~k -] A-t k ~ tET'VT EO[~2] < + (14) O (15) 00 00 and k~l ak~k(t) converges in which, because of (14), is equal a.e.[Leb] on T to L2 (v) to a function f(t) • Hence 12 co = f L ak~k k=l (l.ii) in L2 (V) Conversely, let (S~). Then f = fe range co co -1 L with k=l L (n, F,P ) 2 O in range(s~). fe and by (13) and (15), Ak ~~k(w) -1 Ak ~ 2 co L k=l ~~k in converges in co < + Let co. ~ = \' -1 L Ak k=l ~~k It follows from LZ(n,F,P O). co = L EO[~ ~k]~k k=l such that f that there exists a subsequence in f(t) = Urn k~ Nk L n=l EO[~ ~n] 4J n (t) a.e. [Leb] on T N k = Urn L EO[~ k~ n=l ~n(tHn] a •e. [Leb] on and thus EO[~ f equals a.e.[Leb[ on T a function in )' namely O RKHS(R x(t)]. Let (Z. i) fe range (A). Then there exists a.e.[v] (Z.U) co \' with L -1 k=l a.e.[Leb] on Z ~ < fe T, where f(t) ~ -1 = L Ak k=l = EO[~ ak~k a.e.[v] x(t)] T. It in L2 (V) in co L k=l f(t) L2 (n,F,P O)· on T, i.e. ak~k = E[~ It f = x(t)] follows that A~ and 0 We now consider the case where to both probability measures ~l such that a.e. [Leb] on range(s~). Then f = and as in (l.ii) we obtain co co and fe range(A). + T, hence HO(x) f€ range(SO). Conversely, let Ak on ~e k follows as in (l.i) that and T. Po are also Guassian [15]. and {x(t,w), teT} Pl. is Gaussian with respect Then the probability measures ~O It is well known that both pairs of Gaussian 13 measures (PO,P l ) (~O'~l) and are either equivalent or singular [9]. Necessary and sufficient conditions for ~l ~ ~O are given in [16]. RCt,s), and thus ~ PI Po are given in [18] In the particular case where So = Sl = S, RO(t,s) and for = Rl(t,S) = these conditions are: i f and only if m € RIaIS (R) (16) i f and only i f m € ral1ge(S~) (17) The relationship between the equivalence or singularity of the pairs of measures (PO,P l ) (~O'~l) and is given in Theorem 4. This theorem demon- strates that a detection or discrimination problem;., ,.,hich is defined on the n, probability space functions H can be treated on the Hilbert space of square integrable = L2 (v) , where powerful analytic tods are available, only when case (iii) can be excluded. S, belongs to the class This is the case, for instance, if the process as it is shown in Theorem 2. However, in general, it may be that the two processes are singular and yet their induced measures on H = L2 (v) are equivalent. THEOREH 4. abilities Po PROOF. measures to {x(t,w), t€T} If and PI' is Gaussian with respect to both prob- then one of the following will always be satisfied (i) PI ~ Po and ~l ~ ~O (ii) PI ~ Po and ~l ~ ~O (iii) PI ~ Po and ~l ~ ~o In view of Proposition 1 and the fact that both pairs of Gaussian (PO,P l ) and (~O'~l) are either equivalent or singular, it suffices prove that case (iii) is possible. This is shown by the following example. 14 Take (n,A,p) ~ T = [a,b], - We can find a probability space and independent, zero mean and unit variance, Gaussian random vari~l(w), ~2(w), ~3(w) abIes +~. < a <b < properly included in the completion of denotes the sub- a-algebra of F(~1';2'~3)' {xi(t,w), t€T}, d(t)~l(w) for aSt<c d(c)~2(w) for t = c d(t)~3(w) for c<tsb i=O,l, by and all wen for all where and d(t) and ~ d(t) 0 m(t) on T. Then d(t) € L (v) ) • 2 PO' PIon properly contained in = Rl(t,s) RO(t,s) given m(t) , v Leb ~ t€T and Then Gaussian measures F i ~ 0, are measurable, respectively. J.I 0 ' J.I l and and thus ~ RKHS(R ). O f€ range (S~) o and f~ Gi is Note that It is shown in Lemma 3 that d(t), one can always find a function m v€N Take are induced on Gi = Fi (~l' ~3) For this choice of PI ~ 3. There always exist real valued functions such that xl t€T, d(c) (so that Proposition 2 does not apply). given m € range(si) {;j,j€J}. is to be constructed it suffices it follows from (16) and (17) that we have LE~4A all o and thus (16) and (17) apply. RO(t,s), i.e. such that X (n,F) It is easily seen that B(H». j€J) and (note that if an example for a given = L2 (V), clearly be Wen Also induce Guassian probabilities (H t€T, are defined and finite valued for all a.e.