Multivariate tempered stable random fields

arXiv:2010.10169v1 [math.PR] 20 Oct 2020 MULTIVARIATE TEMPERED STABLE RANDOM FIELDS D. KREMER AND H.-P. SCHEFFLER Abstract. Multivariate tempered stable random measures (ISRMs) are constructed and their corresponding space of integrable functions is characterized in terms of a quasi-norm utilizing the so-called Rosinski measure of a tempered stable law. In the special case of exponential tempered ISRMs operator-fractional tempered stable random fields are presented by a moving-average and a harmonizable representation, respectively. Dedicated to the late Mark M. Meerschaert. 1. Introduction Multivariate tempered stable distributions are obtained by modifying the Lévy measure of a multivariate stable law by a certain tempering function q(r, θ), see (2.1) below. This results in tempering the large jumps of the corresponding Lévy process. See [13, 21] for the basic theory of tempered stable laws. A prominent subclass of tempered stable laws is obtained by using exponential tempering q(r, θ) = qλ (r) = e−λr for some λ > 0. These exponential tempered stable laws have what is called semi-heavy tails, meaning that the tails decay like a power law for in an intermediate regim and then decay exponentially fast near infinity. They have various applications, see e.g. [1, 2, 5, 15, 18, 19]. Symmetric α-stable random measures M(dx) are used to construct a large class of stable processes in terms of stochastic integrals via Z X(t) = ft (x) M(dx) for appropriate kernel functions ft . Prominent examples are fractional stable processes given by either a moving-average or a harmonizable representation, see [22] and the literature cited there. By modifying the kernel function of fractional stable processes, so-called tempered fractional stable motions are obtained, see [6]. The purpose of this paper is to construct fractional tempered stable fields and processes and to study their basic properties. In doing so, we are not modifying the kernel function, but utilizing so-called tempered stable random Date: October 21, 2020. 2010 Mathematics Subject Classification. 60E07; 60G52; 60G57; 60H05. Key words and phrases. Tempered stable distributions; Independently scattered random measures; Stochastic integrals; Tangent fields. 1 2 D. KREMER AND H.-P. SCHEFFLER measures. That is, the random noise M(dx) gets tempered, whereas the kernel functions remains unchanged. Using the recently developed general theory of multivariate independently scattered random measures (abbreviated by ISRM) and their corresponding stochastic integrals (see [11]), in Section 2 we introduce multivariate tempered stable ISRMs and characterize theirs space of integrable functions utilizing a quasi-norm based the so-called Rosinski measure of the underlying tempered stable law. The most prominent example is given by the exponentially tempered ISRM Mλ , see Example 2.7 below. Using those we then give examples of operatorfractional tempered stable fields, using ideas and methods of [10, 12]. It turns out that the range of possible parameters of those fields is larger than their corresponding stable fields. In Section 4, we analyze what happens if the exponential tempering parameter λ tends to zero. It turns out that in this case Mλ (and the constructed random fields) converge in distribution to theirs stable counterpart M0 . Moreover, it is shown that operator-fractional tempered stable random fields given by the moving-average representation behave locally like operator-fractional stable random fields, using the notion of localisability. 2. Preliminary Results We start with some notation. Write min(a, b) = a ∧ b and max(a, b) = a ∨ b, respectively. Let k·k be the Euclidian norm on Rd with inner product h·, ·i. Denote the set of all linear operators on Rd by L(Rd ) with I = Id being the identity operator on Rd , represented as d × d matrices in each case. Then, by a little abuse of notation, we also write k·k for the operator norm on L(Rd ) (which is induced by the vector space norm k·k). Denote the transposed vector of x by xt and the adjoint matrix of D by D ∗ , respectively. Finally, all occurring random vectors are defined on an underlying probability space (Ω, A, P) and we equip Rd with its Borel σ-Algebra B(Rd ) (accordingly for L(Rd )). 2.1. Tempered stable distributions and random measures. Consider µ to be a tempered α-stable distribution (abbreviated by T αS) on Rd in the sense of Definition 2.1 in [21], where the stability index 0 < α < 2 is fixed throughout the paper. That means that µ is infinitely-divisible without Gaussian part and that its Lévy measure φ in polar coordinates is of the form (2.1) φ(dr, dθ) = r −α−1 q(r, θ) dr σ(dθ). Here σ is a finite measure on S d−1 := {x ∈ Rd : kxk = 1} and q : (0, ∞) × S d−1 → (0, ∞) is a Borel function such that, for every θ ∈ S d−1 , we have: n ∂ (i) q(·, θ) is completely monotone, i.e. (−1)n ∂r n q(r, θ) > 0 for all r > 0 and n ∈ N. (ii) limr→∞ q(r, θ) = 0. 3 We call q the tempering function of the T αS distribution µ. It allows to define a measure R on Rd , which is often referred to as the Rosinski measure. The details can be found in [21]. Actually, it is uniquely determined by the relation Z Z ∞ (2.2) φ(A) = 1A (rx)r −α−1 e−r dr R(dx), A ∈ B(Rd ). Rd 0 It fulfills R({0}) = 0, Z Rd (kxk2 ∧ kxkα ) R(dx) < ∞, R which implies that (1 ∧ kxk2 ) R(dx) < ∞. So R is a Lévy measure, too. Also recall the Lévy-Khintchine-Formula (see Theorem 3.1.11 in [17]) to verify that µ ∼ [a, 0, φ], where the vector a ∈ Rd is a shift. In other words, the characteristic function of µ is given by µ b(u) = exp(ψ(u)), where  Z  ihu, xi ihu,xi (2.3) ψ(u) = iha, ui + e −1− φ(dx), u ∈ Rd 2 1 + kxk Rd is the corresponding log-characteristic function (abbreviated by LCF). Remark 2.1. (i) If q additionally fulfills limr↓0 q(r, θ) = 1 for every θ ∈ S d−1 , then µ is called a proper T αS distribution in R theαterminology of [21]. Theorem 2.3 in [21] states that µ is proper if and only if kxk R(dx) < ∞. (ii) It is usual to assume that µ is full, i.e. not concentrated on any hyperplane of Rd . Using Proposition 3.1.20 in [17] together with (2.2) it follows that µ is full if and only if R is not concentrated on any d − 1 dimensional subspace of Rd . In this case, following the notation in [17], we say that R is M-full. (iii) The function ψ can sometimes be represented more explicitly, cf. [13] for the case d = 1. Moreover, Theorem 2.9 in [21] presents integral representations for ψ in terms of the measure R. However, they are based on a different function within the integral in (2.3), which leads to a different shift vector. See Remark 8.4 in [23] for details. (iv) The previous distinction is negligible if µ is symmetric. Because in this case φ is also symmetric and a = 0, which allows to rewrite (2.3) as Z (2.4) ψ(u) = (coshu, xi − 1) φ(dx), u ∈ Rd . Rd In what follows we want to use µ in order to construct a new class of T αS random measures. For this purpose let (S, Σ, ν) be a σ-finite measure space and S := {A ∈ Σ : ν(A) < ∞}. Note that S is a so-called δ-ring whose generated σ-algebra equals Σ (see [11]). Recall that a mapping M : S → {X : Ω → Rd | X random vector} is called an Rd -valued iindependently scattered) random measure on S if the following conditions hold true: (RM1 ) For every finite choice A1 , ..., Ak of disjoint sets in S the random vectors M(A1 ), ..., M(Ak ) are stochastically independent. 