THESIS math“ I Mchlzau State University This is to certify that the dissertation entitled Infinitely divisible measures on multi;Hilbertien spaces and a Levy-Ito decomposition presented by Milan J. Merkle has been accepted towards fulfillment of the requirements for Ph - D - degree in _S.ta_tis_tj_cs_ waits/ask Major professor ' V. Mandrekar Date—AugusL1._1384_ MSU is an Affirmative Action /Equul Opportunity Institution 042771 }V1ESI_J RETURNING MATERIALS: Place in book drop to LlBRARJES remove this checkout from .‘nnuzyn-IL. your record. FINES will be charged if book is returned after the date stamped below. INFINITELY DIVISIBLE MEASURES 0N MULTI-HILBERTIEN SPACES AND A LEVY-ITO DECOMPOSITION By Milan J. Merkle A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree Of DOCTOR OF PHILOSOPHY Department of Statistics and Probability 1984 2;. _.¢"‘_: JQC r“ 317‘? ABSTRACT INFINITELY DIVISIBLE MEASURES ON MULTI-HILBERTIEN SPACES AND A LEVY-ITO DECOMPOSITION By Milan J. Merkle In this work, we give a representation of infinitely divisible (ID) laws on duals of multi-Hilbertien spaces and discuss the convergence. These results give a unified approach to the existant work on infinitely di- mensional Hilbert spaces and on nuclear spaces. The convergence of ID laws can be used to prove the weak convergence of homogeneous processes with independent increments. This is applied to a problem from Neuro- biology, and the results obtained are generalization and improvement of a recent work of G. Kallianpur [16]. The last chapter is devoted to the processes with independent increments on duals of multi-Hilbertien spaces. Levy-Ito decomposition on Hilbert spaces is Obtained. ii ACKNOWLEDGEMENTS The author of this work wishes to thank all faculty members at Department of Statistics and Probability, Michigan State University, for their teaching, understanding and care during the author's two years in the graduate program. Special thanks are due to Dr. V. Mandrekar, whose ideas and suggestions were of greatest value for this work. Many memorable hours are spent with my friends, graduate assis- tants; among other things, drinking American coffee and eating Indian food is what makes life exciting. TABLE OF CONTENTS 0. INTRODUCTION ........................... 1 1. PRELIMINARIES .......................... 3 1.1. Vector spaces and seminorms ................... 3 1.2. Operators on Hilbert spaces ................... 3 1.3. Multi-Hilbertien spaces ..................... 5 1.4. Dual spaces ........................... 9 1.5. Measures on dual spaces ..................... 9 2. REPRESENTATION OF ID LAWS .................... 15 2.1 I(T)-topology and Bochner's theorem ............... 15 2.2. Continuity theorem ....................... 18 2.3 Infinitely divisible probability measures ............ 23 3. CONVERGENCE OF ID LAwS AND HOMOGENEOUS PROCESSES WITH INDEPENDENT INCREMENTS ......................... 33 3.1 Convergence of ID laws ...................... 33 3.2 Homogeneous processes with independent increments ........ 34 3.3 An example ........................... 36 4. PROCESSES WITH INDEPENDENT INCREMENTS .............. 41 4.1 Properties of paths . . . .- ................... 41 4.2. A decomposition of E'-valued processes. ............. 44 4.3. A Levy-Ito decomposition .................... 48 REFERENCES .............................. 59 iv 0. INTRODUCTION The infinitely divisible laws (ID) in the context of infinite- dimensional Hilbert spaces were studied by S.R.S. Varadhan ([23]). For the case of nuclear spaces, a Lévy-Khinchine representation was given by Fernique [6]. Our purpose here is to give a representation of ID laws on a multi-Hilbertien space and discuss the convergence. These results give unified approach to Varadhan's and Fernique‘s work and extend Fernique's work to include convergence of ID laws. Such a unified ap- proach for Bochner theorem for multi-Hilbertien spaces is recently given by K. IuJLI4J. Our results are based on arguments of Fernique for the representation and Levy continuity theorem for the multi-Hilbertien spaces. We note that the method Of Fernique for the Lévy continuity theorem in nuclear spaces fails in the multi-Hilbertien spaces due to an example of Sazonov (cf. 2.1.2). The convergence of ID laws can be used to prove the weak convergence Of additive (homogeneous independent increments) processes. This is shown in the context of an example from Neurobiology. This work includes and improves recent work of G. Kallianpur [16]. In fact, our work is applicable to wider class Of examples. The last chapter is devoted to the Levy-Ito decomposition of infinite- dimensional processes with independent increments (PII). We are unable at this point to prove this theorem for general multi-Hilbertien spaces. In fact, as the reader can see, the results seem to need some work even for the Hilbert space case. However, for the additive processes, such a decomposition follows from the relation of convolution semigroups to ID laws. This, along with the convergence of the related processes, is studied in Chapter 3. Clearly, the results there allow us to extend the 2 work of Kallianpur to multi—neuronal models. The convergence of P11 in this context remains an open problem. But our decomposition is the basic technique needed for this study. In future work we intend to study this problem as a generalization of a recent work of Jacod (ZNV 63, 109-136, 1983). 1. PRELIMINARIES 1.1. Vector Spaces and Seminorms R(N) will denote the set of real (natural) numbers. Throughout this work the phrase "vector space” will mean a real vector space. Let E be a vector space. A real-valued function p defined on E is called a seminorm if for all x, x1, x2 e E and a 6 R : (i) P(X) 3 0 (ii) P(aX) = lalP(X) (iii) P(X1 + x2) f_P(X1) + P(X2) . and (IV) p(x) > O for some x . A seminorm p on E is called a Hilbertien seminorm if (Vxl, x2 6 E) P2(X1 + x2) + P2(X1 - x2) = 2(P2(X1) + P2(X2)) For a Hilbertien seminorm p we define a symmetric bilinear form p(o, -) on E by A 2< 2 p(xls X2) =%(p (X1+ X2) ' p X1 " X2)) p(xl, x2) is called the inner product corresponding tg__p_. A seminorm p is said to be separable if there is a countable set D<: E such that for each x E E and a positive, there exists a d in D such that p(x - d).: e . 1.2 Operators on Hilbert Spaces Let Hi (i = 1, 2) be separable real Hilbert spaces with norms fl.“ and inner products <-, '>i . Let A be a linear mapping from H1 to H2 . 4 A is a compact operator if for every x in H 1 : (1) Ax = Z tnhn n where O < tn + 0 , and en , hn are orthonormal sets in H1 and H2 respectively. A is a Hilbert-Schmidt operator if (1) holds and (2) Xt2l < m for any orthonormal basis {en} in H . For a linear mapping A : H + H we define the adjoint mapping A* by * = * (5) * We say that A is self-adjoint if A = A A is said to be positive (non-negative definite) if (6) .1. aiaj .2 O In] for every finite set Of complex numbers {ai}?=1 and {xi}?