Infinitely Divisible Cylindrical Measures on Banach Spaces

Transcription

1 Infinitely Divisible Cylindrical Measures on Banach Spaces Markus Riedle First version: 13 October 2010 Research Report No. 7, 2010, Probability and Statistics Group School of Mathematics, The niversity of Manchester

2 Infinitely divisible cylindrical measures on Banach spaces Markus Riedle The niversity of Manchester Oxford Road Manchester M13 9PL nited Kingdom October 13, 2010 Abstract In this work infinitely divisible cylindrical probability measures on arbitrary Banach spaces are introduced. The class of infinitely divisible cylindrical probability measures is described in terms of their characteristics, a characterisation which is not known in general for infinitely divisible Radon measures on Banach spaces. Furthermore, continuity properties and the relation to infinitely divisible Radon measures of infinitely divisible cylindrical probability measures are considered. Keywords: infinitely divisible measures, cylindrical measures, cylindrical random variables, cylindrical Levy processes, Banach spaces 1 Introduction Probability theory in Banach spaces has been extensively studied since 1960 and several monographs are dedicated to this field of mathematics, e.g. Araujo and Giné [5], Heyer [7], Ledoux and Talagrand [10], Linde [11] and Vakhania et al [20]. In general, probability theory in Banach spaces is more complicated than in Euclidean spaces or in infinite-dimensional Hilbert spaces and many of the phenomena which occur in probability theory in Banach spaces are related to the theory of Banach space geometry. Cylindrical Wiener processes, which appear rather in the Hilbert space setting, can also be considered in the more general situation of Banach spaces. Defining cylindrical Wiener processes by cylindrical probability measures yields even that most of the results in the Hilbert space situation can be easily rewritten for the Banach space situation, see Kallianpur and Xiong [9], Metivier and Pellaumail markus.riedle@manchester.ac.uk 1

3 [13] or Riedle [16]. Cylindrical probability measures are finitely additive set functions which have projections to Euclidean spaces that are always bona de fide probability measures. In Applebaum and Riedle [1] the cylindrical approach enables us to introduce cylindrical Lévy processes as a natural generalisation of cylindrical Wiener process, a concept which is even new in the Hilbert space situation. The definition of cylindrical Lévy processes in Applebaum and Riedle [1] relies on the introduction of infinitely divisible cylindrical measures. However, since the work [1] is focused on cylindrical Lévy processes and their stochastic integral, no further properties of infinitely divisible cylindrical measures are derived. In this work we give a rigorous introduction of infinitely divisible cylindrical measures and derive some fundamental properties of them. Some of the results give also a new insight on genuine infinitely divisible Radon measures on Banach spaces. On a Hilbert space H it is well known that the class of infinitely divisible measures can be characterised by the set of triplets h, s, ν where h H, s : H H is a trace class operator and ν is a σ-finite measure satisfying u 2 1 νdu <, 1.1 H see for example Parthasarathy [14]. However, in Banach spaces such an explicit description of infinitely divisible measures is not known in general. But it turns out in this work, that the class of infinitely divisible cylindrical measures on an arbitrary Banach space can be described by a set of triplets with some explicit conditions on the entries of the triplets. This result enables us to solve the following problem: even in the finite dimensional case, a probability measure on R 2 which satisfies that all image measures under linear projections to R are infinitely divisible might be not infinitely divisible, see Giné and Hahn [6] and Marcus [12]. However, the question left open if a probability measure on an infinite dimensional space is infinitely divisible if all linear projections to R n for all finite dimensions n N are infinitely divisible. By the characterisation of the set of infinitely divisible cylindrical measures mentioned above we are able to answer this question affirmative. As already indicated, the condition 1.1 might be neither sufficient nor necessary for a σ-finite measure on an arbitrary Banach space to guarantee that there exists an infinitely divisible measure with characteristics 0, 0, ν, see for example = C[0, 1] in Araujo [3]. However, we show in the last part of this work that a σ-finite measure ν satisfying the weaker condition u, a 2 1 νdu < for all a, 1.2 always generates an infinitely divisible cylindrical measure µ. This result reduces the question whether a σ-finite measure generates an infinitely divisible measure to the question whether the infinitely divisible cylindrical measure µ extends to a 2

4 Radon measure. In fact, but without relating it to cylindrical measures, this can be considered as the starting point in some work, e.g. Araujo [4] and Paulauskas [15], where for specific examples of a Banach space the set or subset of all Lévy measures is described explicitly. 2 Preliminaries For a measure space S, S, µ we denote by L p νs, S, p 0 the space of equivalence classes of measurable functions f : S R with fs p µds <. Let be a Banach space with dual. The dual pairing is denoted by u, a for u and a. The Borel σ-algebra in is denoted by B and the ball by B := {u : u 1}. For every a 1,..., a n and n N we define a linear map π a1,...,a n : R n, π a1,...,a n u = u, a 1,..., u, a n. Let Γ be a subset of. Sets of the form Za 1,..., a n ; B : = {u : u, a 1,..., u, a n B} = π 1 a 1,...,a n B, where a 1,..., a n Γ and B BR n are called cylindrical sets. The set of all cylindrical sets is denoted by Z, Γ and it is an algebra. The generated σ- algebra is denoted by C, Γ and it is called the cylindrical σ-algebra with respect to, Γ. If Γ = we write Z := Z, Γ and C := C, Γ. A function µ : Z [0, ] is called a cylindrical measure on Z, if for each finite subset Γ the restriction of µ to the σ-algebra C, Γ is a measure. A cylindrical measure is called finite if µ < and a cylindrical probability measure if µ = 1. For every function f : C which is measurable with respect to C, Γ for a finite subset Γ the integral fu µdu is well defined as a complex valued Lebesgue integral if it exists. In particular, the characteristic function ϕ µ : C of a finite cylindrical measure µ is defined by ϕ µ a := e i u,a µdu for all a. In contrary to probability measures on B there exists an analogue of Bochner s theorem for cylindrical probability measures, see [20, Prop.VI.3.2]: a function ϕ : C with ϕ0 = 1 is the characteristic function of a cylindrical probability measure if and only if it is positive-definite and continuous on every finitedimensional subspace. For every a 1,..., a n we obtain an image measure µ πa 1 1,...,a n on BR n. Its characteristic function ϕ µ π 1 is determined by that of µ: a 1,...,an ϕ µ π 1 a 1,...,an t = ϕ µt 1 a t n a n 2.3 3