4 D. KREMER AND H.-P. SCHEFFLER (RM2 ) For every sequence (An ) ⊂ S of disjoint sets fulfilling ∪∞ n=1 An ∈ S we have that ∞ X M(An ) almost surely (a.s.). M(∪∞ A ) = n=1 n n=1 Using Example 3.7 in [11] we get the following result. Theorem 2.2. Recall that µ ∼ [a, 0, φ], let S and ν be as before. Then there exists an Rd -valued ISRM M on S such that M(A) has an infinitely-divisible distribution given by M(A) ∼ [ν(A) · a, 0, ν(A) · φ] for every A ∈ S. In view of Proposition 2.5 below it is reasonable to call M a tempered (α-)stable ISRM, which is generated by µ. Also note that M, being infinitely-divisible, is closely related to its so-called control measure λM on Σ, cf. Theorem 3.2 in [11]. In particular, λM equals ν up to a positive constant. 2.2. Integration theory. Usually, the main benefit of ISRMs results from the consideration of corresponding stochastic integrals. More precisely, for f : S → L(Rd ), we want to define integrals of the form Z IM (f ) := I(f ) := f (s)M(ds), S leading to Rd -valued random vectors again. Since this integral is achieved by a stochastic limit, conditions for the existence of I(f ) have been studied in [11] (see [20] for the case d = 1). We omit the details. Instead we focus on the main results, adapted to our setting. Therefore, denote the set of all functions f : S → L(Rd ) for which I(f ) is well-defined by I(M). Then Proposition 4.5 in [11] and Theorem 2.3 in [12] yield the following. Theorem 2.3. Let M be as in Theorem 2.2 and define U : L(Rd ) → Rd by  Z  Dx Dx U(D) := Da + φ(dx), D ∈ L(Rd ). − 2 2 1 + kDxk 1 + kxk Rd Then, for f : S → L(Rd ) measurable, the following statements are equivalent: (i) f ∈ I(M). (ii) J1 (f ) + J2 (f ) < ∞ holds true, where Z Z Z (2.5) J1 (f ) := kU(f (s))k ν(ds) and J2 (f ) := (1 ∧ kf (s)xk2 ) φ(dx) ν(ds). d S S R R d ∗ (iii) For every u ∈ R we have that |ψ(f (s) u)| ν(ds) < ∞ and the mapping Z Z d R ∋ u 7→ (coshf (s)∗u, xi − 1) φ(dx) ν(ds) S is continuous. Rd 5 In each caseR it follows that I(f ) is infinitely-divisible. More precisely, I(f ) ∼ [af , 0, φf ], where af := U(f (s)) ν(ds) and φf (A) := (ν ⊗ φ)({(s, x) ∈ S × Rd : f (s)x ∈ A \ {0}}), R Hence, the LCF of I(f ) is given by Rd ∋ u 7→ ψ(f (s)∗ u) ν(ds). (2.6) A ∈ B(Rd ). The notion stochastic integral is first of all justified by the fact that I(M) is a real vector space and that the mapping I(M) ∋ f 7→ I(f ) is linear. Moreover, for f, f1 , f2 , ... ∈ I(M), we have that I(fn ) → I(f ) in probability if and only if Z d ψ((fn (s) − f (s))∗ u) ν(ds) → 0 (as n → ∞). (2.7) ∀u ∈ R : S This and further properties can be found in Proposition 4.3 and Theorem 4.4 in [11]. In addition, we identify elements in I(M) that are identical ν-a.e. because this implies that the resulting integrals are equal a.s. and vice versa. Next we want to understand J2 (f ) (recall (2.5)) in terms of the Rosinski measure R, which is inspired by Lemma 2.10 in [21]. Lemma 2.4. We have that J2 (f ) < ∞ ⇔ H(f ) := Z Z S Rd (kf (s)xkα ∧ kf (s)xk2 ) R(dx) ν(ds) < ∞. Proof. Recall that 0 < α < 2 is fixed and consider for z > 0 the incomplete gamma type functions Z z Z z Z ∞ (2−α)−1 −u 1−α −u γ(2 − α, z) := u e du = u e du and Γ(−α, z) := u−α−1 e−u du, 0 0 z respectively. For instance, see Proposition 12 in [9] together with [24] in order to verify that the following asymptotics hold true: As z → 0 we have that z α−2 γ(2 − α, z) → (2 − α)−1 On the other hand, as z → ∞, γ(2 − α, z) → Γ(2 − α) = Z and z α Γ(−α.z) → α−1 . ∞ 0 u1−α e−u du and z 1+α ez Γ(−α, z) → 1. If we define g(z) := z −2 γ(2 − α, z) + Γ(−α, z), it follows that lim z α g(z) = (2 − α)−1 + α−1 > 0, (2.8) (2.9) z→0 lim z 2 g(z) = lim (γ(2 − α, z) + z 1+α ez Γ(−α, z) × z 1−α e−z ) = Γ(2 − α) > 0. z→∞ z→∞ Combine (2.8)-(2.9) and use the fact that z 7→ g(z)/(z −α ∧ z −2 ) is continuous as well as strictly positive on (0, ∞) to obtain constants C1 , C2 > 0 fulfilling (2.10) ∀z > 0 : C1 (z −α ∧ z −2 ) ≤ g(z) ≤ C2 (z −α ∧ z −2 ). 6 D. KREMER AND H.-P. SCHEFFLER On the other hand, (2.2) yields that Z Z Z ∞ J2 (f ) = (1 ∧ krf (s)xk2 )r −α−1 e−r dr R(dx) ν(ds). S Rd 0 Now fix (s, x) ∈ S × Rd such that f (s)x 6= 0 and let z = kf (s)xk−1 to observe that Z ∞ Z z Z ∞ 2 −α−1 −r −2 1−α −r (1 ∧ krf (s)xk )r e dr = z r e dr + r −α−1 e−r dr = g(z). 0 0 z Then the assertion follows from (2.10).  The function H(f ), which we just introduced, is similar to the approach in [10] and will be useful again later. But before we still have to explain in which sense M and the corresponding integrals have a tempered stable behavior. In this context, note that f = 1A I belongs to I(M) for every A ∈ S. Proposition 2.5. Consider f, f1 , ...fk ∈ I(M). Then the random vector (I(f1 ), ..., I(fk ))t has a T αS distribution. Moreover, the Rosinski measure of I(f ) is given by (2.11) Rf (A) := (ν ⊗ R)({(s, x) ∈ S × Rd : f (s)x ∈ A \ {0}}), which means that (2.12) f φ (A) = Z Rd Z 0 A ∈ B(Rd ), ∞ 1A (ry)r −α−1 e−r dr Rf (dy). Proof. We first assume that k = 1. Recall that I(f ) has no Gaussian part and that its Lévy measure φf is given by (2.6). Using (2.2) together with the definition of Rf from (2.11) it follows for every A ∈ B(Rd ) that Z Z Z ∞ f φ (A) = 1A\{0} (rf (s)x)r −α−1 e−r dr R(dx) ν(ds) d ZS ZR ∞ 0 = 1A (ry)r −α−1 e−r dr Rf (dy), Rd 0 i.e. (2.12) holds true. It remains to show that Z (2.13) (kykα ∧ kyk2 ) Rf (dy) < ∞. Because in this case and based on (2.12), Theorem 2.3 in [21] states that φf really is the Lévy measure of a T αS distribution on Rd . Actually, by (2.11) and in view of Lemma 2.4, condition (2.13) is equivalent to the finiteness of J2 (f ). And since f ∈ I(M) implies that J2 (f ) < ∞ due to Theorem 2.3, this completes the proof for k = 1. In the general case 7 Corollary 4.9 in [11] states that (I(f1 ), ..., I(fk ))t is infinitely-divisible, while a combination of Example 3.7 and Theorem 4.4 in [11] shows that its LCF is given by Rk·d ∋ (u1 , ..., uk )t 7→ (2.14) Z k X ψ S fj (s)∗ uj j=1 ! ν(ds). P Define ψ ′ ((u1, ..., uk )t ) := ψ( kj=1 uj ). Then one can verify that ψ ′ is the LCF of a distribution µ′ on Rk·d, which is also T αS. Since the corresponding notation is somewhat involved, the details are left to the reader. Anyway, it follows from Theorem 2.2 that there exists an Rk·d -valued ISRM, which is generated by µ′ and which we denote by M′ . Moreover, define the block diagonal matrices f ′ (s) := f1 (s) ⊕ · · · ⊕ fk (s) ∈ L(Rk·d) for every s ∈ S. Then, using (2.14), it is easy to check that f ′ ∈ I(M′ ) and that t d (I(f1 ), ..., I(fk )) = Z f ′ (s) M′ (ds), S d where = means equality in distribution. Hence, the argument for (I(f1 ), ..., I(fk ))t having a T αS distribution reduces to the case k = 1 from above.  