[Leb] Gaussian processes. to take F(~j' where A generated by the random variables Define the stochastic processes xO(t,w) = F(;1'~3) on it such that the completion of RKHS(R ). O Po m(t), t€T, d(t) and o and f(t) defined for 15 PROOF. Let g€ RKHS(RO). Then, by Theorem 3.1, there exists some ~ € HO(x) such that g(t) = EO[~ x(t)] for all t€T, and by (12) 00 (18) for all t€T, where Note that for every function f(t) =g(t) a.e. [Leb] on f(t), t€T, f€ RKHS(R ) O such that and T, we have 00 f(t) = L ~~k(t) k=l for all t€T t€T '\, TO all [Leb] and T on n € HO(X). Since by Lemma 2, 2 EO[w (t)] = 0 Leb(T ) = 0, it follows by (18) and (19) that O t€ T '\, TO· f(t) = g(t) have and some (19) + EO[n w(t)] Hence, if we choose to for f € range g(t) but not equal to t € T '\, {to} (S~) f( t), t€T, and n and g(t) f(t) = g(t) on T '\, TO where (for instance to € T '\, TO) we 0 It follows from (16), (17) and Theorem 3 that, in the case where Rl(t,s), for equal almost everywhere f(t O) ri g(t O)' RKHS(R ). O for all RO(t,s) = necessary and sufficient conditions for (i) to (iii) of Theorem 4 are respectively (i) , m (ii) , m ~ range(SO) (iii) , m € ~ Rlms (R ) O h: RKHS(R ) O and h: m € range(S~) • Necessary and sufficient conditions for the general case RO(t,s) ri Rl(t,s) can be obtained in a similar way; they are not given here for space considerations. 16 k f e: range (S~) It is shown in Theorem 3.1 that equal a.e.[Leb] on T to a function in RKHS(R ). O implies that is f By combining (17), Theorem 2 and (16) we obtain for the mean function the following COROLLARY. {x(t,w), te:T} implies 3. Let If with respect to both probabilities is Gaussian and belongs to the class and then PI' k m e: range(S~) m e: RKHS (R ) • O A CLASS OF Sr·100TH SQUARE INTEGRABLE NARTINGALES {x(t,w), te:T} be a real, measurable, second order stochastic process defined on the probability space functions r(t,s) Denote by H(x) included in and R(t,s); The subspace H(x) (n,F,p) T the subspace of {x(t,w), te:T}. i, with autocorrelation and covariance is any interval on the real line. L (n,F,p) 2 H(x,smooth) so that Pi=P). is called smooth if and only if spanned by the random variables defined in Lemma 1 is always N is as in Definition I for one (in the present case value of the index te:T} S, Po The stochastic process H(x) = H(x, {x(t,w), smooth) [4]. In the following the class of square integrable martingales will be considered. A stochastic process matringale if a.s. for all s ~ t in by the random variables T, where {x(t,w), te:T} is a square integrable for all and if F s te:T, E[x(t,w)IF] is the sub-a-algebra of {x(u,w),ue:T,u~s}. F generated If is a square integrable martingale then we have the following. exists a monotone nondecreasing function 2 E[{x(t) - x(s)} ] = F(t) = x(s,w) Note that all stochastic processes with orthogonal increments are square integrable martingales. t€T} s F(t) - F(s) on {x(t,w), There T such that (20) 17 for all s ~ t in T. [12]. Since F(t) function, the left and right limits tET, and the set able. F(t-) and F(t+) D of points of discontinuity of exist at every point F(t) is at most count- It follows from (20) that the left and right mean square limits and x(t+ ,w) exist at every point points of mean square discontinuity of x(t,w) let is a monotone nondecreasing tET x(t,w). and that It is also easily seen that is not weakly continuous at the points of F(t) f F(t-). Then it follows