PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 80(94) (2006), 121–140 DOI:10.2298/PIM0694121H

REGULAR VARIATION FOR MEASURES ON METRIC SPACES

Henrik Hult and Filip Lindskog

Abstract. The foundations of regular variation for Borel measures on a complete separable space S, that is closed under multiplication by nonnegative real numbers, is reviewed. For such measures an appropriate notion of convergence is presented and the basic results such as a Portmanteau theorem, a mapping theorem and a characterization of relative compactness are derived. Regular variation is defined in this general setting and several statements that are equivalent to this definition are presented. This extends the notion of regular variation for Borel measures on the Euclidean space R^{d} to more general metric spaces. Some examples, including regular variation for Borel measures on R^{d}, the space of continuous functions C and the Skorohod space D, are provided.

1. Introduction

In many areas of applied probability one encounters a Borel measure ν on R^{d} with the following asymptotic scaling property: for some α > 0 and all λ > 0

lim_{t→∞} \frac{ν(λtA)}{ν(tE)} = λ^{−α} lim_{t→∞} \frac{ν(tA)}{ν(tE)}, (1.1)

where E is a fixed reference set such as E = {x ∈ R^{d} : |x| ⩾ 1}, tE = {tx : x ∈ E}, and A may vary over the Borel sets that are bounded away from the origin 0 (the closure of A does not contain 0). Typically ν is a probability measure but other classes of measures, such as Lévy- or intensity measures for infinitely divisible distributions or random measures, with this property appear frequently in the probability literature. The asymptotic scaling property implies that the function c(t) = 1/ν(tE) is regularly varying (at ∞) with index α and that c(t)ν(tA) converges to a finite constant μ(A) as t → ∞. Then (1.1) translates into the scaling property μ(λA) = λ^{−α}μ(A) for λ > 0. A measure satisfying (1.1) is called regularly varying (see Section 3 for a precise definition). The name is motivated by t ↦ ν(tA) being a regularly varying function (with index −α) whenever μ(A) > 0. To motivate the approach we suggest for defining regular variation for measures on general spaces, it is useful to first consider the most commonly encountered way to define regular variation for measures on R^{d}. In order to use the well established notion of vague convergence (see e.g. [17]), instead of R^{d} one considers the space [−∞, ∞]^{d}∖{0}. The reason is that sets that are bounded away from 0 in R^{d} become topologically bounded (or relatively compact) in [−∞, ∞]^{d} ∖ {0}. This means that regular variation for a measure ν on R^{d} can be defined as the vague convergence ν(t·)/ν(tE) → μ(·) as t → ∞, where μ is a nonzero measure on [−∞, ∞]^{d} ∖ {0}. There exist definitions of regular variation for measures on other spaces that are not locally compact. For instance, on the Skorohod space D[0, 1], regular variation is formulated using polar coordinates and a notion of convergence for boundedly finite measures (see [13] and [12]). Also in this case the original space is changed by introducing “points at infinity” in order to turn sets bounded away from 0 in the original space into metrically bounded sets. It is the aim of this paper to review the foundations of regular variation for measures on general metric spaces without considering modifications of the original space of the types explained above.

2000 Mathematics Subject Classification: Primary 28A33.

The first author acknowledges support from the Sweden–America Foundation.

The first step towards a general formulation of regular variation is to find an appropriate notion of convergence of measures. Recall that a sequence of bounded (or totally finite) measures μ_{n} on a separable metric space S converges weakly to a bounded measure μ as n → ∞ if lim_{n→∞} ∫ f dμ_{n} = ∫ f dμ for all bounded and continuous real functions f. By the Portmanteau theorem, an equivalent formulation is that lim_{n→∞} μ_{n}(A) = μ(A) for all Borel sets A with μ(∂A) = 0, where ∂A denotes the boundary of A. A convenient way to modify weak convergence to fit into a regular variation context is to define convergence μ_{n} → μ by lim_{n→∞} ∫ f dμ_{n} = ∫ f dμ for all bounded and continuous real functions f that vanish in a neighborhood of a fixed point s_{0} ∈ S (the origin). The foundations of this notion of convergence, including a Portmanteau theorem, a mapping theorem and characterizations of relative compactness, are presented in Section 2.