If the T αS distributed generator µ is proper things become R more practicable. For this purpose let L1 (ν) := {g : S → R | g is measurable and |g(s)| ν(ds) < ∞} and recall Remark 2.1 (i). Corollary 2.6. Assume that µ is proper. Then we have: (i) kf kα ∈ L1 (ν) implies that J2 (f ) < ∞ and, provided that f ∈ I(M), also that the T αS random Rvector I(f ) is proper. (ii) Moreover, if kxk≥1 kxk2 R(dx) < ∞, a sufficient condition for J2 (f ) < ∞ is given by (kf kα ∧ kf k2 ) ∈ L1 (ν). R Proof. The fact that µ is proper yields kxkα R(dx) < ∞. Recall from Lemma 2.4 that R α J2 (f ) < ∞ if and only if H(f ) = (kyk ∧ kyk2) Rf (dy) < ∞, where H(f ) is obviously bounded by Z Rd α f kyk R (dy) = Z Z S Rd α kf (s)xk R(dx) ν(ds) ≤ Z S α kf (s)k ν(ds) Z Rd kxkα R(dx). 8 D. KREMER AND H.-P. SCHEFFLER Hence, (i) follows from Proposition 2.5, since we have that I(f ) is proper if and only if R kykα Rf (dy) < ∞. In a similar way we obtain (ii) due to the following computation: Z Z H(f ) = (kf (s)xkα ∧ kf (s)xk2 ) R(dx) ν(ds) d S R Z Z α 2 ≤ (kf (s)k ∧ kf (s)k ) ν(ds) (kxkα ∨ kxk2 ) R(dx) d R  Z Z ZS α 2 α 2 (2.15) kxk R(dx) + kxk R(dx) . = (kf (s)k ∧ kf (s)k ) ν(ds) kxk≤1 S kxk≥1  R Note that the finiteness of kxk≥1 kxk2 R(dx) is equivalent to the finiteness of R R kxk2 φ(dx) or Rd kxk2 µ(dx), respectively (see Proposition 2.7 in [21]). kxk≥1 2.3. The full and proper-symmetric case. From now on we only consider the case that the generator µ is full as well as proper and symmetric. In view of Remark 2.1 this implies that f ∈ I(M) ⇔ J2 (f ) < ∞ together with I(f ) ∼ [0, 0, φf ]. In particular, the conditions of Corollary 2.6 (i) are fulfilled. The following example fits into this setting and also illustrates part (ii) of Corollary 2.6. Example 2.7. Fix λ > 0 and explicitly consider the proper tempering function q(r, θ) = qλ (r) := e−λr in (2.1). Then, if σ is symmetric, φ = φλ and µλ ∼ [0, 0, φλ] are also symmetric. Moreover, using the construction from (2.3)-(2.5) in [21], we observe that the corresponding Rosinski measure Rλ is concentrated on the set λ · S d−1 , where it equals the image measure λ(σ), defined by λ(σ)(A) := σ(λ−1 · A) (cf. Example 1 in [21]). It follows that Z Z 2 kxk Rλ (dx) = kλθk2 σ(dθ) = λ2 σ(S d−1 ) < ∞. kxk≥1 S d−1 Thus, in view of Corollary 2.6 (ii), (kf kα ∧ kf k2) ∈ L1 (ν) implies that f ∈ I(Mλ ), where Mλ denotes the ISRM that is generated by µλ . In a similar way it follows from Lemma 2.4 that I(Mλ ) does not depend on λ > 0. We remark that the previous example will be very important later. Now consider f, f1 , f2 , ... ∈ I(M) and observe that I(fn ) − I(f ) = I(fn − f ) ∼ [0, 0, φfn−f ] in the present case. Hence, as an alternative to criterion (2.7), we have that I(fn ) → I(f ) in probability if and only if the convergence φfn −f → 0 in the space M (that is the space of all measures on Rd that assign finite measure to sets bounded away from zero) holds true. That means φfn −f (A) → 0 for all A ∈ B(Rd ) bounded away from zero. See Theorem 3.1.16 in [17] for details. Lemma 2.1 in [11] shows that this is also equivalent to J2 (fn − f ) → 0. On the other hand, relation (2.10) from the proof of Lemma 2.4 revealed that (2.16) C1 H(f ) ≤ J2 (f ) ≤ C2 H(f ) 9 holds true for all measurable f . Hence, as n → ∞ we have that (2.17) I(fn ) → I(f ) in probability ⇔ Similarly to (2.13) in [10] this motivates to introduce Z h(D) := (kDxkα ∧ kDxk2 ) R(dx), Rd together with H(f, δ) := Z S H(fn − f ) → 0. D ∈ L(Rd ) h(δ −1 f (s)) ν(ds) ∈ [0, ∞], δ > 0. Recall that R is M-full due to Remark 2.1. Then we obtain the following observation. Lemma 2.8. Under the previous assumptions there exists a constant K > 0 such that (kDkα ∧ kDk2 ) ≤ K h(D) holds true for every D ∈ L(Rd ). Proof. Fix 1 ≤ i, j ≤ d. Denote the corresponding entry of the matrix D by Di,j and let ej be the j-th unit vector in Rd . Since R is M-full, we have that R({zej : z 6= 0}) > 0. In particular, using that R is a Lévy measure, there exists some 0 < cj ≤ 1 such that √ √ 1/α Fj := {zej : |z| ≥ cj } fulfills 0 < R(Fj ) < ∞. In view of cj < cj it follows that 2 |Di,j |α ∧ Di,j ≤ kDej kα ∧ kDej k2 Z √ 1/α −1 (kcj Dej kα ∧ k cj Dej k2 )1Fj (x) R(dx) = (cj R(Fj )) Z −1 (kDxkα ∧ kDxk2 )1Fj (x) R(dx) ≤ (cj R(Fj ))−1 h(D). ≤ (cj R(Fj )) Note that i as well as j were arbitrary and that all norms on L(Rd ) are equivalent. Hence, up to a positive constant, K can be chosen as max({(cj R(Fj ))−1 : 1 ≤ j ≤ d}).  We call a mapping k·kM : I(M) → [0, ∞) a quasi-norm (on I(M)) if it has the usual properties of a norm, except a possible weakening of the triangular inequality. That is, there exists some A ≥ 1 such that (2.18) ∀f1 , f2 ∈ I(M) : kf1 + f2 kM ≤ A(kf1 kM + kf2 kM ) holds true. Recall that we identify elements in I(M) that are identical ν-a.e. Theorem 2.9. (a) The following identities hold true: I(M) = {f : S → L(Rd ) | f is measurable and H(f, δ) < ∞ for all δ > 0} = {f : S → L(Rd ) | f is measurable and H(f, δ) < ∞ for some δ > 0}. 10 D. KREMER AND H.-P. SCHEFFLER (b) kf kM := inf{δ > 0 : H(f, δ) ≤ 1} defines a quasi-norm on I(M). Moreover, for every f ∈ I(M) and δ > 0, we have α  2 α  2   kf kM kf kM kf kM kf kM ∧ ≤ H(f, δ) ≤ ∨ . (2.19) δ δ δ δ (c) There exists some L1 ≥ 1 such that, for every f ∈ I(M) and δ > 0, we have (2.20) P(kI(f )k ≥ δ) ≤ L1 H(f, δ). Additionally, if 0 < p < α, there exists a constant L2 = L2 (p) > 0 such that E(kI(f )kp ) ≤ L2 kf kpM . (2.21) (d) The vector space I(M) is complete with respect to k·kM and, for f, f1 , ... ∈ I(M), we have the characterization I(fn ) → I(f ) in probability ⇔ kfn − f kM → 0. Proof of Theorem 2.9. Note that H(f ) = H(f, 1) and verify that part (a) follows from Lemma 2.4 after a verification of (2.22) ∀δ > 0 : (δ −α ∧ δ −2 )H(f, 1) ≤ H(f, δ) = H(δ −1 f, 1) ≤ (δ −α ∨ δ −2 )H(f, 1). We now prove part (b). Obviously, f = 0 implies that H(f, δ) = 0 for all δ > 0, which yields kf kM = 0. Conversely, if kf kM = 0, we derive from (2.22) that H(f, 1) = 0, i.e. h(f (s)) = 0 for almost every s ∈ S. In view of Lemma 2.8 we obtain for those s that f (s) = 0. Also note that the we have H(ρf, δ) = H(f, |ρ|−1δ) for all ρ 6= 0 and δ > 0. Hence, the homogeneity property kρf kM = |ρ| kf kM (ρ ∈ R) holds true by definition of k·kM . It remains to prove (2.18). For this purpose recall that ku1 + u2 k2 ≤ 2(ku1 k2 + ku2k2 ) (u1, u2 ∈ Rd ) and that the mapping [0, ∞) ∋ z 7→ (1 ∧ z) is sub-additive. Using (2.16) we compute for all δ > 0 and f1 , f2 ∈ I(M) that H(f1 + f2 , δ) ≤ C1−1 J2 (δ −1 (f1 + f2 )) ≤ 2C1−1 (J2 (δ −1 f1 ) + J2 (δ −1 f2 )) ≤ 2C2 /C1 (H(f1 , δ) + H(f2, δ)). Now choose A ≥ 1 such that H(f, Aδ) ≤ (4C2 /C1 )−1 H(f, δ) holds true for every δ > 0 and f as before, which is possible due to (2.22). Since H(f, ·) is decreasing, it follows for ε > 0 arbitrary that H(f1 + f2 , A(kf1 kM + kf2 kM + ε)) ≤ 2C2 /C1 (H(f1, A(kf1 kM + kf2 kM + ε)) + H(f2 , A(kf1 kM + kf2 kM + ε))) ≤ 2C2 /C1 (H(f1, A(kf1 kM + ε)) + H(f2 , A(kf2 kM + ε))) ≤ 2−1 (H(f1 , kf1 kM + ε) + H(f2 , kf2 kM + ε)) ≤ 1, where in the last step we used the definition of k·kM . Letting ε → 0, the same argument shows (2.18) and hence that k·kM is a