=1 a finite subset of H . In the following theorem we give a summary of some well-known facts (see, for example [9], Chapter 1.) Theorem 1. (i) If A and B are Hilbert-Schmidt Operators, then AB is a nuclear Operator. Conversely, every nuclear Operator is a product of two Hilbert-Schmidt operators. (ii) A compact positive Operator A : H + H is nuclear if and only if it is a trace operator. (iii) If A is nuclear (or Hilbert-Schmidt) Operator, then A* is an operator of the same type. (iv) If A : H + H is a positive nuclear operator then A15 is positive Hilbert-Schmidt Operator. (v) A positive compact operator A : Ht+ H has the representation. Ax = E tn en where en is an orthonormal basis for H consisting of eigenvectors of A and tn , O < tn l O are eigenvalues of A , Aen = tn en . (vi) Let H and H2 be Hilbert spaces and A : H1 + H2 1 be a linear mapping. A is a Hilbert-Schmidt operator if and only if v 2 Z. ”Aenuz < 00 n for at least one (equivalently: for all) orthonormal basis {en} in H1 . A is nuclear Operator if and only if §|2. < m , In the following section we describe multi-Hilbertien spaces, following Ito [14]. 1.3. Multi-Hilbertien Spaces Let E be a vector space. The family of all separable Hilbertien seminorms on E will be denoted by Biti- It is easy to see that if P, P1,...,pn e HSN then for every positive c , cp e HSN and (pi + ... + p§)% E HSN . For p, q e HSN we define the relation 3 : p 2 q if and only if p(x) §_cq(x) for some c e R and all x e E . Let Ep denote E with topology given by the seminorm p . Let ker p = {x e E : p(x) = 0} . Then Ep/ker p is a pre-Hilbert space; its completion ED is a Hilbert space. If p s q then the identity map E + E can be extended to a continuous linear Operator E2 + E . Notice that i (x) = x on E , 1M : P ms and the indices p, q are just pointing the topology for completion. Define the relation < ___ : p < q if and only if p 3 q and i __ HS HS '3") is a Hilbert-Schmidt operator. In view of Theorem 3.1.(vi), p < q if and only if 2P2(en) < 0° where {en} is an orthonormal basis in Eq Let P = {pd}jed be a subfamily of HSN on E such that (i) If p e P and p S q or q t p then q 6 P (ii) If P9q E P , p(xn) + 0 , x is q-Cauchy then q(xn) + O . ll Define a locally convex topology T on E by its neighborhood basis at zero: I . ' ' 1 (1) tx E E . pj1(x) < 1, ... ,pjk(x) < 1 , 31, ... ,Jk 6 J. k = l, 2, .... 7 The topology 1 given by (1) is called a multi-Hilbertien topology generated by P Space E with a multi-Hilbertien topology T , (E, T) is called a multi- Hilbertien space. If a multi-Hilbertien topology T is generated by P = {pj} we will JEJ ' = .1. often write T {pJ’JEJ If T ‘ “’fljea a countably Hilbert space. In this case, without loss of generality, we where J is a countable set. we say that (E, T) is may assume (2) pl < p2 < p3 < ... L For, if not, then the topology given by qj = (pi + pg + ... + p?)2 , j = 1, 2, ... coincide with the topology given by {pi} and {qj} satisfies condition (2). In a countably Hilbert space we will use notations En, En, E5 , rather than EP , etc. ... D If the topology 1 is given by countably many norms {pn} (i.e. ker pn = {0}) , then we can define a metric d by L. n 1+Pn(x1-x2) (3) d(x1. x2) E is said to be complete if (E, d) is complete. E is complete if and only if ([27]) (4) E = p E pn We say that (E,t)is a nuclear space if for every D E T there is a q E T such 8 that péécq. Notice that, by Theorem 3.1.(i), E is nuclear if and only if for every P E T there is a q 6 T such that ip q is nuclear operator. A nuclear space in which condition (4) holds is called a nuclear Fréchet SEBCE. Suppose that {e3}:=1 is a complete orthonormal basis is En , n = 1, 2, ... . Then E is a nuclear space if and only if for every n = 1, 2, ... there is an m > n such that (Theorem 3.1.(vi)). (s) g p§ = (1 r (1+t2)”|f(k)(t)l2dt)2 k=O R Let r = {pn}:1 . Then (s, r) is a nuclear space, with En = {f E Cm(R) : pn(f) < ac} and the inner products: n pn(fl, f2) = ; é (1+t2)" f1(k)(t) f2(k)(t)dt Remark. A separable Hilbert space H is nuclear if and only if dim H < w . Let now (E, T) be a multihilbertian space, I = {pj} We say that jet] a set Bc: E is bounded if for each 3 6 J the set of numbers {p.(x) : X E B} is bounded. 1.4 Dual Spaces Let (E, T) be a multi-Hilbertien space. The set of all (T-) continuous linear functionals on E is denoted by E; (or E' if there is no con- fusion about the topology on E ). If F e E' , x 6 E, then F evaluated at x is denoted by . In this dual notation it is understood that for fixed F (or x), is a function of x (or F). Also, separates points in both E and E' . The strong topology on E' is given by seminorms HFHB = SUP [l , B-bounded set in E . X63 Let E' denote the topological dual of Ep . E6 is a Hilbert space p with the norm p(F) = sup [| . P( X):1 We have E' = U E' . P p In the strong topology, E' is an inductive limit of {E6}p€T_[14J. If T is countable, the set A is bounded (compact) in E' if and only if it is bounded (compact) in some E5 , p E T [81. 1.5 Measures on dual spaces Let E' be a topological dual of a topological vector space E . Let A be a given Borel set in Rn . Let (1) Z={FEE':(,, ... , ) e A} Set Z defined by (1).for some x1, ... ,xn €.E is called the cylinder n O set with base A and generating elements x1, ... ,x 10 Another approach to defining a cylinder set is the following: Let Y be a finitely dimensional subspace of E . Let Y0 denote the annihilator of Y , i.e., the set of all F E E' such that = O for' x E Y Consider the factor space E'/YO. It is isomorphic to Y‘ , thus finitely dimensional. Let Ac: E'/YO be a Borel set. The set of all F E E' which are carried into elements of A by the natural mapping E' + E'/YO is called the cylindar set with base A and generating subspace YO . These two definition are equivalent and define the same object ([9]). Note that a cylinder set may have more than one representation in terms of base and generating subspace. It is easy to see that the cylinder sets form an algebra of'sets—c lin- der algebra. By a cylinder set measure in E' we mean a nonnegative function M defined on the cylinder algebra with the following properties: (i) If 2:321. where zin Zj=P if i763“, andall z}. are i=1 generated by the same set x1, ... ,xn , then M(Z) = E M(Zi) . i=1 (ii) For any cylinder set Z , M(Z) = inf M(U) , where U runs through all open cylinder sets containing Z . Let M be a cylinder set measure. For Y CIE a finite dimensional subspace we define (2) MY“) = M(Z) where A is a Borel set in E'/YO and Z is the cylinder with base A 11 and generating subspace YO. So, (2) defines a Borel measure on E'/YO which is regular, i.e. (3) MY(A) = 13f M(U) where U runs through all open sets containing A . Measures MY are compatible in the following sense. Let Y1<: Y2 . Let T be the natural mapping E'/Y3‘+ E'/Yi . Then (4) MY1(A) = MY2(T'1(A)) Conversely, if the system of measures {MY} is given, satisfying condi- tion (3) and (4), then there is a unique cylinder set measure M such that (2) holds for every Y . If M is a cylinder set measure, then for disjoint cylinder sets n n 1, ... ,Z , we have M( U Z.) = Z M(Z.) , which follows from (i) n i=1 ‘ i=1 1 and the fact that for finitely many cylinder set there always exists a Z common generating set. However, a cylinder set measure may not be countably additive. If it is so, then it can be extended to a measure on cylinderci-algebra C, which is defined to be the smallest o-algebra that contains cylinder algebra. We say that the measure M is a probability measure if M(E') = 1 . Let now E be a countably Hilbert space. Then ([14]) the cylinder o-algebra Z coicides with the Borel<:-algebra generated by the strongly open sets. Every probability measure on E' is regular. Let (E, T) be a multi-Hilbertian space. A probability measure P on (El, E) is called separable if there exists a countably Hilbert topology r' c:*: such that 12 P(E'T.) =1 A probability measure P on E' is called infinitely divisible (ID) if, for every n e N , P can be represented as n-th convolution power of some other probability measure Pn . In terms of random variables, X is an ID random variable if for every n e N it can be represented as a sum Of n independent identically distributed (iid) random variables. We say that the sequence Mn Of measures converges weakly to a measure M (Mn = M) if for every continuous bounded real function f defined on E' we have, as n + w : é'f(F)dMn(F) .