To define regular variation for measures on S, the space S has to be closed under multiplication by nonnegative real numbers λ ∈ R_{+}. Moreover, the map (λ, x) ↦ λx from R_{+} × S into S should be continuous, there should exist an element 0 ∈ S so that 0x = 0 for all x ∈ S, and the metric d on S should satisfy d(0, λ_{1}x) < d(0, λ_{2}x) for all λ_{1}, λ_{2} ∈ R_{+} with λ_{1} < λ_{2} and all x ∈ S∖{0}. The last assumption means that for x ∈ S ∖ {0} the distance to the origin for a point on the ray {λx : λ ∈ R_{+}} is strictly increasing in λ. Under these additional assumptions the notion of regular variation for measures on S is introduced in Section 3. An alternative approach to define regular variation for measures on S would be to identify S with a product space and use polar coordinates. However, we do not pursue such alternative approaches here. It is worth noticing that the formulation of regular variation proposed here is equivalent to usual formulations of regular variation for measures on R^{d}, C[0, 1] and D[0, 1] found in e.g. [3, 20, 21, 12, 13].

The advantage of the construction proposed here is two-fold. Firstly, it provides a general framework for measures on metric spaces. In particular, it is irrelevant whether the space is locally compact. Secondly, there is no need to introduce artificial compactification points. These points lead to annoying difficulties that blur the elegant mathematics underlying the theory. For example, any mappings h : S → S′ would have to be modified to cope with the compactification points.

In Section 2 we introduce the space M_{0} of measures on a complete separable metric space S. The measures in M_{0} assign finite mass to sets bounded away from s_{0}, i.e. s_{0} is not contained in the closure of the set, where s_{0} is a fixed element in S. Convergence in M_{0} is closely related to, but not identical to, weak convergence. We derive relevant fundamental results and characterize relative compactness in M_{0}. In Section 3 we consider regular variation for a sequence of measures and for a single measure in M_{0}. A number of statements that are equivalent to the definition of regular variation for a measure in M_{0} is presented. In Section 4 we give examples that illustrate the framework developed in Sections 2 and 3, and provide references to related work. In particular we derive results for R^{d}, for the space of continuous functions, and for the space of càdlàg functions. All the proofs are found in Section 5.

Regular variation conditions for probability measures on R^{d} appear frequently in the literature. Natural examples are the studies of the asymptotic behavior of partial sums of independent and identically distributed terms (the general central limit theorem, see e.g. [22], [19]), of componentwise partial maxima and point processes (extreme value theory, see e.g. [20]) and of solutions to random difference equations (see e.g. [16]). Regular variation also appears naturally in necessary and sufficient conditions in central limit theorems in Banach spaces, see e.g. [1] and [10] and the references therein. In the space C([0, 1]; R^{d}) of continuous functions it is used to characterize max-stable distributions and convergence in distribution of normalized partial maxima (see [11] and [12]). In the space D([0, 1]; R^{d}) of càdlàg functions the framework of regularly varying measures has been employed to study the extremal behavior of heavy-tailed stochastic processes and the tail behavior of functionals of their sample paths (see [13] and [15]).

2. Convergence of measures in the space M_{0}

Let (S, d) be a complete separable metric space. We write 𝒮 for the Borel σ-algebra on S and B_{x,r} = {y ∈ S : d(x, y) < r} for the open ball centered at x with radius r.

Let 𝒞_{b} denote the class of real-valued, bounded and continuous functions on S, and let M_{b} denote the class of finite Borel measures on 𝒮. A basic neighborhood of μ ∈ M_{b} is a set of the form {ν ∈ M_{b} : | ∫ f_{i} dν − ∫ f_{i} dμ| < ε, i = 1, …, k}, where ε > 0 and f_{i} ∈ 𝒞_{b} for i = 1, …, k. Thus, M_{b} is equipped with the weak topology. The convergence μ_{n} → μ in M_{b}, weak convergence, is equivalent to ∫ f dμ_{n} → ∫ f dμ for all f ∈ 𝒞_{b}. See e.g. Sections 2 and 6 in [5] for details. Fix an element s_{0} ∈ S, called the origin, and set S_{0} = S ∖ {s_{0}}. The subspace S_{0} is a metric space in the relative topology with σ-algebra 𝒮_{0} = {A : A ⊂ S_{0}, A ∈ 𝒮 }. Let 𝒞_{0} denote the real-valued bounded and continuous functions f on S_{0} such that for each f there exists r > 0 such that f vanishes on B_{0,r}; we use the notation B_{0,r} for the ball B_{s_{0},r}. Let M_{0} be the class of Borel measures on S_{0} whose restriction to S ∖ B_{0,r} is finite for each r > 0. A basic neighborhood of μ ∈ M_{0} is a set of the form {ν ∈ M_{0} : | ∫ f_{i} dν − ∫ f_{i} dμ| < ε, i = 1, …, k}, where ε > 0 and f_{i} ∈ 𝒞_{0} for i = 1, …, k. Similar to weak convergence, the convergence μ_{n} → μ in M_{0} is equivalent to ∫ f dμ_{n} → ∫ f dμ for all f ∈ 𝒞_{0}.