quasi-norm. Finally, (2.19) is obvious in the case kf kM = 0. Else we observe that H(f, kf kM ) = 1 by continuity of δ 7→ H(f, δ). Then, 11 similar as before, we can write H(f, δ) = H((δ/kf kM )−1 f, kf kM ) such that (2.19) follows from (2.22). For the proof of part (c) recall that |1 − exp(z)| ≤ |z| for any z ∈ C with Re z ≤ 0 and that the LCF of I(f ) is given by (2.14) for k = 1. Combine this with Lemma 2.7 in [10] to obtain a constant C > 0 such that, for every δ > 0, the relation Z Z d (2.23) P(kI(f )k ≥ δ) ≤ Cδ |ψ(f (s)∗u)| ν(ds) du kuk∞ ≤δ−1 S holds true. Here k·k∞ denotes the maximum norm on Rd . At the same time there exists some C0 ≥ 1 fulfilling k·k ≤ C0 k·k∞ . Also recall (2.4) and, for every z ∈ R, that | cos(z) − 1| ≤ 2(1 ∧ z 2 ). From the Cauchy-Schwarz inequality it follows for every kuk∞ ≤ δ −1 that Z Z Z ∗ |ψ(f (s) u)| ν(s) ≤ | coshf (s)∗ u, xi − 1| φ(dx) ν(s) S S Rd Z Z ≤2 (2.24) (1 ∧ kuk2 kf (s)xk2 ) φ(dx) ν(s) d S R Z Z 2 (1 ∧ kδ −1 f (s)xk2 ) φ(dx) ν(s) = 2 C02 J2 (δ −1 f ). ≤ 2 C0 S Rd Let L1 := 2C02 C C2 to verify that (2.16) together with (2.23) implies (2.20). For the second statement of part (c) fix 0 < p < α. Then it is well-known that Z∞ E(kI(f )kp ) = p δ p−1 P(kI(f )k ≥ δ) dδ, 0 where (2.19)-(2.20) provide the inequality P(kI(f )k ≥ δ) ≤ 1[0,kf kM ] (δ) + L1 (kf kM /δ)α 1(kf kM ,∞) (δ), δ > 0. Now, as in the proof of Theorem 2.8 in [10], we observe that (2.21) holds true for a constant L2 > 0. The details are left to the reader. In order to show that I(M) is complete, let (fn ) ⊂ I(M) be a Cauchy-sequence with respect to k·kM . Then we have to find some f ∈ I(M) such that kfn − f kM → 0 as n → ∞, or at least along a suitable subsequence, since (fn ) is Cauchy and since k·kM fulfills (2.18). Hence, without loss of generality and throughout following the idea of the proof of Theorem 5.2.1 in [7], it can be assumed that kfm − fn kM ≤ 2−N (2.25) for all m, n ≥ N . Recall the definition of H(·, 1) and define the sets Aj := {s ∈ S : h(fj+1 (s) − fj (s)) > j −4 } for all j ∈ N. Using (2.19) and (2.25), it follows that P∞ j −4 ν(An ) ≤ H(fj+1 − fj , 1) ≤ 2−αj , j ∈ N, which implies that j=1 ν(Aj ) < ∞. Thus B := lim supj→∞ Aj is a ν-null set. Let f (s) := 0 for every s ∈ B. Conversely, for s ∈ B c = S \ B, there exists some N(s) ∈ N fulfilling s ∈ / Aj for all j ≥ N(s). Furthermore, Lemma 2.8 allows us to assume that N(s) is chosen large 12 D. KREMER AND H.-P. SCHEFFLER enough such that we have kfj+1(s) − fj (s)k ≤ 1 for every j ≥ N(s). Actually, if we use Lemma 2.8 again, this implies for all s ∈ B c and m, n ≥ N(s) that kfm (s) − fn (s)k ≤ = ∞ q X j=N (s) ∞ q X j=N (s) ≤ √ k(fj+1(s) − fj (s))k2 k(fj+1(s) − fj (s))kα ∧ k(fj+1 (s) − fj (s))k2 ∞ ∞ q X X √ K h(fj+1 (s) − fj (s)) ≤ K j −2 . j=N (s) j=N (s) It follows that kfm (s) − fn (s)k → 0 as N(s) tends to ∞, which shows that (fn (s)) is Cauchy with respect to the operator norm k·k. Denote the corresponding limit by f (s) and observe that f : S → L(Rd ) is measurable with fn (s) → f (s) ν-almost everywhere as n → ∞. Moreover, (2.18) and (2.25) imply for every n ∈ N that kfn kM ≤ A(kf1 kM + 2−1 ). Also note that D 7→ h(D) is continuous by the dominated convergence theorem. Hence, Fatou’s Lemma and (2.19) yield that Z H(f, 1) = lim inf h(fn (s)) ν(ds) ≤ lim inf H(fn , 1) ≤ lim inf (kfn kαM ∨ kfn k2M ) < ∞, S n→∞ n→∞ n→∞ i.e. f belongs to I(M) due to part (a). Similarly, it follows from (2.25) for every n ∈ N that H(fn − f, 1) ≤ lim inf (kfn − fm kαM ∨ kfm − fm k2M ) ≤ 2−αn . m→∞ In view of (2.19) this shows that kfn − f kM → 0 as n → ∞. The additional statement of part (d) is just a consequence of (2.17) and (2.19).  Remark 2.10. Since I(f ) has a T αS distribution it is well-known that E(kI(f )kp ) is finite for all 0 < p < α. However, this can also be true for p ≥ α, depending on the behavior of Rf (see Proposition 2.7 in [21]). But in this case we would need sharper estimates in order to perform an appropriate proof of (2.21). We finish this section with a useful observation, which, for D = f (s), we implicitly encountered within the foregoing proofs. More precisely, recall (2.2) and (2.24) together with the proofs of Lemma 2.4 and Corollary 2.6 (particularly (2.15)) as well as Example 2.7. Corollary 2.11. In the situation of Example 2.7 we denote the LCF of µλ by ψλ . Then there exists a constant T > 0 such that |ψλ (D ∗ u)| ≤ T (1 + kuk2 )(kDkα ∧ kDk2 ) holds true for every D ∈ L(Rd ) and u ∈ Rd . 13 3. Examples of multivariate tempered stable random fields: Moving-average and harmonizable representation Throughout this section fix λ > 0 and, for the rest of this paper, we explicitly consider the n-dimensional Lebesgue measure ν(ds) = ds on (S, Σ) = (Rn , B(Rn )). Then, until further notice (in Section 3.2), let µλ and Mλ be as in Example 2.7, where σ is a finite, symmetric measure on S d−1 . In addition, we assume that σ is M-full, since this ensures that the generator µλ is full. Example 3.1. Consider the case n = 1 and let X(t) := Mλ ([0, t]) for every t ≥ 0. If we recall (RM1 )-(RM2 ) it is easy to see that the resulting Rd -valued stochastic process X = {X(t) : t ≥ 0} is a Lévy process. And since X(1) has a T αS distribution it follows that X is a T αS Lévy process in the sense of [21]. Note that there are interesting results in [21] concerning the so-called short time and long time behavior of T αS Lévy processes, respectively. However, in what follows, we treat the general case n ≥ 1. More precisely and in order to take advantage of our previous efforts, we will investigate Rd -valued random fields of the form X = {X(t) : t ∈ Rn }, where X(t) = I(ft ) for every t ∈ Rn and a suitable integrand (or kernel function) ft ∈ I(Mλ ). In this context, note that we benefit from Proposition 2.5 in the following sense, where 0 < α < 2 is still fixed. Definition 3.2. A random field X = {X(t) : t ∈ Rn } is called tempered (α-)stable (abbreviated as a T αS random field ) if all finite-dimensional distributions are T αS. 3.1. Moving-average representation. Our multivariate approach needs some preparation. Fix some matrix E ∈ Q(Rn ), where Q(Rn ) := {E ∈ L(Rn ) : All real parts of the eigenvalues of E are strictly positive}. Recall that exp(E), i.e. the matrix exponential of E is given by ∞ X Ej exp(E) = with cE := exp((log c)E), j! j=0 c > 0. For instance, see Proposition 2.2.2 in [17] for basic properties about the matrix exponential that we will use in sequel. This allows to define so-called generalized polar coordinates with respect to E. More precisely, Lemma 6.1.5 in [17] states that there exists a norm k·kE on Rn such that for SEn−1 := {x ∈ Rn : kxkE = 1} the mapping Ψ : (0, ∞) × SEn−1 → Rn \ {0}, Ψ(c, θ) = cE θ is a homeomorphism. In other words, every x 6= 0 can be uniquely written as x = τ (x)E l(x) for some radial part τ (x) > 0 and some direction l(x) ∈ SEn−1 . In this context, a function ϕ : Rn → C is called E-homogeneous if ϕ(cE x) = c ϕ(x) for all c > 0 and x 6= 0. On the other hand, letting β > 0, the authors in [4] say that a function ϕ : Rn → [0, ∞) is (β, E)-admissible if the following conditions hold true: 14 D. KREMER AND H.