5 for all t = t 1,..., t n R n. If µ 1 and µ 2 are cylindrical probability measures on Z their convolution is the cylindrical probability measure defined by µ 1 µ 2 Z = µ 1 Z u µ 2 du, for each Z Z. Indeed if Z = π 1 a 1,...,a n B for some a 1,..., a n, B BR n, then it is easily verified that µ 1 µ 2 Z = µ 1 π 1 a a1,...,an µ 2 π 1 a a1,...,an B. A standard calculation yields ϕ µ1 µ 2 = ϕ µ1 ϕ µ2. For more information about convolution of cylindrical probability measures, see [17]. The k-times convolution of a cylindrical probability measure µ with itself is denoted by µ k. 3 Infinitely divisible cylindrical measures A probability measure ζ on BR is called infinitely divisible if for every k N there exists a probability measure ζ k such that ζ = ζ k k. It is well known that infinitely divisible probability measures on BR are characterised by their characteristic function. The characteristic function is unique but its specific representation depends on the chosen truncation function. Definition 3.1. A truncation function is any measurable function h : R R which is bounded and satisfies h = Id in a neighborhood Dh of 0. In general the set of admissible truncation functions is much larger but this is not our concern here. Given a truncation function h a probability measure ζ on BR is infinitely divisible if and only if its characteristic function is of the form ϕ ζ : R C, ϕ ζ t = exp imt 1 2 r2 t 2 + ψ h s, t ηds 3.4 R for some constants m R, r 0 and a Lévy measure η, which is a σ-finite measure η on BR with η{0} = 0 and s 2 1 ηds <. R The function ψ h is defined by ψ h : R R C, ψh s, t := e ist 1 iths. In this situation we call the triplet m, r, η h the characteristics of ζ. If h is another truncation function then m, r, η h is the characteristics of ζ with respect to h, where m := m + R h s hs ηds. 4

6 The integral exists because h s hs = 0 for s Dh Dh and h is bounded. From Bochner s theorem and the Schoenberg correspondence it follows that the function t ψ h s, t ηs R is negative-definite for all Lévy measures η. By choosing η = δ s0, where δ s0 denotes the Dirac measure in s 0 for a constant s 0 R, we conclude that t ψ h s 0, t is negative-definite for all s 0 R. 3.5 For an arbitrary Banach space a Radon probability measure µ on B is called infinitely divisible if for each k N there exists a Radon probability measure µ k such that µ = µ k k. We generalise this definition to cylindrical measures: Definition 3.2. A cylindrical probability measure µ on Z is called infinitely divisible if there exists for each k N a cylindrical probability measure µ k such that µ = µ k k. Bochner s theorem for cylindrical probability measures, [20, Prop.VI.3.2], implies that a cylindrical probability measure µ on Z is infinitely divisible if and only if for every k N there exists a characteristic function ϕ µk of a cylindrical probability measure µ k such that ϕ µ a = ϕ µk a k for all a. One might conjecture that a cylindrical probabability measure µ is infinitely divisible if every image measure µ a 1 is infinitely divisible for all a. But this is wrong already in the case = R 2 which is shown by Giné and Hahn [6] and Marcus [12]. They constructed a probability measure µ on BR 2 such that all projections µ a 1 are infinitely divisible for all linear functions a : R 2 R but µ is not infinitely divisible. However, we will show in this work that a cylindrical probability measure or a Radon probability measure µ is infinitely divisible if and only if µ π 1 a 1,...,a n is infinitely divisible for all a 1,..., a n and n N. 3.6 A cylindrical probability measure µ satisfies 3.6 if and only if for each k N and all a 1,..., a n, n N there exists a characteristic function ϕ of ξk,a1,...,an a probability measure ξ k,a1,...,a n on BR n such that ϕ µ π 1 a 1,...,an t = ϕ ξk,a1,...,an t k for all t R n. 3.7 Every infinitely divisible cylindrical probability measure µ satisfies 3.6 because for each t = t 1,..., t n R n we have ϕ µ π 1 a 1,...,an t = ϕ µt 1 a t n a n = ϕ µk t 1 a t n a n k k = ϕ µk πa 1 t 1,...,an 5

7 for all a 1,..., a n and all k N. In particular, we obtain for n = 1 that µ a 1 is infinitely divisible and thus, ϕ µ a = ϕ µ a 11 = exp im a 1 2 r2 a + R e is 1 is 1 BR s η a ds 3.8 for some constants m a R, r a 0 and a Lévy measure η a on BR. This representation can be significantly improved as we have shown in Applebaum and Riedle [1]. Not only for completeness of this work but also because we formulate the result under the weaker condition 3.6 we give the details of the proof here. Despite the weaker condition 3.6 the proof remains the same. Theorem 3.3. Let µ be a cylindrical probability measure on Z which satisfies 3.6. Then its characteristic function ϕ µ : C is given by ϕ µ a = exp iwa 12 qa + e i u,a 1 i u, a 1 BR u, a νdu, 3.9 where w : R is a mapping, q : R is a quadratic form and ν is a cylindrical measure on Z such that ν πa 1 1,...,a n is the Lévy measure on BR n of µ πa 1 1,...,a n for all a 1,..., a n, n N. Proof. For fixed a 1,..., a n let ν a1,...,a n denote the Lévy measure on BR n of the infinitely divisible probability measure µ πa 1 1,...,a n. Define the family of cylindrical sets where G := {Za 1,..., a n ; B : a 1,..., a n, n N, B F a1,...,a n }, F a1,...,a n := {α, β] R n : ν a1,...,a n α, β] = 0, 0 / [α, β]} and α, β] denotes the boundary of the n-dimensional interval α, β] := {x 1,..., x n R n : α i < x i β i, i = 1,..., n} for α = α 1,..., α n R n, β = β 1,..., β n R n. Our proof relies on the relation 1 lim 1 B x µ πa 1 t k 0 t 1,...,a n t k dx = 1 B x ν a1,...,a n dx 3.10 k R n R n for all sets B F a1,...,a n. This can be deduced from Corollary in [18] which states that 1 lim fx µ πa 1 t k 0 t 1,...,a n t k dx = fx ν a1,...,a n dx 3.11 k R n R n 6