+ £.f(F)dM(F) For a measure M on E' we define its characteristic functional as a complex-valued function defined on E by f(x) = f eidM(F) . EI The characteristic functional Of an ID random variable will be called an ID characteristic functional A set M Of measures on E' is said to be (weakly) relatively compact if every sequence Mn in M contains a weakly convergent subsequence. A set M of measures on E' is called pigpp if for every 5 > 0 there is a compact set K such that M(KC) < e for every M 6 M . A result in [30] confirms the validity of Prohorov's theorem in E' , i.e., a sequence Mn of separable probability measures is relatively compact if and only if it is tight. 13 We observe that a sequence of separable measures Mn is weakly con- vergent to a measure M if and only if it is relatively compact and the sequence fn of characteristic functionals fn of Mn converges point- wise to the characteristic function f of M . This follows from the fact that characteristic functional determines the measure uniquely. Example: (Gaussian measures) Let C be a complex valued bilinear func— tion defined on E x E , satisfying C(x, x)“: O , C(xl, x2) = Clxz, x1), continuous in both arguments, and non-degenerate (C(x, x) = O =1x = 0) Let Y be a n-dimensional subspace of E . We define a measure gY: 1 9Y(A) = n J exp (‘%‘ C(y. NW (21% where dy is the Lebesgue measure in Y corresponding to the inner product C . Finitely dimensional Euclidean space Y with inner product C is isomorphic to Y' which is isomorphic to E/YO . Therefore, there is natural isomorphism TY between Y and E'/YO . Now define a measure GY on E'/YO by _ -1 (2) eY(e) - gymY (8)) Now we have a set of finite dimensional measures. It can be shown [9] that (2) defines a compatible set of measures; thus, a cylinder set measure G on E' is determined. We call it centered Gaussian cylinder measure . If E is a nuclear space, every cylinder measure is countably additive, therefore it can be extended to cylinder<3-algebra. In the case of general countably Hilbert space, a sufficient condition for countable additivity of G is that for some n , the identity map- 14 ping from En into EC is Hilbert-Schmidt operator, where EC is E topologized by the norm C%(x, x) The characteristic functional of centered Gaussian measure G defined as above is PM = exp <-% co. m The function C is called covariance and, as we have seen,it uniquely determines a centered Gaussian measure. If C is not non-degenerate, i.e., if for all x in some linear subspace X we have C(x, x) = O , then C is nondegenerate on E/X , so a Gaussian measure G1 on X0 can be constructed following the procedure defined above. Then we define a Gaussian measure on E' by C(A) = G1(A 0 x0) Finally, if G is a centered Gaussian measure, ‘%= the measure that gives mass 1 to some element F of E' , then a noncentered Gaussian measure is defined by (SF * G , where * denotes the convolution. The characteristic functional of noncentered Gaussian measure <fi; * G is f(x) = eXP(i - % C(x, X)) Gaussian measure is an infinitely divisible measure. 2. REPRESENTATION OF ID LAWS 2.1. I(r)--Topology and Bochner's Theorem 1. Definition. Let (E,r) be a multi-Hilbertien space. We denote by 1(1) the Hilbert-Schmidt topology induced by all those Hilbertien semi- norms which are < to some seminorm in T , i.e. HS (1) 1(1) = {g e HSN : q < in for some p e 1} HS (Recall that, by the convention in 1.3., (1) means that I(T) is generated by the set of seminorms on right hand side.) If E is a Hilbert Space, then I(t)-topology coincides with so called S-topology which is proven to be of importance in studying characteristic functionals. In fact, there is a complete analogy between the role of S-topology in Hilbert spaces and the role of I(t)-topology in dual spaces. 2. Notation. Let (H, | )' be a separable Hilbert space. Let T, TH’ TN, 5, be topologies defined by: T = {I'll TH = {p : p(x) = flAx” , A is a Hilbert—Schmidt operator} TN = {p : p(x) = % , A is a positive nuclear operator} 5 = {p : p(x) = %, A is a positive compact, trace class operator} Operators A in the definition of S-topology are usually called S-operators. 15 16 = S = 1(1) 3. Lemma. TH TN Proof. Let us first show TH = Let A be a Hilbert-Schmidt Oper- TN . ator. Let p(x) = “Ax” . Then p2(x) = = , and A*A is nuclear by Theorem 1.2.1. Conversely, let A be a positive nuclear operator. Again by Theorem 1 2.1., A% is positive Hilbert-Schmidt operator; so p(x) = = ”Agxu TN = 5 follows from Theorem 1.2.1.(ii) To show I(r) = TH , let pA(x) = “Ax“ , where A is a Hilbert-Schmidt operator. Then pA(x) 3 NA“ - :1in and z P§(e1.) = 2 ”Ab,“2 < .. , so PA :5 T . So, TH c: UT) Conversely, if P 6 1(1) , P fig T , then P(x) 3_ c ~Hx“ for some c > O . There is a map A : Hp + H such that p(x) = flAxH and E “Aeiflz = Z P2(ei) < w , so A is a Hilbert-Schmidt Operator. Let (D, F, P) be a probability space. In the space of random variables defined on it introduce the topology by the following neighborhood basis at zero: (8) U(eni) = {X : P(on [X(w)| : a) < D} The Obtained topological space we shall denote by LO(Q, F, P) . From (8) it follows that Xn + O in L if and only if Xn + O in P-proba- O bility. Without difficulties we can prove that Xn + O in L0 if and only if E(min (lek, 1)) + O for every k > O , and if and only if X E(1+|X ) + O . Here X = Y if and only if X = Y a.e. [P], i.e. we are considering equivalence classes. 17 4. Definitions. Let (E,T) be a multi-Hilbertian space. (i) By a random linear functional we mean a linear mapping X:(E,T) + H3(D, F, P) . (ii) A random linear functional X is called separable if there exists a countable Hilbertien topology 1': T such that X e E‘T. E E'T (iii) A random linear functional X is called regular if for every x E E , X(X)(w) = where Xw 6 E; for every w . (iv) Random linear functionals X and Y are said to be equivalent if, for every x in E , P(X(x) = Y(x)) = 1 . (v) We say that X is a version of Y if P(X(x) = Y(x) for all x E E) = 1 . 5. Theorem [14]. A random linear functional X has a t-regular separable version if and only if X is I(t)-continuous (i.e. if the mapping X: (E,I(r)) + H>(Q’ F, P) is continuous). 