-P. SCHEFFLER (i) ϕ(x) > 0 for all x 6= 0. (ii) ϕ is continuous and for any 0 < A < B there exists a constant C > 0 such that, for A ≤ kyk ≤ B, (3.1) τ (x) ≤ 1 ⇒ |ϕ(x + y) − ϕ(y)| ≤ Cτ (x)β . Here, τ (x) = τE (x) is the radial part of x in terms of the generalized polar coordinates with respect to E. Now we are able to state the first main result of this section, where the two last-mentioned properties are actually combined. See Theorem 2.11 and Corollary 2.12 in [4] for useful examples of such functions. Also recall that I = Id is the identity operator on Rd . Theorem 3.3. Let E be as before and denote its trace by q. Assume that ϕ : Rn → [0, ∞) is an E-homogeneous, (β, E)-admissible function for some β > 0. Moreover, consider a d × d-matrix D ∈ Q(Rd ) such that the maximum of the real parts of the eigenvalues of D, denoted by H, fulfills   1 1 . (3.2) H <β+q − α 2 Then the random field X = {X(t) : t ∈ Rn }, defined by Z h i q q D− α I D− α I X(t) := ϕ(t − s) Mλ (ds), − ϕ(−s) Rn exists. Furthermore: (a) X is a T αS random field in the sense of Definition 3.2. (b) X is stochastically continuous. (c) X has stationary increments. That is, for any h ∈ Rn , f dd {X(t + h) − X(h) : t ∈ Rn } = {X(t) : t ∈ Rn }, f dd where = denotes equality of all finite-dimensional distributions. (d) If αq is not an eigenvalue of D, then X(t) is full for every t 6= 0. Proof. Note that X(0) = 0 a.s. Then fix t ∈ Rn \ {0} and observe that ϕ(x) = 0 ⇔ x = 0. It follows that (3.3) q q ft (s) := ϕ(t − s)D− α I − ϕ(−s)D− α I , s ∈ Rn is well-defined for every s ∈ / {0, t}. However, the cases s ∈ {0, t} are negligible since, up to a constant, the n-dimensional Lebesgue measure equals the control measure of Mλ in the present case. Then it suffices to verify that (kft (s)kα ∧kft (s)k2 ) ∈ L1 (ds) due to Example 2.7. For this purpose denote the generalized polar coordinates with respect to E by (τ (·), l(·)) 15 and let τ (0) = 0. Recall the proof of Theorem 2.5 in [14] and that, if H < β holds true instead of (3.2), there it has been shown for some suitably chosen γ > 0 that kft (s)kα = kft (s)kα 1{τ (s)≤γ} + kft (s)kα 1{τ (s)>γ} ∈ L1 (ds). Actually, for the verification of kft (s)kα 1{τ (s)≤γ} ∈ L1 (ds) the quoted proof only makes use of the assumptions on ϕ together with D ∈ Q(Rd ). Conversely, it uses H < β in order to establish the accuracy of kft (s)kα 1{τ (s)>γ} ∈ L1 (ds). Now it is easy to verify that condition (3.2) allows us to slightly modify the corresponding estimates from the proof of Theorem 2.5 in [14] and hence to obtain that kft (s)k2 1{τ (s)>γ} ∈ L1 (ds) holds true. The details are left to the reader. Anyway, in view of (3.4) (kft (s)kα ∧ kft (s)k2 ) ≤ kft (s)kα 1{τ (s)≤γ} + kft (s)k2 1{τ (s)>γ} , s ∈ Rn , this proves the existence of X. Then part (a) follows from Proposition 2.5. Now let us prove part (b). For this purpose fix t ∈ Rn and observe that, due to (2.17), we have to show that H(ft+h − ft ) → 0 as h → 0. Hence, if we recall Example 2.7 together with (2.15), the claimed stochastic continuity would follow if Z
kft+h (s) − ft (s)kα ∧ kft+h (s) − ft (s)k2 ds → 0 Rn holds true as h → 0. Thus by (3.3), a change of variables and (3.4), it suffices to prove that Z
(3.5) kfh (s)kα 1{τ (s)≤γ} + kfh (s)k2 1{τ (s)>γ} ds → 0 (h → 0), Rn actually leading back to the above-mentioned modification and its corresponding estimates. We omit the details, which, using the dominated convergence theorem, can be found in the proof of Theorem 2.5 in [14] again and which eventually give (3.5). Moreover, by linearity of the stochastic integral and (2.14), the proof of part (c) merely reduces to the foregoing change of variables. Finally, fix t 6= 0 and recall from Remark 2.1 in [14] that ft (s) is invertible for s in a subset of Rn with positive Lebesgue measure provided that αq is not an eigenvalue of D. Then the assertion of part (d) follows from Proposition 2.6 (a) in [12].  Remark 3.4. Assume that (3.2) is particularly fulfilled with H < β. Then the foregoing proof showed that the function kft (s)kα from (3.3) belongs to L1 (ds). In this case Corollary 2.6 (a) implies that X(t) has even a proper T αS distribution. 3.2. Harmonizable representation. Although Theorem 3.3 provides a large class of multivariate tempered stable random fields by now, we want to present a further class of this type in the sequel. The resulting random fields use a so-called harmonizable representation, which is also popular in the context of classical α-stable settings, that is without the use of tempering. For instance see [4, 14, 22], just to mention a few. It is also mentionable that, by investigating the scalar-valued α-stable moving-average and harmonizable representation from [4], Proposition 6.1 in [3] already established different path properties between both representations. Note that Remark 3.6 as well as Corollary 3.7 below will emphasize these 16 D. KREMER AND H.-P. SCHEFFLER differences. In what follows we have to use kernel functions taking values in L(Cd ), that is the space of d × d-matrices with entries from the complex numbers C. Moreover, this also requires the use of a complex-valued tempered stable ISRM. Essentially, we do so by modifying the underlying generator. More precisely, let σ̃ be a symmetric and finite as well as M-full measure on S 2d−1 = {x ∈ R2d : kxk = 1}. For λ > 0 as before consider the tempering function q(r, θ) = qλ (r) = e−λr again, but based on the polar coordinates in R2d now. Accordingly to (2.1) this allows to define a Lévy measure φ̃λ on R2d . Let µ̃λ ∼ [0, 0, φ̃λ ] with LCF ψ̃λ be the generator of an R2d -valued ISRM in the sense of Theorem 2.2 (still with ν(ds) = ds on e λ. (S, Σ) = (Rn , B(Rn ))), denoted by M e λ can be naturally identified with an Cd -valued ISRM, which we denote by MC Note that M λ hereinafter and which, similarly as before, allows for the use as integrator in a corresponding integration theory. We refer the reader to [11] for details. In any case, Proposition 4.11 in [11] states for a measurable function g : Rn → L(Cd ) that the Rd -valued random vector Z (3.6) Re g(s) MCλ (ds) Rn is well-defined (as a stochastic limit) if and only if   Re g −Im g e λ ), (3.7) g̃ := ∈ I(M 0 0 where g̃ ∈ L(R2d ). Then we can state the second main result of this section. Theorem 3.5. Let E ∈ Q(Rn ), denoting its trace by q again and its smallest real part of the eigenvalues by a. Assume ϕ : Rn → [0, ∞) to be a continuous and E ∗ -homogeneous function fulfilling ϕ(x) > 0 for all x ∈ Rn \ {0}. Also consider D ∈ L(Rd ), where h and H denote the minimum and the maximum of the real parts of the eigenvalues of D, respectively. Moreover, let the assumption   1 1 (3.8) q <h≤H<a − 2 α hold true. Then the random field Y = {Y (t) : t ∈ Rn }, defined by Z