8 for all bounded and continuous functions f : R n R which vanish on a neighborhood of 0. The relation 3.10 can be seen in the following way: let B = α, β] be a set in F a1,...,a n for α, β R n. Because 0 / B there exists ε > 0 such that 0 / [α ε, β+ε] where α ε := α 1 ε,..., α n ε and β+ε := β 1 +ε,..., β n +ε. Define for i = 1,..., n the functions g i : R [0, 1] by g i t = 1 α i t ε 1 αi ε,α i ]t + 1 αi,β i ]t + 1 t β i ε 1 βi,β i +ε]t, and interpolate the function x 1 α,β] x for x = x 1,..., x n R n by fx 1,..., x n := g 1 x 1... g n x n. Because 1 B f 1 α ε,β+ε] we have 1 1 B x µ πa 1 t 1,...,a n t k dx 1 fx µ πa 1 k R n t 1,...,a n t k dx k R n and fx ν a1,...,a n dx 1 α ε,β+ε] x ν a1,...,a n dx = ν a1,...,a n α ε, β + ε]. R n R n Since f is bounded, continuous and vanishes on a neighborhood of 0, it follows from 3.11 that 1 lim sup 1 B x µ πa 1 t k 0 t 1,...,a n t k dx ν a1,...,a n α ε, β + ε] k R n By considering α + ε, β ε] α, β] we obtain similarly that 1 ν a1,...,a n α + ε, β ε] lim inf 1 B x µ πa 1 t k 0 t 1,...,a n t k dx k R n Because ν a1,...,a n B = 0 the inequalities 3.12 and 3.13 imply Now we define a set function ν : Z [0, ], νza 1,..., a n ; B := ν a1,...,a n B. First, we show that ν is well defined. For Za 1,..., a n ; B G equation 3.10 allows us to conclude that 1 νza 1,..., a n ; B = lim 1 B x µ πa 1 tk 0 t 1,...,a n t k dx k R n 1 = lim 1 B x µ t k πa 1 tk 0 t 1,...,a n dx k R n 1 = lim 1 B π a1,...,a tk 0 t n u µ t k du k 1 = lim µ t k Za 1,..., a n ; B. tk 0 t k 7

9 It follows that for two sets in G with Za 1,..., a n ; B = Zb 1,..., b m ; C that νza 1,..., a n ; B = νzb 1,..., b m ; C, which verifies that ν is well defined on G. Having shown that ν is well-defined on G for fixed a 1,..., a n we now demonstrate that its restriction to the σ-algebra Z, {a 1,..., a n } is a measure so that it yields a cylindrical measure on Z. Define a set of n-dimensional intervals by Because ν a1,...,a n H := {α, β] R n : 0 / [α, β]}. is a σ-finite measure the set H\F a1,...,a n = {α, β] H : ν a1,...,a n α, β] 0} is countable. Thus, the set F a1,...,a n generates the same σ-algebra as H because the countably missing sets in F a1,...,a n can easily be approximated by sets in F a1,...,a n. But H is known to be a generator of the Borel σ-algebra BR n which implies that G a1,...,a n := {Za 1,..., a n ; B : B F a1,...,a n } generates Z, {a 1,..., a n }. Furthermore, G a1,...,a n is a semi-ring because F a1,...,a n is a semi-ring. Secondly, ν restricted to G a1,...,a n is well defined and is a pre-measure. For, if {Z k := Z k a 1,..., a n ; B k : k N} are a countable collection of disjoint sets in G a1,...,a n with Z k G a1,...,a n then the Borel sets B k are disjoint and it follows that ν Z k = ν πa 1 1,...,a n B k = ν k 1 k 1 = ν a1,...,a n B k = k 1 π 1 a 1,...,a n ν a1,...,a n B k = k=1 B k k 1 νz k. Thus, ν restricted to G a1,...,a n is a pre-measure and because it is σ-finite it can be extended uniquely to a measure on Z, {a 1,..., a n } by Carathéodory s extension theorem, which verifies that ν is a cylindrical measure on Z. By using 3.8 we define wa := m a and the function q : R +, qa := r 2 a and it remains to show that q is a quadratic form. According to Proposition IV.4.2 in [20] there exits a probability space Ω, A, P and a cylindrical random variable X with cylindrical distribution µ. By the Lévy-Itô decomposition in R it follows that Xa = wa + r a W a + s Ñads + s N a ds P -a.s., < s <1 s 1 k=1 8

10 where W a is a real valued centred Gaussian random variable with EW 2 a = 1, N a is an independent Poisson random measure on R\{0} and Ña is the compensated Poisson random measure. By applying 3.14 to Xa, Xb and Xa + b for arbitrary a, b we obtain Similarly, for t R we have r a+b W a+b = r a W a + r b W b P -a.s r ta W ta = tr a W a P -a.s By squaring both sides of 3.16 and then taking expectations it follows that the function q satisfies qta = t 2 qa. Similarly, one derives from 3.15 that r 2 a+b = r2 a + r 2 b + 2ρa, b, where ρa, b := Covr aw a, r b W b. Equation 3.15 yields for c ρa + c, b = Covr a+c W a+c, r b W b = Covr a W a + r c W c, r b W b = ρa, b + ρc, b, which implies together with properties of the covariance that ρ is a bilinear form. Thus, the function F : R, F a, b := qa + b qa qb = 2ρa, b is a bilinear form and q is verified as a quadratic form. Definition 3.4. A cylindrical measure ν : Z [0, ] is called a cylindrical Lévy measure if ν a 1 is a Lévy measure on BR for all a. From 3.9 we can easily derive a representation of the characteristic function ϕ µ of an infinitely divisible cylindrical probability measure µ for an arbitrary truncation function h. Since h = Id on Dh one can define p : R, pa := wa + h u, a u, a 1BR u, a νdu. It follows from 3.9 that ϕ µ a = exp ipa 12 qa + where the kernel function ψ h is defined by ψ h u, a νdu, 3.17 ψ h : R C, ψ h t := e it 1 iht for an arbitrary truncation function h. 9