6. Theorem (Ito, [14]--Generalized Bochner's theorem) Let (E,t) be a multi-Hilbertien space. Let f be a complex-valued function defined on E such that (i) f is positive definite, (ii) f(0) = 1 . (iii) f is I(t)-continuous at O . Then (iv) f is the characteristic function of a separable probability measure P on E’ . Conversely, (iv) implies (i), (ii) and (iii). 7. Remark. If E is a Hilbert space, then by Lemma 3, I(T) = S , so Theorem 8 reduces to a well-known result of Sazonov [261. On the 18 other hand, when E is a nuclear space, thus 1(1) = r , this result is given in [20]. In the next section, we generalize Lévy's continuity theorem to multi- Hilbertien spaces. 2.2 Continuity theorem The convergence of characteristic functions on real line implies con- vergence Of corresponding probability measures. In infinitely dimensional spaces we have to impose some conditions for relative compactness. 1. Theorem{[23], Ch. V1). Let H be a Hilbert space. Let {Pk} be a sequence of probability measures on H , and fk the corresponding sequence of characteristic functionals. {Pk} is relatively compact if and only if for every e > O , and for every k = 1, 2, ... (i) There exists a S-operator Sk such that 1 - Refk (x) §_ + E (ii) spp Z (Skei’ 9i) < w l o o o o V - (111) lim sup_é — O , N k l-N where {e1} is an orthonormal basis in H . 2. Example.(Sazonov, [26]) It is well known that, on real line, a convergent sequence Of characteristic functions is equicontinuous. This example shows that it is not true in a Hilbert space. However, as shown in Meyer's paper [20], it remains true in a nuclear space. Let 1/i2j2 if l#J tij = 2 19 j = 19 2, ' 1/i if l=J Then (1) 321 tjj < C < w (2) lim sup 2 t1 = O N i j=N J (3) z sup t-- = m i=1 i ‘3 Define operators Sk , k = 1, 2, ... by Skek = tkj ej , where ej , j = 1, 2, ... is an orthonormal basis in an infinitely dimensional Hilbert space H . Let Pk be probability measures on H with characteristic functions fk(x) = exp ('%-) , k = 1, 2, ... By Theorem 1, (2) and (3) imply weak convergence Of fk . Suppose now that {fk} is equicontinuous at O in S(=I(T))-topology. Then for every 2 > 0 there is an S-operator 58 such that < 1 implies 1 - fk(x) < c , for all k . By definition of fk it follows < ri(e) whenever < 1 for all k , n(e) + O . Let t6 j = . 3 For every real r , if = rzth j < n(e) , then 2 . . 2 < 1 , but then r t€,j O N+m i J=N For an orthonormal sequence {e1} in H , define the operators Sk and S by Skej = tkjej ; Sej = uj , J, k = 1, 2, ... By (4) we have + for every x in H . Define ik(X) = exp t~% } and f(x) = exp {-%-} . Then clearly fk(x) + f(x) for every x . But by (6), the corresponding sequence of measures is not weakly convergent. Until further notice, E will denote a milti-Hilbertien space, with topology T . 4. Theorem. Let Mk be a sequence of separable probability measures on E', let fk be the corresponding sequence of Characteristic functionals. Assume the following: (i) There is a function f , 1(t)-continuous at O , such that (7) fk(x) + f(x) , for every x in E . (ii) For every 8 > 0 there is a sequence of I(T)-seminorms pk and a 7-seminorm q such that lfi< HS q for every k and 21 (8) 1- Re fk(x) _<_ e + pi(x) (9) SEP i PE (ea) < w (10) lim sup 02° pi (e1) = O , N k i=N where {8i} is an orthonormal basis in Eq . Then there is a separable probability measure M on E' such that Mk a M . Conversely, if Mk = M , where M is a separable probability measure, then (7)-(10) hold. Proof. Assume (7)-(10). By Theorem 1, the set of measures induced by Mk in Eq is relatively compact. So there exists a q-compact (thus t-compact) set K such that Mk(KC) < 1 - c . So, Mk is a tight sequence for which fk(x) + f(x) pointwise. By Theorem 1.5, there is a separable probability measure M such that f is the characteristic functional of M ; thus Mk = M . Conversely, let Mk = M . Then there is a compact set K ggEl. such that M (KC)< 8/2 for every k = 1, 2, ... . k For every x and every k we have 1 - Re fk(x) f (1 - cos )de(F) IA f (1 - cos )de(F) + e K %-r 2de + e K IA K is compact in E;. ; so there is a q such that K is compact in Eq . 22 2 _ 1 2 Let pk(X) - §-£ dMK(F) . If {ei} is an orthonormal basis in Eq then F)dMK(F) < 1-sup 5 52( 2 FEK Epidei) =7} l: 2(E) < e because a is bounded on K , being a continuous function. Also we have 2 - Pk(x) 5 qzlx) .1. £q2(F)de(F) _<_ c - q2(X) . so pk Esq , for every k . Thus, (8) and (9) are proved . To prove (10) note that E pE(ei) = g f 2de(F) i=N N K 5_ sup 2 2 + O , as N +-m , FEIC N by compactness of K . 5. Corollary. Let fk , k = 1, 2, ... , be a sequence Of character- istic functionals of separable probability measures Mk , k = 1, 2, ... on E' , which is I(t)-equicontinuous at O , i.e., there is a semi- norm p in 1(r) such that for every 5 > 0 there is a o>~o so that for all k = 1, 2, ... we have: (11) p(x)_<_5=1-Ref(x):e. Then {Mk} is a relatively compact sequence. Proof. Assume (11). Then conditions (7)-(10) are satisfied for P = %-- p , k = l, 2, ... , and q such that p < q . Then from k HS the proof of Theorem 4 it follows that {Mk} is relatively compact. 6. Remark. Theorem 4 and Corollary 5 remain true for any sequence of finite separable measures Mn such that sup Mn(E;.) < m . n L 23 2.3. Infinitely divisible probability measures Let (E,T) be a multi-Hilbertian space. We have defined infinitely divisible probability measures in Chapter 1. By Theorem 1.5., a separable probability measure on E' is uniquely determined by its characteristic function, so we have 1. Theorem. Let M be a separable probability measure on E‘ , with the characteristic function f . M is an ID measure if for every n there is a characteristic function fn such that s (12) f"(x) = f(x) . for every x in E . The following theorem can be proved in the same way as for the real ran- dom variables. 2. Theorem. (i) ID characteristic functional never vanishes. (ii) If f is an ID characteristic functional, then fS is ID characteristic functional, for every 5 > O . (iii) The sum of finite number of 1D random variables is ID. (iv) Neak limit of ID measures is an ID measure; moreover, if only fk + f pointwise, and fk are ID character- istic functionals, then f is an ID characteristic functional. 3. Examples. (i) Let (13) f(x) = exp tirelaN(P) - 1) , where M is a separable probability measure on E‘ , and C any real number. The function defined by (13) is positive definite, I(t)-continuous and 24 f(0) = 1 . So, by Theorem 1.5 it is the characteristic functional of a separable probability measure on E'-Poisson measure. It satisfies (12), so it is ID measure. (ii) Let (14) f(x) = exp(i - %-p2(x)) , where F is a fixed element in E' , p is a I(t)-seminorm. By Bochner‘s theorem there is a unique separable probability measure on E' determined by (14)--Gaussian measure. It is clearly an ID measure. In Chapter 1, the construction of a Gaussian measure, starting from cylinder measures is given. 4. Theorem. A function g is the logarithm of an ID characteristic functional f (of a separable probability measure on E' ) if and only if (15) 9(0) = O (16) g is I(t)—continuous at O , and for all finite sets of complex numbers .. v _ _ . 1 AI - {ai}ieI such that iél’ai - 0 , and xI _ {XIIIEI Xi E E , we have (17) Z aiajg(Xi - Xj) :_0 i,jeI .Epppf. The assertion follows directly from the Theorem 1.5. In fact, (17) is the consequence of positive definiteness of f and Schoenberg's theorem [19], (16) follows from I(t)-continuity of f , and (15) from f(0) = 1 . 