q (3.9) Y (t) := Re eiht,si − 1 ϕ(s)−D− α I MCλ (ds), Rn exists in the sense of (3.6). Furthermore: (a) Y is tempered stable. (b) Y is stochastically continuous. (c) Y (t) is full for every t 6= 0. Proof. Denote the generalized polar coordinates with respect to E t by (τ1 (·), l1(·)) and observe that Lemma 4.2 in [14], although stated for D ∈ Q(Rd ), remains true for D ∈ L(Rd ) 17 due to (2.2) in [12]. Then, using the assumption H < a, it follows exactly as in the proof q of Theorem 2.6 in [14] that kskα kϕ(s)−D− α I kα 1{τ1 (s)<1} ∈ L1 (ds). Conversely, assuming that h > 0 (which is nothing else than D ∈ Q(Rd )), the aforementioned proof in [14] q showed that kϕ(s)−D− α I kα 1{τ1 (s)≥1} ∈ L1 (ds). Actually, by a slight modification it turns out that we just need the assumption h > q( 21 − α1 ) from (3.8) in order to establish that q kϕ(s)−D− α I k2 1{τ1 (s)≥1} ∈ L1 (ds) instead. If we combine our findings it follows that the function q q Ψ(s) := kskα kϕ(s)−D− α I kα 1{τ1 (s)<1} + kϕ(s)−D− α I k2 1{τ1 (s)≥1} , (3.10) s ∈ Rn belongs to L1 (ds), where the case s = 0 is negligible again. Now fix t ∈ Rn . Then, in view of (3.7) and Euler’s formula, we define  q  q (cosht, si − 1)ϕ(s)−D− α I − sinht, siϕ(s)−D− α I , s ∈ Rn (3.11) g̃t (s) := 0 0 and observe by equivalence of all norms on L(R2d ) that there exists some K0 > 0 fulfilling q kg̃t (s)k ≤ K0 (| cosht, si − 1| + | sinht, si|)kϕ(s)−D− α I k, (3.12) s ∈ Rn . Recall the idea from (3.4) and the inequality (a + b)α ≤ 2(aα + bα ). Then we obtain another constant K1 > 0 such that, for every s ∈ Rn , kg̃t (s)kα ∧ kg̃t (s)k2 is bounded by q q K1 (| cosht, si − 1|α + | sinht, si|α )kϕ(s)−D− α I kα 1{τ1 (s)<1} + kϕ(s)−D− α I k2 1{τ1 (s)≥1} . Finally, using the Cauchy-Schwarz inequality together with some routine estimates for sin(·) and cos(·), we verify that kg̃t (s)kα ∧ kg̃t (s)k2 ≤ K2 (1 + ktkα )Ψ(s), (3.13) s ∈ Rn holds true for some constant K2 > 0 and due to (3.10). In view of (3.7) and Example 2.7 (which remains true accordingly) this proves the existence of Y. We now prove part (a) and consider k ∈ N as well as t1 , ..., tk ∈ Rn arbitrary. It follows from Remark 4.13 in [11] that the random vector (Y (t1 ), ..., Y (tk ))t is infinitely-divisible and that its LCF is given by (3.14)  !t  Z k k X X ∗ q ∗ q ψ̃λ  (coshtj , si − 1)ϕ(s)−D − α I uj , − u 7→ sinhtj , siϕ(s)−D − α I uj  ds, Rn j=1 j=1 where u = (u1 , ..., uk )t ∈ Rk·d . Recall the specific form of g̃t from (3.11) and observe that (Y (t1 ), ..., Y (tk ))t can be regarded as projection of the Rk·(2d) -valued random vector R fλ for every t ∈ Rn . More precisely, for every (Z(t1 ), ..., Z(tk ))t , where Z(t) := g̃t dM t k·2d w = (w1 , ..., wk ) ∈ R with wj = (uj , vj )t (for j = 1, ...k) we can rewrite (3.14) as ! Z k X ψ̃λ g̃t (s)∗ wj ds. Rn j=1 18 D. KREMER AND H.-P. SCHEFFLER In any case, (Y (t1 ), ..., Y (tk ))t inherits the T αS-property from (Z(t1 ), ..., Z(tk ))t due to Proposition 2.5. For the proof of part (b) we fix t ∈ Rn and have to show that Y (t+h)−Y (t) converges in distribution to zero as h → 0. For this purpose define   cosht, siId sinht, siId A(t, s) := ∈ L(R2d ) − sinht, siId cosht, siId and recall from Proposition 4.3 in [11] that the integral in (3.6) is also linear (as a function of g). Then, according to (3.14) and similarly as in the proof of Proposition 5.2 in [12], it R d can be computed that the LCF of Y (t + h) − Y (t) is given by R ∋ u 7→ ψ̃λ (ζ(t, h, s, u)) ds, where  t q q ∗ −D ∗ − α I −D ∗ − α I ζ(t, h, s, u) : = A(t, s) (coshh, si − 1)ϕ(s) u, − sinhh, siϕ(s) u (3.15) = (g̃h (s)A(t, s))∗ (u, v)t and the last-mentioned identity holds true for every v ∈ Rd . For fixed u ∈ Rd and due to Lévy’s continuity theorem it remans to show the convergence Z (3.16) ψ̃λ (ζ(t, h, s, u)) ds → 0 (h → 0). Observe for every s ∈ Rn that we have ψ̃λ (ζ(t, h, s, u)) → 0 by continuity of ψ̃λ . At the same time it is obvious that Corollary 2.11 remains true accordingly for ψ̃λ . Hence, if we combine Corollary 2.11 with (3.15) (say for v = 0), it follows that |ψ̃λ (ζ(t, h, s, u)| ≤ T (1 + kuk2 )(kg̃h (s)A(t, s)kα ∧ kg̃h (s)A(t, s)k2 ) ≤ T (1 + kuk2 )(kg̃h (s)kα ∧ kg̃h (s)k2 ). Note that, in the second step, we used the sub-multiplicativity of the operator norm together with the face that kA(t, s)k = 1. Then the dominated convergence theorem gives (3.16) due to (3.13). Finally, the argument for part (c) is mostly presented in the proof of Proposition 5.2 (c) in [12] and therefore left to the reader.  Remark 3.6. Besides the fact that we demand ϕ to be E ∗ -homogeneous, we can drop condition (3.1) compared to Theorem 3.3. In addition, we get without further assumptions that Y (t) is full for every t 6= 0. Let us also emphasize that we did not require that D belongs to Q(Rd ), which will be explained in Section 4 more into detail. The following observation indicates another difference between both representations. Corollary 3.7. Let the assumptions of Theorem 3.5 be fulfilled. Additionally, assume that the measure σ̃ is invariant under all operators A ∈ T (2d), i.e. A(σ̃) = σ̃, where    (cos β)Id (sin β)Id T (2d) := : β ∈ [0, 2π) . −(sin β)Id (cos β)Id Then Y has stationary increments. 19 Proof of Corollary 3.7. Fix t, h ∈ Rn and observe that, by swapping the roles of t and h Y (h) is given by Rd ∋ u 7→ R in the proof of Theorem 3.5, the LCF of Y (t + h) −−λr does not depend on θ, it is ψ̃λ (ζ(h, t, s, u)) ds. Moreover, since q(r, θ) = qλ (r) = e easy to compute that the given assumption implies that ∀A ∈ T (2d) ∀w ∈ R2d : ψ̃λ (A∗ w) = ψ̃λ (w). R Hence, since A(0, s) = I2d , we observe for every u ∈ Rd that ψ̃λ (ζ(h, t, s, u)) ds equals R ψ̃λ (ζ(0, t, s, u)) ds, which is nothing else than the LCF of Y (t). Finally, in view of (3.14), the fdd-extension is obvious and therefore left to the reader. This shows that Y has stationary increments.  3.3. The special cases d = 1 and n = 1. Certainly, our multivariate approach is very general and its notation therefore rather involved. For many applications it is often enough to consider the cases d = 1 or n = 1 (or even d = n = 1). For those readers we briefly want to remark some useful observations and examples: • Recall that we denote the greatest real part of the eigenvalues of D by H, which, in case d = 1, is just a real number. Then H = h is referred to as the so-called Hurstindex in literature. Sometimes this Hurst-Index can help to determine possible scaling properties (see Section 4 below) as well as sample path properties of the corresponding random fields. • Consider the situation from Example 2.7 again, particularly for d = 1 and α 6= 1. In this case we have σ = c(ε−1 + ε1 ) for some c > 0, where εx denotes the point measure in x. Then it follows from Remark 2.8 in [13] that the ψλ can be computed as ψ(u) = −2c Γ(−α)[λα − Re (λ + iu)α ], u ∈ R. • On the other hand, in case n = 1, it appears natural to consider the function ϕ(·) = |·|, which is 1-homogeneous and (1, 1)-admissible. Then for the existence of the movingaverage representation X = {X(t) : t ∈ R} from Theorem 3.3 it only remains to ensure that 0 < H < α1 + 21 holds true. Moreover, X(t) is full for every t 6= 0 provided that H 6= α1 . Conversely, in the context of Theorem 3.5, condition (3.8) is equivalent to 21 − α1 < h ≤ H < 1 in this case. 4. The role of the tempering parameter e λ , MC and the respective Recall the definitions from Section 3, particularly those of Mλ , M λ generators. In either case we call λ > 0 the tempering parameter. This section will essentially deal with the behavior of the random fields from Section 3 as λ → 0. For this purpose we first have to define further generators and corresponding random measures that deal with the case λ = 0 in a reasonable way. 20 D. KREMER AND H.