11 Definition 3.5. Let h be an truncation function and let µ be an infinitely divisible cylindrical probability measure on Z with characteristic function Then we call the triplet p, q, ν h the cylindrical characteristics of µ. Analogously to the one-dimensional situation after Definition 3.1 one can convert the cylindrical characteristcs p, s, ν h into p, s, ν h if h is another truncation function. It follows from 3.17 that the characteristic function ϕ µ a 1 of the probability measure µ a 1 on BR is for all t R given by ϕ µ a 1t = ϕ µ at = exp ipat 1 2 qat2 + R ψ hst ν a 1 ds This representation of ϕ µ a 1 does not coincide with the representation 3.4 because the functions ψ h and ψ h do not coincide. Thus, we can not read out directly the characteristics of µ a 1 from Lemma 3.6. Let µ be a cylindrical probability measure on Z which satisfies 3.6 and has therefore the characteristic function ϕ µ : C, ϕ µ a = exp ipa 12 qa + e i u,a 1 i u, a 1 BR u, a νdu. Then µ a 1 has the characteristics pa, qa, ν a 1 h for all a. Proof. As above we can rewrite the characteristic function ϕ µ a 1 in the form 3.18 for the given truncation function h. In order to write ϕ µ a 1 in the standard form 3.4, we introduce the function p : R R defined by { pat + pa, t := R t hs hst ν a 1 ds, if t 0, 0, if t = 0. Note, that the integral is well defined because for each t 0 we have t hs hst = 0 for all s Dh 1 t Dh and because h is bounded. By defining the function ψ h : R R C, ψh s, t = e ist 1 iths, we can rewrite the characteristic function 3.18 of µ a 1 for all t R: ϕ µ a 1t = exp i pa, t 1 2 qat2 + ψ h s, t ν a 1 ds. R By Theorem 3.3 the Lévy measure of the infinitely divisible probability measure µ a 1 is given by ν a 1 for each a. Thus, there exist some constants 10

12 m a R and r a 0 such that m a, r a, ν a 1 h is the characteristics of µ a 1. For all t R it follows that ϕ µ a 1t = exp i pa, t 1 2 qat2 + ψ h s, t ν a 1 ds R = exp im a t 1 2 r2 at 2 + ψ h s, t ν a 1 ds, R which results in pa, t = m a t = pa, 1t = pat. Consequently, we have ϕ µ a 1t = exp ipat 1 2 qat2 + ψ h s, t ν a 1 ds, R which completes the proof. Recalling the Lévy-Khintchine decomposition for infinitely divisible measures on Banach spaces we could expect from 3.17 that a exp ipa, a exp ψ h u, a νdu are characteristic functions of cylindrical measures on Z, respectively. But the following example shows that we can not separate the drift part p and the cylindrical Lévy measure ν in order to obtain cylindrical measures. Example 3.7. Let l : R be a linear but not necessarily a continuous functional and λ > 0 a constant. We will see later in Example 3.10 that ϕ : C, ϕa := exp λ e ila 1 is the characteristic function of an infinitely divisible cylindrical probability measure. In order to write ϕ in the form 3.17 let ν be the cylindrical measure on Z defined by { λ, if la 1,..., la n B, νza 1,..., a n ; B := 0, else, for every a 1,..., a n, B BR n and n N. Then we can represent ϕ by ϕa = exp ipa + ψ h u, a νdu, where pa := λhla. Since a expipa is not positive-definite in general there does not exist a cylindrical measure with this function as its characteristic function. 11

13 The Example 3.7 leads us to the insight that some necessary conditions guaranteeing the existence of an infinitely divisible cylindrical probability measure with cylindrical characteristics p, 0, ν rely on the interplay of the entries p and ν. The following result gives some properties of the entries p, q and ν of the cylindrical characteristics, respectively, but also the interplay of p and ν. For later purpose, we formulate the result under the weaker condition 3.6. Lemma 3.8. Let µ be a cylindrical probability measure on Z which satisfies 3.6 and has therefore the characteristic function ϕ µ : C, ϕ µ : C, ϕ µ a = exp ipa 12 qa + ψ h u, a νdu. If h is a continuous truncation function it follows that: a a κa := ipa + ψ h u, a νdu is negative-definite. b for every sequence a n a in a finite dimensional subspace V equipped with we have: i pa n pa; ii qa n qa. iii s 2 1 ν a 1 n ds s 2 1 ν a 1 ds weakly; Proof. a: We define the functions ϕ i : C, i = 1, 2 by ϕ 1 a := exp 1 2 qa, ϕ 2a := exp κa. Since q is a quadratic form the function ϕ 1 is the characteristic function of a cylindrical probability measure µ 1, see [20, p.393]. Since ϕ µ = ϕ 1 ϕ 2 and both ϕ µ and ϕ 1 are characteristic functions of cylindrical probability measures, it follows that ϕ 2 is also the characteristic function of a cylindrical probability measure, say µ 2. For fixed a 1,..., a n and k N define the functions ϕ a1,...,a n : R n C, ϕ a1,...,a n t := ϕ 1 a 1,...,a n : R n C, ϕ 1 a 1,...,a n t := ϕ 2 a 1,...,a n : R n C, ϕ 2 a 1,...,a n t := ϕ µ π 1 a 1,...,an t 1/k ϕ µ1 π 1 a 1,...,an t 1/k ϕ µ2 π 1 a 1,...,an t 1/k. It is easy to see that µ 1 πa 1 1,...,a n is a Gaussian measure on BR n and thus, it follows that ϕ 1 a 1,...,a n is the characteristic function of a probability measure on BR n. By condition 3.6 the function ϕ a1,...,a n is the characteristic function of a probability measure on BR n and because ϕ a1,...,a n = ϕ 1 a 1,...,a n ϕ 2 a 1,...,a n 12