by (20) that t n t t and ~ E H(x) D is the set of D. Indeed let tED x(t,w) f x(t-,w) is not orthogonal to and in x(t)-x(t-) we have lim E[x(tn )~] = E[x(t-)~] f E[x(t)~] n which shows that x(t,w) is not weakly continuous at tED. The following condition will be considered: (Cl) At every point of mean square discontinuity (tED) the square integrable martingale equals either its left or its right mean square limits or equivalently, (C2) At every point of discontinuity (tED), F equals either its left or its right limit. THEOREM 5. If a square integrable martingale satisfies (el) then it is smooth. PROOF. and f(t) = 0 a.e. [Leb] assume that f(t) By [4, Theorem 4.1] it suffices to prove that if f(t) = 0, T, then f( t) = 0 for all tET. = E [~ x( t) ] for all tET, where = 0 for all tET Leb(T') on 'V T', where there exists a sequence Leb(T') {t } n n f E RKHS(r) ~ E H(x), Indeed and that = o. Now let tET' . Since of points in T 'V T' con- 18 verging to t; in particular there always exist increasing (decreasing) such sequences. We clearly have tim f(t) = O. n n and we obtain If f(t) creasing sequence tED, =0 processes (C2) If ~ t D, it follows that by (Cl), either t ~ f(t) = or D by choosing an increasing or de- respectively. Thus =0 f(t) for all tET and 0 the theorem is proven. REfvJARK. n as when {t} n n = O. F(t ) The proof of Theorem 5 applies to all second order stochastic {x(t,w), tET} For all and tET x(t,w) satisfying the condition equals either which are thus smooth. x(t-,w) the mean square limits x(t-,w) and x(t+ ,w) exist x(t+ ,w) or This follows from the fact that if (C2) is satisfied then the set of mean square discontinuity points of x(t,w) is at most countable [6, Lemma 1]. We now proceed to characterize the reproducing kernel Hilbert space and the range of the square root of the covariance operator of a square integrable martingale. These spaces play an important role in the Gaussian case, as it is seen from Theorem 4 and the conditions (i)', (ii) , and (iii)'. It will be assumed that the following simplifying condition is satisfied: (~3) T is of the form x(O,w) = 0 [O,a], a < + 00, or [O,a), a:S + 00, and a.s. A simple stochastic integral with respect to a square integrable martingale . will also be used here [5,7]. We define F' by for all and we denote by m or the usual way, m{(s,t]} T. For all f E L (dF') 2 dF' tET the measure induced on = F'(t) - F'(s) = F(t+) (21) • (T,B(T» - F(s+) the stochastic integral by for all ITf(t)dX(t,W) F' s:s t in in is a well 19 defined random variable in L (n,F,p) 2 and has the property (22) THEOREI~1 (~l) 6. For a square integrable martingale satisfying and (C3) the following are true: (1) H(x) (2) If RKHS(R) = {~(w) = JTg(t)dX(t,W), E[x(t)] = = {f(t) = I: ° for all g(u)dF'(u) where the integral in (24) is over F(t). {x(t,w), tET} [O,t) g E L (dF')} 2 tET, then for all for (23) tET tET, g E L2 (dF')} ~ D and for tED (24) with = F(t-), and over [O,t] for tED with F(t) = F(t+); and range(s~) = {f(t) = t J° g(u)dF'(u) a.e.[Leb] on T, gEL (dF')} (25) 2 The results of Theorem 6 are known for the Wiener process, for which F(t) = t •. The characterization of the reproducing kernel Hilbert space (24) is also known for mean square continuous processes with orthogonal increments [10]. As it is suggested by Theorem 6, a number of results established for the Wiener process can be extended in an appropriate way to all square integrable Gaussian martingales satisfying certain conditions (like ~l) and (C3». Results in this direction will be included in a forthcoming paper. PROOF OF THEOREM 6. (1) Let L(x) = {~(w) = ITg(t)dX(t,W), g€L (dF')}. 