-P. SCHEFFLER Remark 4.1. Roughly speaking, for q ≡ 1 in (2.1) we get back the class of α-stable distributions on Rd , although this q is no valid tempering function anymore. Anyway, let µ0 ∼ [0, 0, φ0], where φ0 (dr, dθ) := r −α−1 dr σ(dθ) and note that µ0 is symmetric again. Similarly to Theorem 2.2 there exists an Rd -valued random measure, which is generated by µ0 and which we denote by M0 . We call M0 a symmetric α-stable (abbreviated by SαS) ISRM. Note that M0 corresponds to Example 2.4 in [12]. There it has been shown that a measurable function f : S → L(Rd ) is integrable with respect to M0 (i.e. f ∈ I(M0 )) if and only if Z Z Z Z S S d−1 kf (s)θkα σ(dθ) ν(ds) = Rn S d−1 kf (s)θkα σ(dθ) ds < ∞. In view of Lemma 2.4 and Example 2.7 it easily follows that I(M0 ) ⊂ I(Mλ ) = I(M1 ) for every λ > 0. Finally, in a similar way as before we obtain an R2d -valued SαS random e 0 (generated by µ̃0 with LCF ψ̃0 ) and a corresponding Cd -valued one, denoted by measure M MC0 . We omit the details. 4.1. A note on different kinds of tempering. Let us emphasize that a main advantage of our approach is based on the fact that we use T αS distributions as a generator for the underlying random measures. By doing so we combine both the heavy tailed behavior of classical α-stable distributions and Gaussian trends. Hence, concerning the corresponding stochastic integrals, the observation of Remark 4.1 above is not surprising. Namely we enlarge the class of possible kernel functions when using T αS generators instead of α-stable ones. Let us illustrate this effect by means of the kernel functions from Section 3. For this purpose note that the following statement is due to Theorem 2.5 and Theorem 2.6 in [14], respectively, except for a negligible difference. That is, the authors in [14] assume σ (and σ̃) to be uniformly distributed on S d−1 (and S 2d−1 ), at least up to a constant, i.e. the LCF of µ0 is given by ψ0 (u) = −ρkukα for some ρ > 0. Proposition 4.2. (a) Assume that the conditions of Theorem 3.3 are fulfilled with H < β instead of (3.2). Then the random field X0 = {X0 (t) : t ∈ Rn } exists, where Z h i q q X0 (t) := ϕ(t − s)D− α I − ϕ(−s)D− α I M0 (ds), t ∈ Rn . Rn (b) Assume that the conditions of Theorem 3.5 are fulfilled for some D ∈ Q(Rd ) (i.e. (3.8) reduces to H < a). Then the random field Y0 = {Y0 (t) : t ∈ Rn } exists, where Z
q Y0 (t) := Re eiht,si − 1 ϕ(s)−D− α I MC0 (ds), t ∈ Rn . Rn In both cases, X0 and Y0 are SαS random fields, respectively. Here the notion of SαS random fields is analogous to Definition 3.2. Moreover, as expected, the assumptions under which X0 and Y0 exist are more restrictive compared to their natural 21 counterparts from Section 3 and we lose the T αS property. However, the remaining properties from Theorem 3.3 and Theorem 3.5 still apply to X0 and Y0 accordingly. Actually, the random fields from Section 3 have a true tempered stable character that arises from Proposition 2.5. Let us be more specific what we mean by the word true: Recall the main results from [6] and note that there a different kind of tempering is considered, which extends the univariate idea from [16]. That is, the authors in [6] somehow add a tempering function to the actual kernel function, say f (·), while they abstain from the use of T αS random measured as integrator. This has two consequences. On the one hand, it follows that the behavior of f (s) is restrained for large values of s. However, on the other hand, the resulting stochastic integrals have no T αS distribution. Also note that, in [6], the aforementioned approach has been performed both for a movingaverage representation and for a harmonizable representation. We focus on the first one. Then, taking into account the different notation in [6], it is easy to verify that we obtain the following observation as a by-product of Theorem 3.1 in [6] (namely for B = α1 I). The details are left to the reader. Example 4.3. Fix λ > 0 and let ϕ : Rn → [0, ∞) be an E-homogeneous function for some E ∈ Q(Rn ), where q denotes the trace of E again. Also consider D ∈ Q(Rd ) arbitrary. Then the random field X̂ = {X̂(t) : t ∈ Rn }, defined by Z h i q q X̂(t) := e−λϕ(t−s) ϕ(t − s)D− α I − e−λϕ(−s) ϕ(−s)D− α I M0 (ds), Rn exists, where M0 is as above. Moreover, X̂ is a SαS random field. Recall from the proof of Theorem 2.5 in [14] that, in the context of Proposition 4.2 (a), condition H < β is needed in order to control the behavior of ft (s) from (3.3) for large values of s. Hence, as already announced before, this condition disappears when tempering in the sense of [6]. For the same reason we observe in Example 4.3 that ϕ has not to be admissible anymore. 4.2. Scaling properties and tangent fields. In what follows we will show that the random fields from Section 3 have an intimate relation to those from Proposition 4.2. Lemma 4.4. We have that ∀ρ, λ > 0 ∀u ∈ Rd : ψλ (ρ u) = ρα ψλ/ρ (u). Similarly, the statement holds true for ψ̃λ . Proof. In view of µλ ∼ [0, 0, φλ], where φλ is symmetric, we obtain that Z Z ψλ (ρu) = (coshρu, xi − 1) φλ (dx) = (coshu, xi − 1) (ρφλ )(dx). Rd Rd 22 D. KREMER AND H.-P. SCHEFFLER It remains to show that (ρφλ ) = ρα · φλ/ρ . Fix A ∈ B(Rd ). Then, due to (2.1) and by a change of variables, we compute that Z Z ∞ (ρφλ )(A) = 1ρ−1 A (rθ)r −α−1 e−λr dr σ(dθ) d−1 ZS Z0 ∞ = 1A (ρrθ)r −α−1 e−λr dr σ(dθ) d−1 S Z 0Z ∞ = ρα 1A (rθ)r −α−1 e−λr/ρ dr σ(dθ) = ρα · φλ/ρ (A). S d−1 0  In order to be more specific we denote by Xλ = {Xλ (t) : t ∈ Rn } and Yλ = {Yλ (t) : t ∈ Rn } the moving-average and harmonizable representation corresponding to Mλ and MCλ attained in Section 3, respectively. Then the next result states scaling properties for Xλ and Yλ and is therefore of independent interest. Proposition 4.5. (4.1) (a) Under the assumptions of Theorem 3.3 we have that ∀c > 0 : f dd {Xλ (cE t) : t ∈ Rn } = {cD Xc αq λ (t) : t ∈ Rn }. (b) Conversely, under the assumptions of Theorem 3.5 we have that (4.2) ∀c > 0 : f dd {Yλ (cE t) : t ∈ Rn } = {cD Yc− αq λ (t) : t ∈ Rn }. Proof. For the proof of part (a) recall ft (·) from (3.3) and observe for (almost) all s ∈ Rn that, by E-homogeneity of ϕ, we have (4.3) q fcE t (s) = cD− α I ft (c−E s). Now fix k ∈ N and t1 , ..., tk ∈ Rn . Then, due to (2.14), (4.3), Lemma 4.4 and by a change of variables, we compute that the LCF of (Xλ (cE t1 ), ..., Xλ (cE tk ))t is given by ! Z k X (4.4) Rk·d ∋ (u1 , ..., uk ) 7→ ψλ fcE tj (s)∗ uj ds Rn = Z j=1 ψλ Rn = c−q = Rn k X ftj (c−E s)∗ c D∗ uj j=1 Z Rn Z c q −α ψc αq λ ψc αq λ k X ftj (c−E s)∗ c j=1 k X j=1 ftj (s)∗ c D∗ uj ! D∗ ! uj ds, ds ! ds 23 which, again by (2.14), is just the LCF of (cD Xc αq λ (t1 ), ..., cD Xc αq λ (tk ))t . This gives (4.1). Concerning part (b) recall the representation of Y (t) = Yλ(t) from (3.9). Using the E ∗ homogeneity of ϕ (which, in general, is different from the function ϕ in part (a)) we obtain that (4.5) ∗− q I α = c α (cosht, cE si − 1)φ(cE s)−D ∗− q I α = c α sinht, cE siϕ(cE s)−D (coshcE t, si − 1)φ(s)−D and, similarly, that sinhcE t, siϕ(s)−D (4.6) q q ∗ ∗ ∗ ∗ ∗− q I α ∗− q I α cD ∗ cD . ∗ Note that the trace of E ∗ equals q again. Hence, based on (3.14) and (4.5)-(4.6), we can proceed as in the proof of part (a) in order to verify the accuracy of (4.2). The details are left to the reader.  