14 it follows that also ϕ 2 a 1,...,a n is the characteristic function of a probability measure on BR n. Thus, the cylindrical measure µ 2 satisfies 3.6 and has the characteristic function ϕ 2 a = exp κa. Consequently, we can assume in the following that q = 0. With this assumption we show a by applying Schoenberg s correspondence, see [20, Propertyh, p.192], for which we have to show that a exp 1 k κa is positive-definite for all k N and that κ is Hermitian, i.e. κ a = κa for all a. To prove positive-definiteness, fix k N, a 1,..., a n and z 1,..., z n C and let e i denote the i-th unit vector in R n. By 3.7 there exists a characteristic function ϕ of a probability measure ξ ξk,a1,...,an k,a 1,...,a n on BR n such that k ϕ µ π 1 t = ϕ t ξk,a1 for all t R n. a 1,...,an,...,an Consequently, we have n z i z j exp 1 k κa i a j = i,j=1 = = 0, n z i z j ϕ µ a i a j 1/k i,j=1 n i,j=1 1/k z i z j ϕ µ π 1 e i e a j 1,...,an n z i z j ϕ e ξk,a1,...,an i e j i,j=1 where the last line follows from the fact that ϕ ξk,a1 is a characteristic function,...,an on R n. Next, we want show that κ is Hermitian. Since rewriting the characteristic function for different truncation functions does not effect the function κ we can fix hs = s 1 BR s for s R which yields ψ h s, t = ψ h s, t for all s, t R. By Lemma 3.6 we obtain for all t R that ϕ µ a 1t = ϕ µ a 1 t = exp ip a t + R = exp ip a t + R = exp ip a t + R ψ h s, t ν a 1 ds ψ h s, t ν a 1 ds ψ h s, t ν a 1 ds, 13

15 which implies p a = pa. It follows that κ a = ip a ψ h u, a νdu = ipa ψ h u, a νdu = κa for all a, which completes the proof of a. To see b let a n a in a finite-dimensional subspace V and let the truncation function h be continuous. Then Bochner s theorem implies that lim ϕ n µ a 1 n t = lim ϕ µta n = ϕ µ ta = ϕ µ a 1t 3.19 n for all t R. By Lemma 3.6 the measures µ a 1 n are infinitely divisible with characteristics pa n, qa n, ν a 1 n. It follows from 3.19 that the infinitely divisible measures with characteristics pa n, qa n, ν a 1 n converge weakly to µ a 1 which has the characteristics pa, qa, ν a 1. Applying Theorem VII.2.9 and Remark VII.2.10 p.396 in Jacod and Shiryaev which characterise the weak convergence of infinitely divisible measures in terms of their characteristics implies pa n pa and qa n δ 0 ds + s 2 1 ν a 1 n ds qa δ 0 ds + s 2 1 ν a 1 ds weakly. But since q is a quadratic form and V is a finite-dimensional space we have qa n qa which is property ii and which yields iii. Theorem 3.9. Let ν : Z [0, ] be a set function and p, q : R some functions and let h be a continuous truncation function. Then the following are equivalent: a there exists an infinitely divisible cylindrical probability measure µ with cylindrical characteristics p, q, ν h ; b the following is satisfied: 1 p0 = 0 and pa n pa for every sequence a n a in a finite dimensional subspace V equipped with ; 2 q : R is a quadratic form; 3 ν is a cylindrical Lévy measure; 4 a κa := ipa + ψ h u, a νdu is negative-definite. 14

16 In this situation, the characteristic function of µ is given by ϕ µ : C, ϕ µ a = exp ipa 12 qa + ψ h u, a νdu and µ = µ 1 µ 2 where µ 1 and µ 2 are cylindrical probability measures with characteristic functions ϕ µ1 a = exp 1 2 qa and ϕ µ 2 a = exp κa. Proof. a b: The properties 2 and 3 are stated in Theorem 3.3 and the properties 1 and 4 are derived in Lemma 3.8. The property p0 = 0 is an immediate consequence of Bochner s theorem and the fact that q0 = 0. b a: Property 2 implies that ϕ q : C, ϕ q a := e 1 2 qa is the characteristic function of a Gaussian cylindrical probability measure µ 1, see [16] or [20, p.393]. Since also 1 k q is a quadratic form for every k N it follows that ϕ q 1/k is the characteristic function of a cylindrical measure which verifies µ 1 as infinitely divisible. Thus, we can assume q = 0 in the sequel. We have to show that the functions 1 ϕ k : C, ϕ k a := exp ipa + ψ h u, a νdu k are the characteristic function of a cylindrical probability measure for each k N. The case k = 1 shows that there exists a cylindrical measure µ with characteristic function ϕ 1 and the cases k 1 show that µ is infinitely divisible. Note firstly, that the integral in the definition of ϕ k exists and is finite because of condition 3. Obviously, ϕ k 0 = 1 by 1. Property 4 implies by the Schoenberg correspondence for functions on Banach spaces property h, p.192 in Vakhania et al that ϕ k is positive-definite. In order to show the last condition of Bochner s theorem let V be a finite-dimensional subspace, say V = span{b 1,..., b d } for b 1,..., b d and a n a 0 in V. Then, Z, {b 1,..., b d }, ν is a standard measure space. Let f : R R be a bounded continuous function and define g n : R, g n u := f u, a n u, a n 2 1 for n N {0}. It follows by 3 that each g n L 1 ν, Z, {b 1,..., b d } and g n u f 1 + c u, a for a constant c > 0. Lebesgue s theorem of dominated convergence implies that g n u νdu = g 0 u νdu, lim n 15