Theorem 2.1. μ_{n} → μ in M_{0} if and only if ∫ f dμ_{n} → ∫ f dμ for each f ∈ 𝒞_{0}.

The proof of this and all subsequent results are found in Section 5.

For μ ∈ M_{0} and r > 0, let μ^{(r)} denote the restriction of μ to S ∖ B_{0,r}. Then μ^{(r)} is finite and μ is uniquely determined by its restrictions μ^{(r)}, r > 0. Moreover, convergence in M_{0} has a natural characterization in terms of weak convergence of the restrictions to S ∖ B_{0,r}.

Theorem 2.2. (i) If μ_{n} → μ in M_{0}, then μ_{n}^{(r)} → μ^{(r)} in M_{b}(S ∖ B_{0,r}) for all but at most countably many r > 0. (ii) If there exists a sequence {r_{i}} with r_{i} ↓ 0 such that μ_{n}^{(r_{i})} → μ^{(r_{i})} in M_{b}(S∖B_{0,r_{i}}) for each i, then μ_{n} → μ in M_{0}.

Weak convergence is metrizable (for instance by the Prohorov metric, see e.g. p. 72 in [5]) and the close relation between weak convergence and convergence in M_{0} in Theorem 2.2 indicates that the topology in M_{0} is metrizable too. Theorem 2.3 shows that, with minor modifications of the arguments in [7, pp. 627–628], we may choose the metric

d_{M_{0}}(μ, ν) = ∫_{0}^{∞} e^{−r}p_{r}(μ^{(r)}, ν^{(r)})[1 + p_{r}(μ^{(r)}, ν^{(r)})]^{−1} dr, (2.1)

where μ^{(r)}, ν^{(r)} are the finite restriction of μ, ν to S ∖ B_{0,r}, and p_{r} is the Prohorov metric on M_{b}(S ∖ B_{0,r}).

Theorem 2.3. The metric d_{M_{0}} makes M_{0} a complete separable metric space.

Many useful applications of weak convergence rely on the Portmanteau theorem and the Mapping theorem. Next we derive the corresponding versions of these results for convergence in M_{0}. A more general version of the Portmanteau theorem below can be found in [2].

For A ⊂ S, let A^{∘} and A^{−} denote the interior and closure of A, and let ∂A = A^{−} ∖ A^{∘} be the boundary of A.

Theorem 2.4. (Portmanteau theorem) Let μ, μ_{n} ∈ M_{0}. The following statements are equivalent.

(i) μ_{n} → μ in M_{0} as n → ∞, (ii) lim_{n→∞} μ_{n}(A) = μ(A) for all A ∈ 𝒮 with μ(∂A) = 0 and 0 ∉ A^{−}, (iii) lim sup_{n→∞} μ_{n}(F) ⩽ μ(F) and lim inf_{n→∞} μ_{n}(G) ⩾ μ(G) for all closed

F ∈ 𝒮 and open G ∈ 𝒮 with s_{0} ∉ F and s_{0} ∉ G^{−}.

Remark 2.1. Consider statement (iii) above. In contrast to weak convergence of probability measures neither of the two statements implies the other. Take c ∈ (0, ∞) ∖ {1} and x ∈ S_{0}. Let μ_{n} = δ_{x} and μ = c δ_{x}, where δ_{x}(A) = 1 if x ∈ A and 0 otherwise. If c < 1, then the second statement in (iii) holds but not the first. If c > 1, then the first statement in (iii) holds but not the second. See also Remarks 3 and 4 in [2].