5. Remark. Making an appropriate choice of AI and XI in (17), one can show that for every x E E 25 (18) Re g(x) §_O (19) g(-X) = 6117'. 6. Theorem. The class of ID Characteristic functionals coincide with the class of Poisson characteristic functionals and their pointwise limits. Proof. By Example 3 and Theorem 2, Poisson characteristic functionals and their limits are ID. Conversely, let f be an ID characteristic functional. Then for all x in E , n(f1/"(x) - 1) +109 f(x) (n ....) l/n so, f(x) = lim exp (n(f1/n(x) - 1)) . By Theorem 1, f is a char- n acteristic functional of some separable probability measure Mn , so fl/n(x) = reldMn(F) and f(x) = lam exp n f (ekF’X> - 1)dMn(F) , which is a limit of Poisson characteristic functionals. Now we give some elementary inequalities that will be used later. 7. Lemma. For any real t,s : (i) |eit-1| §_2 min (t2,1) §_2 min (t,1) (ii) lelt-i-itl i t2/2 (iii) 1 - sin t/t _3 c - min (t2,1) (iv) )(elS-is) — (alt-it): : ls-t Proof: elementary. 26 8. Notation. For q 6 1 define the following quantity (finite or not) (20) _ E(F) = sup [] 4(X)gl Let M be a positive separable measure on E'-{O} such that (21) ,1 min (32(E), 1)dM(F).< a . E-{O} For a G in E' and r 6 1(1) define (22) 9(X) gEG.r.M.qJ(X) .-. i - r2(x) + f(ei- 1 - i - 1£4(F):IJ)dM(F> Function defined by (22) will be referred to as gEG,r,M,q] or only 9 . He will also use the following notation: (23), h(F,x) = ei - 1 - i - 116(F):11 9. Lemma. 9 is I(1)-continuous at 0 . Proof: G e E' implies G e E5 for some p e 1 . Let {ei} be an orthonormal basis in ED . 2 Then 2 = 5(6) , thus l| is a 1(1)-continuous seminorm, so i is I(1)-continuous. Since r 6 1(1) , it remains to show the 1(1)-continuity of the integral in (22). We have: ( 1 - 1 - i )dM(F) (24) fh(F,x)dM(F) = . (ekF’X> - 1)dM(F) +~r e 4(F):1 ~J q(F)>1 Let us show that both terms above are I(1)-continuous. - 1)dM(F)l 5.1 min (2,1)dM(F) I q(F)>1 Restricted to the set where E(F)>1, M is a positive finite measure; 27 let M(E(F)>1) = m . By separability of M , there is a topology <5 determined by an increasing family of seminorms qn such that EA i E3 . n So, for given 5 we can find n such that M(F E E'qn) E {5/2 i.e., if {ek} is qn-orthonormal basis: M()2 = 0P) < 5/2 3 k _ a Let r = r(c) be a real number such that (26) M(Z2 _>_ r) 5 c/2 . Then, with Fk = , we have (27) f min(2,1)dM '5 f 2 - 11) FE < r1dM(F) P(F)>1 4(F)>1 + f 11) FE 3 r3dM(F) . 2 .. 2 ._ 2 2 - Let p (e,x) - p (x) - f - 112 Fk < rJdM(F) R(F)>l Then ) p2(ek) < r . m ; so p is an I(1)-seminorm ; by (25) and (27) we have ) x (e‘ - 1)dM(F)) 3 p2(x) + e . fi(F)>1 so the first term in (24) is I(1)-continuous. By Lemma 7.(ii): ei (28) l f 2 5(F)<1 - 1 - i )dM(F)[ 5_ %IO(F;FIX> < dM(F) . Now, by definition of 5 , it follows l| .5 5(F)Q(X) . so by the assumption (21), the right hand side of (28) is finite. Moreover, F) < on the set where 6( 1 , we have F 6 EA , so the expression on the 28 lefthand side of (28) is 1(1)-continuous. 10. Corollary. 9 is logarithm of an ID characteristic functional of a separable probability measure on E' Proof. We shall use Theorem 4. Condition (15) is satisfied; (16) follows from Lemma 9 and (17) can be easily checked. 11. Lemma. For given q e 1 , there is 1-1 correspondence between functions 9 defined by (22) and triplets [G, r, M] . Proof: Let t e R . We have i_ i-1 )Ree 2 1I_<_ )9 2 Igmin (2. 2/t2). t t 5 2O (X) - min (52(F). 1) . for a large enough t . So, by dominated convergence theorem, 1'2 - { (e‘- 1) - dM(F) + O as t ...... q F)>1 Similarly, if2 - { (eKF’tX> - 1 - i) dM(F) + O as t .... 'c‘i F)_<_1 So, we have 2 _ . -2 r (x) - lim - Reg(tx) - t (3 O by (18)) , t-mo which shows that r is uniquely determined by g . Let now x,y 6 E . The following formula holds: (29) f (1 - cos)eidM(F) = r2(y) + g(x) - g(x+y) : g(x-y) . 29 We shall show that M is uniquely determined by (29). For x = O and y being fixed, we have from (29) that the measure N defined by (30) dN = (1 - cos)dM is a positive finite measure on El (more precisely, its extension to E') Since N has a I(1)-continuous characteristic functional, it follows that N is a separable finite measure, uniquely determined by (30). So, M is uniquely determined by N on any set on which 1 - cos is bounded away from O . Since y runs through E , by regularity we conclude that M is uniquely determined. Finally, G is determined by M, q and r , from (22). The following two lemmas are proved in 161, in a different context: 12. Lemma. Let f be an ID characteristic functional. 1/n Let fn = f Suppose that, for some c > O and some p 6 1(1), p < q , we have: HS (31) p(x) : 1 = 1 - f(x) 1 e . Then (32) n(l-Re fn(x)) §_8c(1+p(x)) , for all x , and (33) 1' min (q2(F), 1)dun 3 482: , for all n , where Un is the measure that corresponds to nfn . 13- _Lemma- Let Q = {F: q(F):1} - {O} . Let K(F,x) = _ ~-2 i , . . - q (F)(e ' 1 ‘ l ) . Let Mn be a sequence of pOSitive finite measures on Q , such that Mn(Q)-: 1 and f K(F,X)dMn(F) 30 converges pointwise to a function w(x) . Then there is a positive separable measure M0 on Q , such that MO(Q):1 and an 1(1)-seminorm r , such that w(x) = -r2(x) + f K(F,x)dMO(x) . 14. Theorem. Let g be the logarithm of an ID characteristic functional of a separable probability measure on E' . Suppose q e 1 such that there is a p 6 1(1), p < q , and HS (34) P(x) 3 1 = 19(X)l.: e . for some 3 , 0 < c < %—. Then there is a triplet LG, r, M] satisfying conditions of Notation 8, such that g = gLG,r,M,q1. This representation is unique. Epppf, Let g = log f , where f is an ID characteristic functional of a separable probability measure on E' . Let fn = nfl/n .‘ Let Q be as in Lemma 13. We have log f = g = lam n(fl/n-l) . Let Pn be the measure corresponding to fn . Let Un be Pn restricted to Q : let Vn be Pn restricted to QC-{O}. Denote by an and V" the correspond- ing characteristic functionals. Let f(x) = 1-Z . Then, as n+w we have in(0) - fn(x) n(I - 11/"(x)) = n(1 - (1-2)1/”) n(1 - (l-fii) é *2 - so we conclude that {fn} is equicontinuous at O . Now we have: A fn(O) - Re E(x) = (On(O) - Re On(x)) + (Vn(O) - Re Vn(x)) , and by On(O) - Re On(x) > O , we conclude that fn(0) - Re fn(x) < e = Vn(O) - Re Vn(x) < e A i.e., V is equicontinuous at O . n By (34) and Lemma 12 we have 31 (35) . Vn(E') = Pn(QC - {0}) 5 485: , so by Corollary 2.5. and Remark 2.6. it follows that Vn is relatively compact. Let V be a measure such that V". = V for some subsequence n' . Then Vn' - Vn(0) + V - Vn(0) pointwise. By (34) and Lemma 12, 2 (36) f5 dUn. 5 486 By inequality 2.7.(ii) we have, for every x in E: We”:x> - 1 - i)dUn1(F)I 2 i f dUnI(F) .3 12 e q2(x) . Now let wn(x) = f(eKF’X> - 1 - i)dUn. . The above inequality shows that wn(x) is bounded for every x ; therefore, there is a function w(x) such that for every x , w (x) + w(x) , for some subsequence n" . n" We have obtained so far: g(x) = lim n(fn(x) - 1) n = 11W ((Qn"(X) ‘ Gn"(0)) + (Unu(x) ‘ Unu(o))) n = (V(x) - V(O)) + w(x) + i lim f dUn”(F) n V(x) - V(O) + w(x) + i . By Lemma 13, we have that there is a r 6 1(1) , a separable measure M0 on Q such that w(x) = -r2(x) + f(ekF’X> - 1 - i) . 