17 which shows that s 2 1 ν a 1 n ds s 2 1 ν a 1 ds weakly Condition 3 guarantees for each a, that ϕ µa : R C, ϕ µa t = exp ipat + ψ h s, t ν a 1 ds R is the characteristic function of an infinitely divisible probability measure, say µ a on BR with characteristics pa, 0, ν a 1 h. Then condition 1 together with the weak convergence in 3.20 imply by Theorem VII.2.9 and Remark VII.2.10 p.396 in [8] that ϕ µan t ϕ µa0 t for all t R. Because ϕ k a = ϕ µa 1 k for all a and k N the functions ϕ k are verified as continuous on every finite-dimensional subspace which is the last condition in Bochner s theorem. The remaining part on the form of the characteristic function is derived in Theorem 3.3. Example Now we can show that the function ϕ in Example 3.7 is in fact the characteristic function of an infinitely divisible cylindrical measure. The linearity of l yields that the mapping a pa = λhla is continuous on each finite dimensional subspace of if the truncation function h is continuous. The measure ν satisfies ν a 1 = δ la for each a and is therefore a cylindrical Lévy measure. Since 3.5 yields that f : R C, ft := λ e it 1 is a negative-definite function it follows for z 1,..., z n C, a 1,..., a n that n z i z j κa i a j = i,j=1 n z i z j f la i la j 0. i,j=1 Thus, the map κ is negative-definite which proves the claim due to Theorem 3.9. For a given cylindrical Lévy measure ν there does not exist in general an infinitely divisible cylindrical probability measure with cylindrical characteristics 0, 0, ν. But one might be able to construct a function p : R such that there exists a cylindrical probability measure with cylindrical characteristics p, 0, ν. The following example shows the construction of the function p for a given cylindrical Lévy measure ν with weak second moments. In Section 5 we consider the case if the cylindrical Lévy measure extends to a σ-finite measure on B. Example Let ν be a cylindrical Lévy measure which satisfies u, a 2 νdu < for all a. 16

18 The existence of the weak second moments enables us to define p : R, pa := h u, a u, a νdu. for a continuous truncation function h. With a careful analysis similar to the one in the proof of Theorem 5.1 it can be shown that p is continuous on every finite-dimensional subspace of. From 3.5 it follows that f : R C ft := e it 1 it is negative-definite. For z 1,..., z n C, a 1,..., a n we have: n i,j=1 z i z j ipa i a j + ψ h u, a i a j νdu n = z i z j f u, a i u, a j νdu 0. i,j=1 Theorem 3.9 shows that there exits an infinitely divisible cylindrical measure with cylindrical characteristics p, 0, ν h. We finish this section with characterising infinitely divisible cylindrical probability measures by their finite-dimensional projections. This result enables us to derive the analogue result for infinitely divisible Radon probability measures. Theorem a A set function µ : Z [0, ] is an infinitely divisible cylindrical probability measure if and only if µ πa 1 1,...,a n is an infinitely divisible probability measure for all a 1,..., a n, n N. b A Radon probability measure µ on B is infinitely divisible if and only if µ πa 1 1,...,a n is an infinitely divisible probability measure for all a 1,..., a n, n N. Proof. a Since all image measures µ πa 1 1,...,a n are probability measures on BR n it follows that µ is a cylindrical probability measure. Theorem 3.3 and Lemma 3.8 guarantee that the characteristic function of µ satisfies the conditions in Theorem 3.9. b For the restriction of µ to Z it follows from a that for each k N there exists a cylindrical probability measure µ k such that µ = µ k k. Theorem 1 in [17] 17

19 implies that there exists l in the algebraic dual of such that µ k δ l is a Radon probability measure where { 1, if la 1,..., la n B, δ l Z := 0, otherwise for every Z := Za 1,..., a n ; B Z. Since µ δ k l = µ k k δ k l = µ k δ l k and the right hand side is Radon it follows from [2, Prop ] that δl k is a Radon probability measure which yields l. Since µ k δ l is Radon a further application of [2, Prop ] shows that µ k is a Radon probability measure and µ is verified as an infinitely divisible Radon measure. 4 Continuous infinitely divisible cylindrical measures Continuity of cylindrical measures is defined with respect to an arbitrary vector topology O in. We assume here that the topological space, O satisfies the first countability axiom, that is that every neighborhood system of every point in has a countable local base. In such spaces, convergence is equivalent to sequentially convergent. In particular, equipped with the norm topology satisfies the first countability axiom. Definition 4.1. A cylindrical probability measure µ on Z is called O-continuous if for each ε > 0 there exists a neighborhood N of 0 such that µ {u : u, a 1} ε for all a N. If O is the norm topology we say µ is continuous. A cylindrical probability measure µ is O-continuous if and only if its characteristic function ϕ µ : C is continuous in the topology O, see [19, Th.II.3.1]. This enables us to derive the following criteria: Lemma 4.2. Let µ be an infinitely divisible cylindrical probability measure on Z with cylindrical characteristics p, q, ν h for a continuous truncation function h. Then the following are equivalent: a µ is O-continuous; b for every sequence a n a in, O we have: i pa n pa; ii qa n δ 0 ds + s 2 1 ν a 1 n ds qaδ 0 ds + s 2 1 ν a 1 ds weakly. 18