We conclude this section with a mapping theorem. Let (S, 𝒮) and (S′, 𝒮′) be complete separable metric spaces. We denote by s_{0} and s′_{0} the origin in S and S′, respectively. The open ball in S′ centered at s′_{0} with radius r is denoted by B_{0′,r}. For a measurable mapping h : (S, 𝒮) → (S′, 𝒮′), let D_{h} ⊂ S be the set of discontinuity points of h. Notice that D_{h} ∈ 𝒮, see e.g. p. 243 in [5].

Theorem 2.5. (Mapping theorem) Let h : (S, 𝒮) → (S′, 𝒮′) be a measurable mapping. If μ_{n} → μ in M_{0}(S), μ(D_{h} ∩ S_{0}) = 0, h(s_{0}) = s′_{0} and s_{0} ∉ D_{h}, then μ_{n}h^{−1} → μh^{−1} in M_{0′}(S′).

Consider the following statements for a measurable mapping h : S → S′:

(i) The mapping h is continuous at s_{0} and h(s_{0}) = s′_{0}.

(ii) For every A ∈ 𝒮′ with s′_{0} ∉ A^{−} it holds that s_{0} ∉ h^{−1}(A)^{−} in S. (iii) For every ε > 0 there exists δ > 0 such that B_{0,δ} ⊂ h^{−1}(B_{0′,ε}). By Lemma 2.1 below, they are all equivalent. Hence, we could have chosen to

formulate the mapping theorem with any one of them.

Lemma 2.1. The statements (i)–(iii) are equivalent.

2.1. Relative compactness in M_{0}. Since we are interested in convergence of measures in M_{0} it is essential to give an appropriate characterization of relative compactness. A subset of a topological space is said to be relatively compact if its closure is compact. A subset of a metric space is compact if and only if it is sequentially compact. Hence, M ⊂ M_{0} is relatively compact if and only if every sequence {μ_{n}} in M contains a convergent subsequence.

For μ ∈ M ⊂ M_{0} and r > 0, let μ^{(r)} be the restriction of μ to S ∖ B_{0,r} and M^{(r)} = {μ^{(r)} : μ ∈ M}. By Theorem 2.2 we have the following characterization of relative compactness.

Theorem 2.6. A subset M ⊂ M_{0} is relatively compact if and only if there exists a sequence {r_{i}} with r_{i} ↓ 0 such that M^{(r_{i})} is relatively compact in M_{b}(S ∖ B_{0,r_{i}}) for each i.

Relative compactness in the weak topology is characterized by Prohorov’s theorem. This translates to the following characterization of relative compactness in M_{0}.

Theorem 2.7. M ⊂ M_{0} is relatively compact if and only if there exists a sequence {r_{i}} with r_{i} ↓ 0 such that for each i

sup_{μ∈M} μ(S ∖ B_{0,r_{i}}) < ∞, (2.2)

and for each η > 0 there exists a compact set C_{i} ⊂ S ∖ B_{0,r_{i}} such that

sup_{μ∈M} μ(S ∖ (B_{0,r_{i}} ∪ C_{i})) ⩽ η. (2.3)

A convenient way to prove the convergence μ_{n} → μ in M_{0} is to show that{μ_{n}} is relatively compact and show convergence μ_{n}(A) → μ(A) for sets A in a determining class 𝒜. The next standard result (e.g. Theorem 2.3 in [5]) is useful for identifying a determining class of sets in 𝒮_{0}.

Theorem 2.8. Suppose that 𝒜 is a π-system of sets in 𝒮_{0} and, for each x ∈ S_{0} and ε > 0, there exists an A ∈ 𝒜 for which x ∈ A^{∘} ⊂ A ⊂ B_{x,ε}. If μ, ν ∈ M_{0} and μ = ν on 𝒜, then μ = ν on 𝒮_{0}.

3. Regularly varying sequences of measures

In the first part of this section we introduce the notion of regularly varying sequences of measures in M_{0}, only assuming the general setting presented in Section 2. Then we define regular variation for a single measure in M_{0}. In order to formulate this definition we need to assume further properties of the space S, essentially we assume that S has the structure of a cone. For a measure ν ∈ M_{0} we provide equivalent statements which are all equivalent to ν being regularly varying. These statements extend the corresponding equivalent definitions of regular variation for Borel measures on R^{d} to the general setting considered here.