5'2(F)dMO To conclude the proof, define G by = + f dV(F) 0 any 32 and define measure M on E'-{O} to be M0 on O and V on QC-{O} . Clearly, the condition (21) is satisfied and g = gEG,r,M,q]. Uniqueness is proved in Lemma 11. 3. CONVERGENCE OF ID LAWS AND HOMOGENEOUS PROCESSES WITH INDEPENDENT INCREMENTS 3.1. Convergence of IO laws 1. Definition. We say that a sequence of 1(1)-seminorms {pn} is compact if there exists a 1-seminorm t such that pn < t for all n HS and, for an orthonormal basis {e1} in Et , we have: (1) sup 2 P2(e ) < w . °° 2 2 lim su e. = O ( ) N nP iZN Pn( 1) 2. Theorem. Let Pn and P be separable ID probability measures on E' with the characteristic functionals gn = gnth, rn, Mn’ q] and g = gtearsquj 0 Then Pn : P if and only if (3) lim G = G n n (4) Mn = M on every set {F: E(F) > c} (5) The sequence {tn} of 1(1)-seminorms defined by t§(x) = riot) + NI 2dMn(F) n(F)§1 is compact, (6) for every x , lim Tim' %- f 2dM (F) + r (X) = r2(x) 5+0 n+w E(F) + ; (eKF’X> - 1 - i - 1[q(F):11)dMn(F) E(F):€ + ~f (el‘F’x> - 1 - i)dMn(F) - r§(x) . 4(F):e Now we let new to obtain that first term in (7) converges to i 33 34 and the second term to the corresponding expression with M . Now we take care of the remainder. Use inequality 2.3.7.(ii) and note that if |a| < |b| then la-cl < l-b-cl , for a,b,c complex numbers. So we have (8) { (ekF’X> - 1 - i)dMn(F) - rim + r2(><)l 5 : l-vlg { dMn(X) - rim + r2(x)l . Letting now a + O , and using (6) we have that for every x in E , gn(x) + g(x) , so then characteristic functionals fn converge point- wise to the characteristic functional f of the measure M . The condition (5), together with inequalities in the proof of Lemma 2.9., provides condition (ii) of Theorem 2.4. SO, by Theorem 2.4., we conclude that Pn = P . Conversely, let Pn = P . Using Theorem 2.4. and the proof of Theorem 5.5. of [23] we obtain conditions (3)-(6). 3.2. Homogeneous processes with independent increments Let T be a finite or infinite interval on the real line, starting at O . 1. Definition. (i) A process with independent increments is a family of random variables {X(t)} defined on a probability space (9, F, P), tel" such that for every t1,...,t (O < t < ... < tn), the random variables n -— 1 X(t0) a X(t1) ' X(tO) s --- 9 x(tn) " X(tn_1) are independent. (ii) {X(t)} ‘tET' is said to be a homogeneous process if the distribution of random variable X(t) - X(S)(S < t) depends 0” 35 t - s only. A connection between homogeneous processes with independent increments and ID laws is immediate. Let t E‘T be fixed. By Definition 1, XiéF) - X((k33 t) are iid random variables (k = 1, ..., n) , and n X(t) - X(O) = Z X(%%) - X((k13)t) is an ID random variable. Therefore, k=1 every increment of a homogeneous process with independent increments is an 10 random variable. Let P be the distribution of X(t) - X(O) , t and Q be the distribution of X(O) . It is easy to see that Pt+s = Pt* p 5 group of convolutions. So, the distribution of X and P0 = 60 (Dirac distribution), so {Pt}tET is a semi- t is obtained as Pt*(1; with some additional work it can be shown ([1]) that all finitely dimensional distributions of the process Xt are determined by the semi- group {Pt} and Q . Let now {Xt} be a E'-valued process, and let f be the characteristic t functional of Xt . Until further notice assume that {Xt} is a homo- geneous process with independent increments. By independence we have: (1) ft+S(X) = ft(X) - fS(X) . In a particular case when ft(x) is, for every x , continuous at t = O, we have an especially simple relation. 1. Theorem. Suppose that for every x in E , ft(x) is continuous at t = O . Then (2) f Proof: Let s + O for fixed t in (1) to show that f is continuous 36 for every t . The only continuous solutions of (1) are functions of the form (3) ft(x) = exp t g(X) . for some 9 . This proves (2). Suppose now that f1 is the characteristic functional of a separable probability measure on E' . Then (2) completely determines the process. If X(O) = O with probability 1, then for every t, Xt is an ID random variable; by (3) and results in Chapter 2, we know the form of ft(x), and by Theorem 1.2. we have necessary and sufficient conditions for weak convergence. Note that f1 is an ID characteristic functional. 3.3. An Example In this section we present the solution to a problem in real line, which arises in a stochastical model of neuronal activity. This is an improve- ment and a generalization of results of Kallianpur [16] and Tuckwell [29]. We also discuss a possiblity Of a generalization to infinitely dimensional spaces. Let us first recall some facts about real valued ID random variables. Let X be a random variable with the characteristic function f , and its logarithm g . Suppose that Var X < m . Then 9 is represented in the form -2 (1) g(x) = ti + f(eiux - 1 - iux)u dK(u) , where K is distribution function Of some finite measure on R . The representation (1) is unique and we write 9 = gIG,K1 . Xn = X if and only if Kn(u) + K(u), for every u E R , and Gn + G . 37 If X is normal with mean m and the standard deviation S , then 2 G = m and K(u) = S . ltu > 0] If X is generalised Poisson, i.e., if -A k (2) P(X=a+kh)=%—- ,k=O,1,2,... then G=a+>\h; k(u)=Xh2-1[t:h] Let Ykn(t) k = 1, 2, ..., p , n = 1, 2, ... be independent Poisson processes with parameters Akn ; let Ekn , k = 1, 2, ..., p , n n = 1, 2, ... be real numbers, 0 §_t §_T < m . Define pn (3) Nn(t) = Z 5k” Ykn(t) k-l — _ -1 (4) Nnm - (Nn(t) - ENn(t)) on , pn pn 2 p where ENn(t) - t . E Ekn an , on - ( Z Ekn 1k") k-l k—l Then for Nn(1) we have the representation: 2 . _ pn Akn Ekn 1f u - €kn (5) Gn = g )‘kn Ekn a kn(u) = 0 otherwise , where kn is the point mass function; (U) K = X k (v) . n YE“ n Similarly, Nh(1) has the representation: 6 - 2 . _ ku (ckn/O’n) °Akn If U - -5n— (6) G” = 0 ; kn(U) = 0 otherwise The interest is to investigate the limit behavior Of Nn(t) and Nh(t) . Let W(t) be a standard Brownian motion and let X(t) be the Poisson 38 process with independent increment whose distribution is given by (2) with At in place of A . 1. Theorem. In order that Nb =»W (in the space D(O,T)) 9 it is nec- essary and sufficient that for every 5 > O : (7) X (eknfisn)2 Akn + 0 (n + 00) kzlekn/on[>e 2. Theorem. In order that Nn = X (in D(O,T)) it is necessary and sufficient that, for every 2 > O : p11 (8) X Akn Ekn + a + Ah k=1 2 (9) Z an Ekn + O (n +rw) k:lckn-h[>e ph (10) E Akn + A , for some X . k=1 Proof of Theorem 1 Let us first prove the convergence of one dimensional distributions. Without loss of generality set t = 1 (otherwise we may take Aknt in place of 1k”) . By (6), we have to show that (7) is equivalent to (11) E Hem/on) - (em/on)2 hm -» 1(0) as n + w , for all bounded continuous functions on R . So, assume (11) and let, for 5 fixed, 1 if IULiE f€(u) = C if u = 0 linear in (—c,e) -: Z fc(€kn/On) ° (Ekn/dn)2 Ak T 0 ’ 2 Then 2 (ekn/on) A. k n kn kzlekn/Onl > e 39 and (7) is proved. Assume now (7) and let f be a bounded continuous function. SO for every u , [f(u)] : M and for every ‘n there is an e such that f(O) — n < f(u) < f(O) +11 if u 6 (-€, 8) . From (7) it follows I; f(ekn/On)(€kn/Gn)2 )‘kn ' “0” -<- n . Z (elm/On)2 Akn + l k: cknflj .5 c n l +2M Z (Ekn/On)2)‘kn+n kikh/°n|>5 Since n is arbitrary small, it proves (11). The convergence of finite dimensional distributions follows by indepen- dence of increments: firstly we have (Nh(t1), Nh(t2) - Nh(t1)) = (W(t1), W(t2) - W(t1)) by one-dimensional result; then it follows (Nn(t1). ~n(t,)) = (Nn(tl). Nn(t1) + (Nn(t2) - Nh(t1))) = (w(tl), w(t2)). and for higher dimension by induction. Finally, to show tightness, note that, for t1.i t1: t2 : —- —- 2 —- - 2 _ —- - 2 —- —- 2 E((Nn(t) - Nn(11)) (Nn(t2) - Nn is continuous in probability for every x and (ii) X.