20 Proof. The cylindrical measure µ is O-continuous if and only if its characteristic function ϕ µ : C is continuous in, O, or equivalently, that ϕ µ : C is sequentially continuous. It follows as in the proof of Theorem 3.9 that: ϕ µ a n ϕ µ a for all sequences a n a in, O ϕ µ a 1 n t ϕ µ a 1t for all sequences a n a in, O and t R. By applying Theorem VII.2.9 and Remark VII.2.10 p.396 in [8] and Lemma 3.6 the right hand side is equivalent to the conditions i and ii in b which completes the proof. In Corollary 4.2 it does not follow from a that we can consider separately the quadratic form q and the cylindrical Lévy measure ν in condition b. This is due to the well known fact, that a sequence of infinitely divisible measures on BR can converge weakly such that the small jumps contribute to the Gaussian part in the limit. But because of the representation of an infinitely divisible cylindrical measure as the convolution of two cylindrical probability measures according to Theorem 3.9 this is an important property which motivates the following definition: Definition 4.3. An infinitely divisible cylindrical probability measure with cylindrical characteristics p, q, ν h is called regularly O-continuous if the infinitely divisible cylindrical measures with cylindrical characteristics 0, q, 0 h and p, 0, ν h are O-continuous. Lemma 4.4. Let h be a continuous truncation function. For an O-continuous infinitely divisible cylindrical probability measure µ with cylindrical characteristics p, q, ν h the following are equivalent: a µ is regularly O-continuous; b q : R is continuous in, O; c for every sequence a n a in, O we have: i pa n pa; ii s 2 1 ν a 1 n ds s 2 1 ν a 1 ds weakly. Proof. Let ϕ µ be the characteristic function of µ. Then ϕ µ = ϕ 1 ϕ 2 where ϕ 1 is the characteristic function of the cylindrical measure µ 1 with cylindrical characteristics 0, q, 0 h and ϕ 2 is the characteristic function of the cylindrical measure µ 2 with cylindrical characteristics p, 0, ν h. Since the characteristic function of an infinitely divisible measure does not vanish in any point it follows that continuity of ϕ 1 and ϕ µ results in the continuity of ϕ 2 and analogously if ϕ 2 is continuous. Thus, µ is regularly O-continuous if and only if either µ 1 or µ 2 is O-continuous. Applying Lemma 4.2 to µ 1 shows the equivalence a b and applying Lemma 4.2 to µ 2 shows the equivalence a c. 19

21 Remark 4.5. If is equipped with the norm topology then b in Lemma 4.4 can be replaced by b there exists a positive, symmetric operator Q : such that qa = a, Qa for all a ; Proof. According to Proposition IV.4.2 in [20] there exist a probability space Ω, A, P and a cylindrical random variable X : L 0 P Ω, A with cylindrical distribution 0, q, 0 and with characteristic function a ϕa = exp 1 2 qa. If q is continuous Proposition VI.5.1 in [20] implies that the mapping X : L 0 P Ω, A is continuous. It follows from Theorem 4.7 in [1] that Qab := E[XaXb] for a, b defines a positive, symmetric operator Q :. Obviously, it satisfies qa = Qaa for each a. Example 4.6. Let Ω, A, P be a probability space and let L := Lt : t 0 be a cylindrical process, that is the mappings Lt : L 0 P Ω, A are linear. In Applebaum and Riedle [1] we call L a cylindrical Lévy process if Lta1,..., Lta n : t 0 is a Lévy process in R n for all a 1,..., a n, n N. If L is a cylindrical Lévy process we derive in [1] that it can be decomposed according to Lt = W t + Y t for all t 0 P -a.s., where W t : t 0 and Y t : t 0 are cylindrical processes. Their characteristic functions are for all a given by ϕ W t a := E[expiW ta] = exp 1 2 qat for a quadratic form q : R and ϕ Y t a := E[expiY ta] = exp t ipa + ψ h u, a νdu for a mapping p : R and a cylindrical Lévy measure ν. Obviously, p, q, ν is a cylindrical characteristics. If the corresponding cylindrical measure is regularly continuous, i.e. the cylindrical measures with the characteristic functions ϕ W 1 and ϕ L1 are continuous, it follows that also the mappings W t : L 0 P Ω, A, Y t : L 0 P Ω, A, are continuous, see [20, Prop.VI.5.1]. Moreover, according to Remark 4.5 the quadratic form q is of the form qa = a, Qa for all a and for a symmetric, positive operator Q :. If Q then W is a cylindrical Wiener process in a strong sense, which is usually referred to in the literature, see Riedle [16]. 20

22 5 Lévy measures on Banach spaces In this section we consider the situation that the cylindrical Lévy measure ν extends to a σ-finite measure on B which is also denoted by ν. Theorem 5.1. Let ν be a cylindrical Lévy measure which extends to a σ-finite measure on B with νb c <. Then there exits a regularly continuous infinitely divisible cylindrical probability measure µ with cylindrical characteristics d ν, 0, ν h, where d ν : R, d ν a := h u, a u, a 1B u νdu. Proof. First we show that the integral in the definition of the function d ν is well defined for every truncation function h. Choose a constant c > 0 such that and define for every a the set {t R : t c} Dh Da := {v : v, a c}. For the integrand f a u := h u, a u, a 1 B u it follows for every u that f a u 0 u Da B c D c a B D c a B c. But on these three domains we obtain f a u νdu Da B c and f a u νdu D c a B = B c c< u,a a c< s a c νdu = c νb c <, h u, a u, a νdu hs s ν a 1 ds h + a ν a 1 {s R : s > c} <, because ν a 1 is a Lévy measure on BR and f a u νdu h νdu = h νb c <. D c a B c Now we choose the truncation function h continuous and show by a similar decomposition that d ν is continuous. Let a n a in and choose a constant c > 0 such that B c {t R : t c + ε} Dh. 21

23 for some ε > 0. Let Da = {v : v, a c}. Since for every u B we have u, a n u, a a n a, we can conclude that there exists n 0 N such that u Da B implies that u, a, u, a n Dh for every n n 0. Consequently, we have for f a,n u := h u, a n h u, a u, a n u, a 1 B u and n n 0 the implication: f a,n u 0 u Da B c D c a B D c a B c. As above it can be shown that f a,n is dominated by an integrable function on all three sets and therefore, Lebesgue s theorem on dominated convergence shows that d ν is continuous. It follows for h s := s 1 BR s from 3.5 that f : R C fs 0, t := ψ h s 0, t is negative-definite for each s 0 R. For z 1,..., z n C, a 1,..., a n we have: n z i z j id ν a i a j + ψ h u, a i a j νdu i,j=1 n = z i z j e i u,a i a j 1 i u, a i a j 1 B u νdu = = i,j=1 n z i z j i,j=1 n i,j=1 e i u,a i a j u u 1 i u,a i a j u u 1 BR u z i z j f u, 1 u u, a i u, a j νdu 0. νdu Theorem 3.9 implies that there exists an infinitely divisible cylindrical probbility measure µ with cylindrical characteristics d ν, 0, ν h. In order to show that µ is continuous let a n a 0 in. For a bounded continuous function f : R R define g n : R, g n u := f u, a n u, a n 2 1 for n N {0}. It follows that each g n L 1 ν, B and g n u f 1 + c u, a for a constant c > 0. Lebesgue s theorem on dominated convergence implies that g n u νdu = g 0 u νdu, lim n 22