Recall from e.g. [6] that a positive measurable function c defined on (0, ∞) is regularly varying with index ρ ∈ R if lim_{t→∞} c(λt)/c(t) = λ^{ρ} for all λ > 0. Similarly, a sequence {c_{n}}_{n⩾1} of positive numbers is regularly varying with index ρ ∈ R if lim_{n→∞} c_{[λn]}/c_{n} = λ^{ρ} for all λ > 0 (here [λn] denotes the integer part of λn).

Definition 3.1. A sequence {ν_{n}}_{n⩾1} in M_{0} is regularly varying with index −α < 0 if there exists a sequence {c_{n}}_{n⩾1} of positive numbers which is regularly varying with index α > 0, and a nonzero μ ∈ M_{0} such that c_{n}ν_{n} → μ in M_{0} as n → ∞.

The choice of terminology is motivated by the fact that {ν_{n}(A)}_{n⩾1} is a regularly varying sequence for each set A ∈ 𝒮 with 0 ∉ A^{−}, μ(∂A) = 0 and μ(A) > 0.

We will now define regular variation for a single measure in M_{0}. In order to formulate this definition we need to assume further properties of the space S. Suppose that there is an element 0 ∈ S and let s_{0} = 0 in the definitions of S_{0} and M_{0} in Section 2. Suppose that the space S is closed under multiplication by nonnegative real numbers λ ∈ R_{+} and that the map (λ, x) ↦ λx from R_{+} × S into S is continuous. In particular, we have 0x = 0 for all x ∈ S. Suppose further that the metric d on S satisfies d(0, λ_{1}x) < d(0, λ_{2}x) for all λ_{1}, λ_{2} ∈ R_{+} with λ_{1} < λ_{2} and all x ∈ S_{0}, i.e. the distance to the origin s_{0} = 0 for a point on the ray {λx : λ ∈ R_{+}} is strictly increasing in λ.

Definition 3.2. A measure ν ∈ M_{0} is regularly varying if the sequence {ν(n·)}_{n⩾1} in M_{0} is regularly varying.

In particular, a probability measure P on 𝒮 is regularly varying if the sequence {P (n·)}_{n⩾1} in M_{0} is regularly varying.

There are many possible equivalent ways to formulate regular variation for a measure ν ∈ M_{0}. Consider the following statements.

(i) There exist a nonzero μ ∈ M_{0} and a regularly varying sequence {c_{n}}_{n⩾1} of positive numbers such that c_{n}ν(n·) → μ(·) in M_{0} as n → ∞. (ii) There exist a nonzero μ ∈ M_{0} and a regularly varying function c such

that c(t)ν(t·) → μ(·) in M_{0} as t → ∞.

(iii) There exist a nonzero μ ∈ M_{0} and a set E ∈ 𝒮 with 0 ∉ E^{−} such that

ν(tE)^{−1}ν(t·) → μ(·) in M_{0} as t → ∞. (iv) There exists a nonzero μ ∈ M_{0} such that ν(t[S ∖ B_{0,1}])^{−1}ν(t·) → μ(·) in

M_{0} as t → ∞.

(v) There exist a nonzero μ ∈ M_{0} and a sequence {a_{n}}_{n⩾1} of positive numbers such that nν(a_{n}·) → μ(·) in M_{0} as n → ∞.

Theorem 3.1. (a) Each of the statements (i)–(v) above implies that μ(λA) = λ^{−α}μ(A) for some α > 0 and all A ∈ 𝒮_{0} and λ > 0. (b) The statements (i)–(v) are equivalent.

Several equivalent formulations of regular variation for measures on R^{d}, similar to those above, can be found in e.g. [3] and [21]. Theorem 3.1 extends some of them to measures on general metric spaces. One could also identify S with a product space and formulate regular variation in terms of polar coordinates. However, we have not pursued this approach here.