Remark 4.6. Observe that there is a striking difference in the scaling behavior between the moving-average and the harmonizable representation of T αS random fields considered in Section 3. We need another auxiliary result, which is of independent interest again. For this purpose recall Proposition 4.2 and that the conditions mentioned there (and quoted below) always f dd imply the existence of the associated T αS representations from Section 3. Here, ⇒ means convergence of all finite-dimensional distributions. (a) Assume that the conditions of Theorem 3.3 are fulfilled with H < β Lemma 4.7. f dd instead of (3.2). Then, as λ → 0, we have that Xλ ⇒ X0 . (b) Assume that the conditions of Theorem 3.5 are fulfilled for some D ∈ Q(Rd ). Then, f dd as λ → 0, we have that Yλ ⇒ Y0 . Proof. We only prove part (a) since the proof of part (b) is very similar and, moreover, mostly covered by the ideas in the proof of Theorem 4.10 below. Fix k ∈ N as well as t1 , ..., tk ∈ Rn and u1 , ..., uk ∈ Rd . Then, in view of (4.4) (for c = 1), the assertion would follow from ! ! Z Z k k X X (4.7) ψλ ftj (s)∗ uj ds → ψ0 ftj (s)∗ uj ds (λ → 0) Rn Rn j=1 j=1 due to Lévy’s continuity theorem. Using (2.1) and (2.4) we first observe for every λ > 0 and u ∈ Rd that Z Z ∞ |ψλ (u)| = −ψλ (u) = (1 − coshu, rθi)r −α−1 e−λr dr σ(dθ) d−1 ZS Z0 ∞ ≤ (4.8) (1 − coshu, rθi)r −α−1 dr σ(dθ) = −ψ0 (u) = |ψ0 (u)|. S d−1 0 In particular, the dominated convergence theorem implies for every u ∈ Rd that we have ψλ (u) → ψ0 (u) as λ → 0. At the same time it is well-known that we have |ψ0 (u)| ≤ ρkukα , 24 D. KREMER AND H.-P. SCHEFFLER where ρ > 0 is a constant (see the proof of Example 2.4 in [12]), and that the inequality 1 − cos(a + b) ≤ 2(2 − cos(a) − cos(b)) holds true (see Proposition 1.3.4 in [17]). Combine this with (4.8) to verify the accuracy of ! k k k X X X k−1 ∗ k−1 ∗ ψλ kuj kα kftj (s)kα . (4.9) ftj (s) uj ≤ 2 ψ0 (ftj (s) uj ) ≤ 2 ρ j=1 j=1 j=1 In addition and under the present assumptions, Theorem 2.5 in [14] states that the function on the right-hand side of (4.9) belongs to L1 (ds). Using the dominated convergence theorem again this gives (4.7).  The foregoing observations can be used to go a step further, leading to the notion of so-called tangent fields. Note that, in the univariate case, the idea goes back to [8]. The following definition turns out to be a multivariate extension of it and is essentially due to [10]. Definition 4.8. Fix x ∈ Rn and let X = {X(t) : t ∈ Rn } as well as X′ = {X ′ (t) : t ∈ Rn } be Rd -valued random fields. Then, for E ∈ Q(Rn ) and D ∈ Q(Rd ), we say that X is (E, D)-localisable at x with local form/tangent field X′ if we have that (4.10) f dd {c−D (X(x + cE t) − X(x)) : t ∈ Rn } ⇒ {X ′ (t) : t ∈ Rn } (c → 0). Let us emphasize that, in general, the tangent field X′ as well as the corresponding operators E and D in (4.10) do depend on the point x at which we have localisability. However, this is not the case in our setting as the following main result demonstrates. Theorem 4.9. Assume that the conditions of Theorem 3.3 are fulfilled with H < β instead of (3.2). Then we have for every λ > 0 and x ∈ Rn that the moving-average representation Xλ is (E, D)-localisable at x with tangent field X′ = X0 (not depending on x). Proof. Fix k ∈ N and consider t1 , ..., tk ∈ Rn . Then, using the fact that Xλ has stationary increments together with (4.1), it is easy to check that we have ∀c > 0 : f dd {c−D (Xλ (x + cE t) − Xλ (x)) : t ∈ Rn } = {Xc αq λ (t) : t ∈ Rn }. Thus the assertion follows from Lemma 4.7.  Recall (4.10) and observe for all x, t ∈ Rn that x + cE t contracts to x as c → 0, since E ∈ Q(Rn ) and due to Lemma 4.2 in [14]. Hence, it is crucial to emphasize that the convergence in (4.11) below is meant as c → ∞, which is striking differently. Accordingly, we do no longer use the term tangent field hereinafter. Theorem 4.10. Assume that the conditions of Theorem 3.5 are fulfilled for some D ∈ Q(Rd ). Then we have for every λ > 0 and x ∈ Rn that (4.11) f dd {c−D (Yλ (x + cE t) − Yλ (x)) : t ∈ Rn } ⇒ Y0 (c → ∞). 25 Proof. Let us recall the proof of Theorem 3.5, particularly (3.14) and (3.15). Then, for t1 , ..., tk ∈ Rn as before, we compute that the LCF of (c−D (Yλ (x+cE tj )−Yλ(x)) : 1 ≤ j ≤ k)t is given by R k·d t ∋ (u1 , ..., uk ) 7→ Z k X ψ̃λ Rn E ζ(x, c tj , s, c −D ∗ uj ) j=1 ! ds. At the same time, by definition of A(x, s) and ζ, we can use (4.5)-(4.6) together with Lemma 4.4 to obtain that Z ψ̃ Rn k X ζ(x, cE tj , s, c j=1 −D ∗ uj ) ! ds = Z ψ̃λ Rn = cq = Rn k X E∗ ζ(c−E x, tj , c s, uj ) j=1 Z Rn Z c q α ψ̃c−q/α λ k X E∗ ψ̃c−q/α λ ds ζ(c−E x, tj , c s, uj ) j=1 k X ! ζ(c −E x, tj , s, uj ) j=1 ! ! ds ds, where in the last step we performed a change of variables. Fix u1 , ..., uk ∈ Rd . Then, by Lévy’s continuity theorem and in view of A(0, s) = I2d , it obviously remains to establish that (4.12) Z Rn ψ̃c−q/α λ k X ζ(c−E x, tj , s, uj ) j=1 ! ds → Z ψ̃0 Rn k X ζ(0, tj , s, uj ) j=1 ! ds (c → ∞). First observe for every w ∈ R2d and as λ → 0 that, as in the proof of Lemma 4.7, we have ψ̃λ (w) → ψ̃0 (w). That is, µ̃λ converges to µ̃0 weakly due to Lévy’s continuity theorem again. Then Lemma 3.1.10 in [17] states that the convergence ψ̃λ → ψ̃0 holds even uniformly on compact subsets of R2d . Hence, since c−E x → 0 as c → ∞ (see Lemma 4.2 in [14]), it is easy to check for every s ∈ Rn that we have ψ̃c−q/α λ k X j=1 ζ(c−E x, tj , s, uj ) ! → ψ̃0 k X j=1 ζ(0, tj , s, uj ) ! (c → ∞). 26 D. KREMER AND H.-P. SCHEFFLER Use essentially the same argument as in (4.9) to verify that there exists a constant L > 0 (only depending on k) such that, for every c > 0 and s ∈ Rn , we have ! k k X X −E kζ(c−E x, tj , s, uj )kα ζ(c x, tj , s, uj ) ≤ L ψ̃c−q/α λ j=1 j=1 ≤L (4.13) =L k X j=1 k X j=1 kA(c−E , x)kα kg̃tj (s)kα kuj kα kg̃tj (s)kα kuj kα , where in the second step we benefited from (3.15) together with the sup-multiplicativity of the operator norm. Also note that we used kA(c−E , x)kα = 1 again. Anyway and based on (3.12), it was just the outcome of Theorem 2.6 in [14] that the function in (4.13) belongs to L1 (ds). Eventually, the dominated convergence theorem gives (4.12).  Remark 4.11. 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Dustin Kremer, Department Mathematik, Universität Siegen, 57068 Siegen, Germany Email address: [email protected] Hans-Peter Scheffler, Department Mathematik, Universität Siegen, 57068 Siegen, Germany Email address: [email protected]