24 which shows that s 2 1 ν a 1 n ds s 2 1 ν a 1 0 ds weakly Lemma 4.2 implies that µ is continuous and thus µ is regular continuous by Lemma 4.4. A cylindrical Lévy measure which extends to a σ-finite measure on B is an obvious candidate to be a Lévy measure in the usual sense. We recall the definition from Linde [11]: a σ-finite measure ν on B is called a Lévy measure if a u, a 2 1 νdu < for all a ; b there exists a Radon probability measure µ on B with characteristic function ϕ µ : C, ϕ µ a = exp e i u,a 1 i u, a 1 B u νdu In fact, this is rather a result Theorem in Linde [11] than his definition. Note furthermore, that this definition includes already the requirement that a Radon probability measure on B exists with the corresponding characteristic function. In general, there are no conditions known on a measure ν which guarantee that ν is a Lévy measure. In particular, the condition u 2 1 νdu < is sufficient and necessary in Hilbert spaces, but in general spaces it is even neither sufficient nor necessary, for example in the space of continuous functions on [0, 1], see Araujo [3]. Corollary 5.2. Let ν be a σ-finite measure on B and h be a truncation function. Then the following are equivalent: a ν is a Lévy measure; b there exists an infinitely divisible cylindrical probability measure µ with cylindrical characteristics d ν, 0, ν h and it extends to a Radon measure on B. In this situation, the Radon probability measure with characteristic function 5.22 corresponding to the Lévy measure ν coincides with the Radon extension of µ. 23

25 Proof. It is easily seen that the characteristic function of the cylindrical measure with cylindrical characteristics d ν, 0, ν h is of the form Consequently, b implies a. If ν is a Lévy measure Proposition in [11] guarantees that νb c <. Theorem 5.1 implies that there exists a cylindrical probability measure with cylindrical characteristics d ν, 0, ν h which extends to a Radon probability measure because its characteristic function is of the form Remark 5.3. If ν is a Lévy measure and µ the infinitely divisible Radon probability measure with characteristic function 5.22 one might call the triplet 0, 0, ν the characteristics of µ. However, according to Corollary 5.2 the measure µ considered as an infinitely divisible cylindrical probability measure has the cylindrical characteristics d ν, 0, ν h. This asymmetry illustrates the interaction of the components p and ν of the cylindrical characteristics p, 0, ν of an infinitely divisible cylindrical measure. Even if ν is a Lévy measure and p = d ν then the function a κa := e i u,a 1 i u, a 1 B u νdu is negative-definite by Bochner s theorem and the Schoenmacher correspondence. But although κa = id ν a + ψ h u, a νdu none of the both summands in this representation respectively are negativedefinite in general. In general, the condition b in Corollary 5.2 might be verified by applying Prohorov s theorem, [20, Th.VI.3.2], and proving that the cylindrical measure µ is tight. In Sazanov spaces this is simplified: Remark 5.4. If is a Sazanov space then condition b in Corollary 5.2 can be replaced by: b i there exits an infinitely divisible cylindrical probability measure µ with characteristics d ν, 0, ν; ii a κa = id ν a + ψ h u, a νdu is continuous in an admissible topology. Example 5.5. If is a Hilbert space then the Sazanov topology is admissible. If d ν, 0, ν h is a cylindrical characteristics the function κ is necessarily negativedefinite and if it is also continuous in the Sazanov topology one obtains the well known Lévy-Khintchine decomposition in Hilbert spaces, see [14, Th ]. 24

26 References [1] D. Applebaum and M. Riedle. Cylindrical Lévy processes in Banach spaces. To appear in Proc. of Lond. Math. Soc. [2] V. I. Bogachev. Measure theory. Vol. I and II. Berlin: Springer., [3] A. P. de Araujo. On infinitely divisible laws in C[0, 1]. Proc. Am. Math. Soc., 51: , [4] A. P. de Araujo. On the central limit theorem in Banach spaces. J. Multivariate Anal., 8: , [5] A. P. de Araujo and E. Giné. The central limit theorem for real and Banach valued random variables. Chichester: John Wiley & Sons., [6] E. Giné and M. G. Hahn. On stability of probability laws with univariate stable marginals. Z. Wahrscheinlichkeitstheor. Verw. Geb., 64: , [7] H. Heyer. Probability measures on locally compact groups. Berlin: Springer, [8] J. Jacod and A. N. Shiryaev. Limit theorems for stochastic processes. Berlin: Springer, [9] G. Kallianpur and J. Xiong. Stochastic differential equations in infinite dimensional spaces. Hayward, CA: IMS, Inst. of Math. Statistics, [10] M. Ledoux and M. Talagrand. Probability in Banach spaces. Isoperimetry and processes. Berlin: Springer, [11] W. Linde. Infinitely divisible and stable measures on Banach spaces. Leipzig: BSB B. G. Teubner Verlagsgesellschaft, [12] D. J. Marcus. Non-stable laws with all projections stable. Z. Wahrscheinlichkeitstheor. Verw. Geb., 64: , [13] M. Metivier and J. Pellaumail. Stochastic integration. New York: Academic Press, [14] K. R. Parthasarathy. Probability measures on metric spaces. New York: Academic Press, [15] V. I. Paulauskas. Infinitely divisible and stable laws in separable Banach spaces. I. Lith. Math. J., 18: , [16] M. Riedle. Cylindrical Wiener processes. To appear in Séminaire de Probabilités. 25

27 [17] J. Rosiński. On the convolution of cylindrical measures. Bull. Acad. Pol. Sci., 30: , [18] K.-I. Sato. Lévy processes and infinitely divisible distributions. Cambridge: Cambridge niversity Press, [19] L. Schwartz. Radon measures on arbitrary topological spaces and cylindrical measures. London: Oxford niversity Press, [20] N. N. Vakhaniya, V. I. Tarieladze, and S. A. Chobanyan. Probability distributions on Banach spaces. Dordrecht: D. Reidel Publishing Company,