On R^{d} statements equivalent to regular variation for probability measures have appeared at numerous places in the vast literature on domains of attraction for sums and maxima. The notion of regular variation for measures on R^{d} first appeared in [18], where it was used for multivariate extensions of results in [8] on characterizations of domains of attractions. See Chapter 6 in [19] for a more recent account on this topic. The definition of regular variation for a measure on R^{d} in [18] differs from the one considered here and those in e.g. [21] in the sense that the limiting measure μ in the above statements (i)–(v) may be supported in a proper subspace of S.

4. Examples

In this section we provide some examples of metric spaces on which regularly varying measures are natural in applications. We consider the Euclidean space R^{d}, the space of continuous functions, and the space of càdlàg functions. We review some conditions to check relative compactness in M_{0} for measures on these spaces and provide conditions for determining if a given measure is regularly varying.

4.1. The Euclidean space R^{d}. A fundamental example of a metric space S is the Euclidean space R^{d} with the usual Euclidean norm | · |. The characterization of relative compactness in M_{0} simplifies considerably if M_{0} = M_{0}(R^{d}). Since the unit ball is relatively compact in R^{d}, Theorem 2.7 implies that M ⊂ M_{0}(R^{d}) is relatively compact if and only if sup_{μ∈M} μ(R^{d} ∖ B_{0,r}) < ∞ for each r > 0, and lim_{R→∞} sup_{μ∈M} μ(R^{d} ∖ B_{0,R}) = 0.

Regular variation for measures on R^{d} is often proved by showing convergence to a measure μ ∈ M_{0}(R^{d}) for an appropriate convergence determining class of subsets of R^{d}. If V_{u,S} = {x ∈ R^{d} : |x| > u, x/|x| ∈ S} for u > 0 and Borel sets S ⊂ {x ∈ R^{d} : |x| = 1}, then the collection of such sets satisfying μ(∂V_{u,S}) = 0 form a convergence determining class (see [3]).

Theorem 4.1. Let ν, μ ∈ M_{0}(R^{d}) be nonzero and let {c_{n}} be a regularly varying sequence with index α > 0. If c_{n}ν(nV_{u,S}) → μ(V_{u,S}) as n → ∞ for each u > 0 and Borel set S ⊂ {x ∈ R^{d} : |x| = 1} with μ(∂V_{u,S}) = 0, then c_{n}ν(n·) → μ(·) in M_{0} as n → ∞ and ν is regularly varying.

The sets of the form A_{x} = [0, ∞)^{d}∖{[0, x_{1}]×⋯×[0, x_{d}]}, for x = (x_{1}, …, x_{d}) ∈ [0, ∞)^{d}, form a convergence determining class for regular variation for measures on [0, ∞)^{d} ∖ {0}. This is well known, see e.g. [20].

Theorem 4.2. Let ν, μ ∈ M_{0}([0, ∞)^{d}) be nonzero and let {c_{n}} be a regularly varying sequence with index α > 0. If c_{n}ν(nA_{x}) → μ(A_{x}) as n → ∞ for each x ∈ [0, ∞)^{d} ∖{0}, then c_{n}ν(n·) → μ(·) in M_{0} as n → ∞ and ν is regularly varying.

To show regular variation in function spaces is typically less straight-forward. Similar to weak convergence of probability measures on these spaces, convergence is typically shown by showing relative compactness and convergence for finite dimensional projections of the original measures. In the following two sections we will exemplify applications of the framework set up in Sections 2 and 3 by considering regular variation for measures on the space C([0, 1]; R^{d}) of continuous functions with the uniform topology and the space D([0, 1]; R^{d}) of càdlàg functions with the Skorohod J_{1}-topology.

4.2. The space C([0, 1]; R^{d}). Let S be the space C = C([0, 1]; R^{d}) of continuous functions [0, 1] → R^{d} with the uniform topology given by the supremum norm | · |_{∞}. Tightness conditions for weak convergence on C are well known [5, p. 82] and translate naturally to conditions for relative compactness in M_{0}(C). For x : [0, 1] → R^{d} the modulus of continuity is given by w_{x}(δ) = sup_{|s−t|⩽δ} |x(s)−x(t)|.

Theorem 4.3. A set M ⊂ M_{0}(C) is relatively compact if and only if for each r > 0 and each ε > 0

(4.1)  sup_{μ∈M} μ(S ∖ B_{0,r}) < ∞,
(4.2)  lim_{R→∞} sup_{μ∈M} μ(x : |x(0)| > R) = 0,
(4.3)  lim_{δ→0} sup_{μ∈M} μ(x : w_{x}(δ) ⩾ ε) = 0.