2 It follows by (<3) that x(t-,w) = x(t-,w)-x(O,w) = ITIro,C)(U)dX(U,W) € L(x) (26) 20 t€T. for all x(t+ ,w) Similarly L(x)cH(x) Hence it suffices to show that and ~ ~ ~(w) = H(x) imply ~ IT g(~)dX(U,w) for all = O. and thus, by (el), L(x) € ~ Assume for some H(x)=- L(x) • or equivalently that € L(x) g € L (dF') 2 and ~ ~ H(x). ~ € L(x) Then and by (26) and (22) we have t€T: =< Since the set of functions follows that i.e. ~ = 0 (2) g = 0 in in is dense in {I [0, t) (u), t€T} L (dF') 2 and by (22), 2 E[ ~ ] = L (dF'), 2 II g II ~ (dF' ) it = 0, 2 L (Q,f,P). 2 It follows by the definition of reproducing kernel Hilbert space, (23) and (22) that = E[x(t)~] RKHS(R) = {f(t) = {f(t) where the integral is over Theorem 3.1 and (24). F(min(t,s». to [O,t] t€T, v€N, Let S g€L 2 (dF')} as explained in the statement follows directly from alternative proof of independent interest is the An i.e., = LL*, v~Leb where and = R(t,s) = ret,s) fTF(t)dV(t) < + 00. Then it is L is a bounded linear operator from L (V) defined by 2 (Lg)(t) and has norm [O,t) or for all Note that, as it follows from (20) and (C.3), easily seen that L (dF') 2 g(u)dF'(u) ~€H(x)} t€T, The characterization of range (S~) of the theorem. following. f: = for all I ILl 1 2 = f: g(u)dF'(u) = fTF(t)dV(t) < + a.e.[Leb] on 00. 0 g € L2(dF') It follows by a straightforward extension of a result given in [2, Corollary 2.c] that and hence (25). T, range(s~)= range (L) 21 REFERENCES [1] C. R. Baker, On equivalence of probability measures, Institute of Statistics Mimeo Series No. ?Ol~ University of North Carolina at Chapel Hill. AnaZ.~ [2] C. R. Baker, On covariance operators, SIAM J. Math. [3] B. H. Bharucha and T.T. Kadota, On the representation of continuous to appear. parameter processes by a sequence of random variables, IEEE Trans. Information Theory IT-16 (1970), pp. 139-141. [4] S. Cambanis, Representation of stochastic processes of second order and linear operations, submitted to SIAM J. AppZ. Math. [5] P. Courrege, Integra1es stochastiques et martingales de carree integrable, S~minaire de Theorie du Potentie1, M. Bre1ot, G. Choquet and J. Deny, ed., Institut Henri Poincar~~ Paris, 7e annee, 1962- 63, No.7. [6] H. Cramer, On the structure of purely non-determinirtic stochastic processes, Ark. Mat. 3 4(1961), pp. 249-266. [7] J.L. Doob, Stochastic Processes~ Wiley, New York, 1953. [8] B. Eisenberg, Translating Gaussian processes, Ann. Math. Stat.~ 41(1970), pp. 888-893. [9] J. Feldman, Equivalence and perpendicularity of Gaussian processes, Pacific J. [10] Math.~ 8(1958), 699-708. 1.1. Gikhman and A.V. Skorokhod, On the densities of probability measures in function spaces, Russian Math. Surveys~ 21(1966), pp. 83-156. [11] E. Masry and S. Cambanis, The representation of stochastic processes without loss of information, submitted to Information and ControZ. PotentiaZs~ [12] P.A. Meyer, ProbabiZity and [13] E. Mourier, Elements aleatoires dans un espace de Banach, Ann. Inst. H. Poincare Seat. B~ Blaisdell, Waltham, Mass., 1966. 13(1953), pp. 161-244. 22 [14] E. Parzen, Statistical inference on time series by Hilbert space methods, Time Series Analysis Papers, Holden-Day, San Francisco, 1967, pp. 251-382. [15] B.S. Rajput and S. Cambanis, Gaussian stochastic processes and Gaussian measures, Institute of Statistics Mimeo Series No. 705, University of North Carolina at Chapel Hill. [16] C.R. Rao and V.S. Varadarajan, Discrimination of Gaussian processes, Sankhya Ser. A, 25(1963), pp. 303-330. [17] W.L. Root, Singular Gaussian measures in detection theory, Proc. of Symp. on Time Series Analysis, M. Rosenblatt, ed., Wiley, New York, 1962. [18] Yu. A. Rosanov, Some renlarks on the paper on Gaussian distribution densities and Wiener-Hopf integral equations, Theor. Prob. Appl. 11(1966), 483-485.