(w) e D(EO,T1, E‘) for almost every w . (iii) Xo(w) = O for almost every w . In what follows we shall consider Lévy processes only. Also, we shall assume that (ii) and (iii) hold for every «1, which is not a loss of generality. From Definition 1, it follows that for each x , is a real valued Lévy process, continuous in probability (see Ito [13] for the definitions on the real line). By a jump (or a strong jump) of X(w) at time t we mean the difference Xt(w) - Xt-(od if it is not zero. By a weak jump of X(w) at time t at the point x we mean O we have lim X(w) = X(‘”) . s+t s t- s O . So, again by S Lemma 1, there is an so such that X£”) belongs to some space Eqé for all x in [0, co]. By compactness, there is a finite cover Oi , 1 < i < n , such that all points in 0i belong to the space Eqé . But then all Xt for belong to the space Eq' = Eq' U ... LJEq'. n t e [0,T] = 01 U 02 U ... U On 4. Lemma. Suppose T < w and suppose that, for every .x E E , the has at number of jumps of x> , O :_t §_T is finite. Then there is most M jumps in [O,T] (w is fixed). Proof. Let fT(x) be the number Of jumps of up to time T . For each .x e E , fT(x) is finite. Let us now prove that fT is lower semicontinuous, i.e., if xn + x then 43 (1) lim f (x ) > ft(x) . It clearly suffices to show that, whenever has a jump of size >11 then, starting with some nO all functions have a jump 0f size >n . Suppose now that has a jump of amount >‘n at time t . By assumption lim XS = XS_ . s| < e , s 6 [t-e, t) . By 1-continuity of Xt and Xt- we have (4) | - | < c for n:_nO (5) | - | < c for n-: nO . Then from (3), (4) and (5) it follows: ( - i > n - 3c , t-’ xn and the assertion (1) follows by arbitrarity Of e . By Osgood's theorem [12], page 62, , there exists a 1-norm p such that lft(X)l < M/2 for all x in Br(z) = {x: p(z-x) < r} , for some 2 e E2 and r > O . The function fT(x) has the following properties: 44 (6) ..fT(x + y) _<_ fT(X) + fT(y) (7) fT(CX) = fT(X) which follow immediately from the definition of f So, we deduce: T o tT(x) iT((p'1(x) . rx + z - z) - p(x) - r'l) fT(P'1(x) - rx + z - 2) f (P'1(x) - rx + z) + fT(-z) IA T < M . 5. Lemma. If T < w , then Xt has only countably many jumps. Proof. See Remark 3.3. 4.2. A decomposition of E'-valued processes. Let E be a countably Hilbert complete space. Let E5 = E' - {O} . Denote by C the cylinder (= Borel) o-algebra of E' , and by B the Borel o-algebra of sets in T5 x E5 , where To will denote the interval (O,T1 , for T finite or infinite. Let 8* be the class of all sets A in B such that for some a > O and some x1, x2, ..., xk E E we have 1 1 (1) A c:(O,a) x (F: |I > 3"°"’ [l > 51 . Let Xt be a process with independent increments. Let, for an to fixed, I(w) = {t: Xt f Xt_} and 3(a) = {(t,11Xt), t 6 1(a)} , where (ext = Xt - Xt- . Define the set function N by N(A) = N(A,afl = number of points in A O J(w) N(A) is a finite random variable for A 6 8* . The completed cylinder o-algebra generated by Xu - Xv , s < u , v §_t will be denoted by Bst(X)‘ 45 2. Lemma. 1f Aes" and Ac(s,t1xE5 then N(A) is BSt(X)-measur- able. Proof. Let E(s, t, x1,...,xk, a1,...,ak) = E(s, t, x, a) = (s,t1 x {F e E : > a1 ,..., > ak} , where O §_s < t < T , a = (a1,...ak) 6 Rk k , ai > O for 1 §_i §_k , x = (x1,...,xk) e E Let Q be a countable dense subset of (s,t], including t . We have: {N(E(s,t,x,a))_: 1} = {for some u 6 (s,t1, k = U n U n { .3 a + 15} E Bst(x) p q r.r'eO i=1 p (s+1)/p5r_ k +1}= u {N(E(s,t,x,a)) _>_ k} rleHs,t] n {N(E(r,t.x,a)) > 1} . so, by induction, {N(E(s,t,x,a)) 3 k} 6 Egt (X) , thus N(E(s,t,x,a)) is Bst(X)-measurable. Let 0 denote the class of all sets A in B such that N(A n E(s,t.x,a)) is E;t(x)-measurable. Since N(A n E(x,t,x,a)) is a bounded measure in A , it follows that D is a Dynkin class. By measurability of N(E(s,t,x,a)) we conclude that the class M = {(s,w) X {F e E': .3 bj} , j = 1 ,..., n , n 6 N , 0 < s < w , -m < bj < w} 46 belongs to D . M is a multiplicative class that generates B ; so, by Dynkin's theorem N(A n E(s,t,x,a)) is Bst(X)-measurable for every A in B . Now if we define E'(s,t,x,a) = (s,t] x {F e E' : .3 -a1,..., §_ -ak} we similarly conclude that N(A n E'(s,t,x,a)) is 8gt(X)-measurable, for every A in B . Finally. if A e 8* and A c:(s,t1 x E; . then for some a and X we have A = A n E(s,t,x,a) u A n E’(s,t,x,a) , so N(A) = N(A n E(s,t,x,a)) + N(A n E'(s,t,x,a)) is 8gt(X)-measurable. Let Bc c: 8 denote the algebra of all sets of the form: Borel set in TO x cylinder set in E5 . 3. Lemma. (i) For A e 3* , N(A) is Poisson distributed with a finite parameter. (ii) For A 6 BC , N(A) is either identically equal to w (a.s.) or is Poisson distributed with finite parameter. Proof. Let A(t) = A n (O,t1 x EO , A e 8* Let N(t) = N(t,w) = N(A(t),w) . Clearly, N(t,m) is a right continuous step function in t , increasing, with jumps of amount 1. From Lemma 2 and N(t) - N(s) = N(A(t) - A(s)) = N(A n (s,t1 X ES) , it follows that N(t) is a real Lévy process with independent increments. For every t fixed we have 47 P(N(t) - N(t') e O) 3 P(xt - xt- ,1 O) = O , so N is continuous in probability. So, by results about real processes, N(t) is a Poisson process. For sufficiently large t , A = A(t) , N(A) = N(A(t)) and so N(A) is Poisson distributed. Let A 6 BC . Then there is an increasing sequence An 6 8* such that An i A . Then N(A) = lam N(An) . N(An) is Poisson random variable with expecta- tion An , say. A is increasing sequence. If lim An = m , then for n +00 every k , -1 k . P(N(A):k) 5 P(N(An) is a real Lévy process with independent increments. Let BX(TO x R0) denote cylinder sets in TO x RO that depend on x only. If A e Bx(to>< R0) then the number Of strong jumps of X that take place in A is equal to the t number of jumps of the real process that take place in A . Denoting by N(x, A) the number of jumps of in A we have the following: 4. Lemma. If A e BX(TO>< R0) , then N(A) = N(x, A) . Let n(A) = E(N(A,w)) . Rewriting the Lévy-Ito decomposition for real processes, and using Lemma 3, we have 48 5. Theorem. For every x in E = Zt(x) + Yt(x) , where Zt(x) is a Gaussian process with independent increments, and Yt(x) is given by Yt(x) = lim ( f f dN(s,F) + f f dn (S.F)) k+w O|2p p sI|_<_1 6. Remark. The decomposition in Theorem 5 is not a decomposition in E' , because the process Yt , and consequently Zt may not be linear in x . In the following section we shall prove the complete decomposition in a separable Hilbert space. The general problem remains open. 4.3. A Ler-Ito decomposition Let H be a real separable Hilbert space. We assume that Xt is a H- valued Lévy process with independent increments, and t 6 [O,T], where T is finite or infinite. 1. Theorem. Let n > O . If Xt e D([O,T1,H) then X has only finitely many jumps of the norm bigger than n on any finite subinterval of [0,T1. Epppf. Let us first prove the theorem on [O,T], T < m . Let A be the class of all points in [O,T] such that there are only finitely many jumps of the norm > n in [0,t) . Let s = sup {t : t e A} . By continuity of X at O , and by X0 = O , there is a neighborhood U = (O,c) such that ”Xt” < n/2 for t e U . Then for u,v e U , ”X - Xv“ fi-qu“ + ”Xv” f_n , u so there is no jump in U of size > n . Thus 5 > O . Suppose s < 1 . Then again, there is a neighborhood (5, s+c) in which X does not have jumps exceeding n in the norm; so 5 = T . The above proof goes through if EO,T3 is replaced by an arbitrary finite interval. 49 2. Theorem. If X e D([O,T], H) , then X has only countably many jumps. Proof. Immediate by Theorem 1. 3. Remark. If E is a countably Hilbert space, and Xt an E'=valued Lévy process, it is proved in Lemma 1.2. that for w fixed, Xt(w) belongs to a Hilbert space. SO, Theorem 2 holds for E' . 4. Lemma. Let F = {Bst’ s and Y . Then by Ito's fundamental lemma and Yt are inde- pendent. By a trivial extension of this argument, all finite dimensional processes ( ,..., ) are independent of Yt , and this gives the result. 5. Lemma. A Lévy process Xt whose sample functions Xt(w) are con- tinuous a.s. is a Gaussian process. Proof. By finite dimensional case, all finite-dimensional distributions 50~ of Xt - XS are Gaussian. Therefore, for every s,t , Xt - XS is a Gaussian process. 6. Lemma. If a Lévy process Xt is Gaussian, then its sample functions are continuous almost surely. Proof. Let X be a Gaussian Lévy process. Then for every x , t is a Gaussian process, so is a.s. continuous (null sets de— pend on t ) . By separability of H , there is a countable dense set D = {x1,x2,...} . Then there is 91 czo , such that for every w E 91 and every xi 6 D , is continuous, and P(Ql) = 1 . Now fix aJE DI . We want to show that for every x 6 H , is continuous. Since Xt is a Lévy process, there is a neighborhood U of t such that “xs(o)” < M if s e U , for some M . Then, let 6 > D be given, and let xi 6 I) be such that ”x - xj“ < e . Let s e u be such that < c . Then I| .3 II + |I .3 < e + 2M8 , which shows continuity of . Now, since for every x , is continuous, it follows that Xt(w) is continuous for all m 6 DI . 7. Notation. I(w) {t : Xt(w) # Xt - (w)} {(t, AXt(w)) 9 t E I(w)} AXt(m) = Xt(w) - Xt_(w) T = T - {O} ; HO = H - {O} . C; A 8 V II 51 B = B(TO x H0) is the class of all Borel subsets of Tb x HO . B * = 8*(TO x H0) is the class Of all sets A c:B such that Ac(O,a)x{FeH : IIFII>%}, for some a>O. By Theorem 1, for A c:B* , A r)J(o) is a finite set. Let N(A) = N(Asw) denote the number of points in A n J(w) The proofs of thtefollowing two lemmas are almost identical to those of 2.2. and 2.3. and therefore are omitted. 8. Lemma. If A e 8*(T0 x HO) and A c:(s,t] x HO , then N(A) is Bst(x) measurable. 9. Lemma. For A e B(TO>< HO) , N(A) is either Poisson distributed or identically = m . If A e 8* , then N(A) is Poisson with a finite parameter. Now consider for every x e H and A e 8*: S(A,x) = S(A,x,co)' = 2 M (w), X> ‘ } (t,AXt)eA t (taF)E A n J(w) Clearly, for A E 8* we have k-1 k (1) . S(A,x) = lim :5N(A n {F : e (_, -1} ”*n k n n n = f f dN(t,F) (t,F)EA Define also S(A) = S(A,w) = Z AXt (t,theA This sum is finite, so we have S(A,x) = . 10. Lemma. Lemma 8 holds for S(A,x) and S(A) in place of N(A) . 52 Proof. For S(A,x) clear from relation (1). Then for S(A) it follows from S(A) = 0E1 ei 1: where {e1}:=1 is an orthonormal basis in H . 'I: Set now A(t) = A n [0,t] x HO , A 6 B H(t,A) = X - S(A(t)) . t 11. Lemma. H(t,A) is a Lévy process independent of the process N(A(t)). Proof. Immediate, by Lemmas 4, 8 and 9. D The following three lemmas can be proved the same way as for real processes (see [13], Section 1.6.5. 12. Lemma. Let A1 ,..., An 6 8* be disjoint. Then the following processes are independent: l’l N(Al(t)) ,..., N(An(t)) . H(t, p A.) . 13. Lemma. Lemma 12 holds for S in place of N . 14. Lemma. Let A An be disjoint sets in B . Then 1 ,..., N(Al) ,..., N(An) are independent. Now for A e 8* set _ , k k+1 Amk(x) - {(s,F) e A n {F . E Qfi,-7fi-J} Then (2) = f f dN(s,F) = lim 2 %-N(Amk(x)) A m—w° k Let EN(A) = n(A) . 15. Lemma. If A e 3* , then 53 E e' = exp f ; (em;X> - 1)dn(s,F) A Proof. For each m,k , N(Amk(x)) is Poisson distributed with parameter n(Amk(x)) . 50, k 1 exp -",-n— N(A k(x)) = exp ((‘Tn'i- 1) n(Amk(x))) By relation (2), lim E exp (i gig-N(Amkum *“ k E exp i = lim {1 E exp (%§-N(Amk(x))) mew k ik = lim [7 exp ((em_- 1) n(Amk(x))) m+w k 13. = lim exp (2 (e m - 1) n(Amk(X))) m—1Po k exp f f (ekF’X> - 1) dn(s,F) . A From the above we obtain 16. Lemma. If A E 8* is included in {(s,F) : “F” < m} for some m < w , then E = f f dn(s,F) A Var = f f 2 dn(s,F) A 17. Remark. If (O,t1 x {F : “F” > c} (e > 0) belongs to 6*(TD>< HO) , we have f f dn(s,F) < w , O2 dn(s,F) < .. O O . Then E8 6 BI and, by Lemma 15, E e1 = exp I I (ekF’X> - 1) dn(s,F) . O = E el . E ei So, we have IE ei! )5 eil IA = exp I I (cos - 1) dn(s,F) O| §_1 ; in that case we have cos :3 1 - 2/4 . SO, for every x such that ”x” §_1 : IE e1I .3 exp { -.ff 2 O2 dn(s,F) :_-log IE e1/NXII O , being an ID, never vanishes. This proves (i). The proof of (ii) will be given in Theorem 22. 19. Notations and some facts. Set S1(t,x) = Sl(t,x,w) = f f dN(s,F) O1 Sk(t,x) = Sk(t,x,w) f f dN(s,F) , k > 1 . O = Sk(t,x) . Define also k Tk(t,x) = Tk(t,x,w) = Sl(t,w,x) + jZZ Sj(t,x,w) - ESj(t,x,w) = f f dN(s,F) - f f dn(s,F) . O1 k 1 II II / F_<_IIFII51 Using previous results, it can be shown that Sn(t), n = 1, 2, ... , are independent Lévy processes. By Lemma 18(i), E(Sk(t,x)) is continuous in t ; so Tk(t,x) is a Lévy process for every k and x . 56 Using Komogorov's inequality (see1131 Section 1.7.2.) we have the following 20. Theorem. For every x 6 II, we have the following decomposition: (4) = Zt(x,w) + lim ( f f dN(s,F) k-rPP O dn(s,F)) , O = lim { f1” dN(s,F) + [.1 dn(s,F)} k-wo O1 k 1 1 I) )- / (3.: HP)! 5.1 Proof. For t) and k fixed, the term after the "lim" in (4) is linear in x . Notice that f f dN(s,F) O_1/k is continuous in x , as a finite sum of continuous terms. For the other term, we have 57 I I I dn(s,F)) .5 fix“ ° I I “F“ dn(s,F) O is continuous t in t for every x , it foolows that Z t t is also continuous. Then, by Lemma 5, Zt is Gaussian on H . Zt and N are independent by Lemma 4. For a Lévy process Xt decomposed as in previous theorem, we define x v II E < (‘8' A X v I Var , where Zt is Gaussian process defined by (5). For fixed t , M is linear and continuous in x ; Vt is an I(n-H)- t seminorm. The functions M , V and the measure dn(s,F) are called (following t t’ Ito), the three components of Xt . 22. Theorem. Let X be a Lévy process with three components Mt’ V t t’ Then the characteristic function of X - X , t2 > t1 is given by t2 t1 . 1 (6) f (x) = exp { i - —-(V (x) - V (x)) 1,12 t2 t1 2 t2 t1 + I I (eI 4 1 - i . ltnrug11) dn(s,F)} t1 = + lim Tk(t,x) , k—ioo and these two terms are independent. So, i l E e = E e . E lim e k-tco ' T (t x) _ . Vt 1k’ - exp (i - —%—l) ~Aim E e By previous lemmas we have i Tk(t,x) i . E e k = exp i I f (eKF'X> . 1{IIFII_<_1}) dn(s,F)} O