To prove that a measure ν ∈ M_{0}(C) is regularly varying we typically need to show that for some regularly varying sequence {c_{n}} of positive numbers, (i) {c_{n}ν(n·) : n ⩾ 1} is relatively compact, and (ii) any two subsequential limits of {c_{n}ν(n·)} coincide. The point (ii) holds, similar to the case for weak convergence, if the subsequential limits have the same finite dimensional projections. For (t_{1}, …, t_{k}) ∈ [0, 1]^{k} denote by π_{t_{1},...,t_{k}} the map C ∋ x ↦ (x(t_{1}), …, x(t_{k})) ∈ R^{dk}. The finite dimensional projections of ν ∈ M_{0}(C) are measures of the form νπ_{t_{1},...,t_{k}}^{−1}.

Theorem 4.4. Let ν, μ ∈ M_{0}(C) be nonzero and let {c_{n}} be a regularly varying sequence of positive numbers. Then c_{n}ν(n·) → μ(·) in M_{0}(C) as n → ∞ if and only if for each integer k ⩾ 1 and (t_{1}, …, t_{k}) ∈ [0, 1]^{k}

c_{n}νπ_{t_{1},...,t_{k}}^{−1}(n·) → μπ_{t_{1},...,t_{k}}^{−1}(·) (4.4) in M_{0}(R^{dk}) as n → ∞, and for each r > 0 and each ε > 0

(4.5)  sup_{n} c_{n}ν(n[S ∖ B_{0,r}]) < ∞,
(4.6)  lim_{δ→0} sup_{n} c_{n}ν(x : w_{x}(δ) ⩾ nε) = 0.

Recall that the regular variation statement c_{n}ν(n·) → μ(·) can be replaced by any of the equivalent statements in Theorem 3.1. For those statements there are corresponding versions of the conditions in Theorem 4.4. Regular variation on C can be used for studying extremal properties of stochastic processes. The mapping theorem can be used to determine the tail behavior of functionals of a heavy-tailed stochastic process with continuous sample paths. Another application is to characterize max-stable distributions in C. For f_{1}, …, f_{n} ∈ C let ⋁_{i=1}^{n} f_{i} be the element in C given by (⋁_{i=1}^{n} f_{i})(t) = max_{i=1,...,n} f_{i}(t). A random variable X with values in C is said to be max-stable if, for each integer n ⩾ 1 there are functions a_{n}(t) > 0 and b_{n}(t) such that a_{n}^{−1}(⋁_{i=1}^{n} X_{i} − b_{n}) =^{d} X, where =^{d} denotes equality in distribution and X_{1}, …, X_{n} are independent and identically distributed copies of X. The distribution of X is called simple max-stable if one can choose a_{n}(t) = n and b_{n}(t) = 0 for all t. Max-stable distributions appear as limiting distributions of pointwise maxima of independent and identically distributed stochastic processes. Their domain of attraction can be characterized in terms of regular variation. If Y_{1}, Y_{2}, … are independent and identically distributed with values in C, then n^{−1} ⋁_{i=1}^{n} Y_{i} →^{d} X in C for some X (where X necessarily has a simple maxstable distribution) if and only if the distribution of Y on C is regularly varying and satisfies statement (v) above (before Theorem 3.1) with a_{n} = n. The same characterization holds for random variables taking values in the space D studied below. See [12, Theorem 2.4] and [11] for more details. Theorem 4.4 provides necessary and sufficient conditions for a Borel (probability) measure on C to be regularly varying.

4.3. The space D([0, 1]; R^{d}). Let S be the space D = D([0, 1]; R^{d}) of càdlàg functions equipped with the Skorohod J_{1}-topology. We refer to [5] for details on this space and the J_{1}-topology. In particular, elements in D are assumed to be left-continuous at 1. Notice that if d is the J_{1}-metric then d(x, 0) = |x|_{∞}. Notice also that D is not complete under d but there exists an equivalent metric under which D is complete (see Section 12 in [5]). Notice also that D is not a topological vector space since addition in D is in general not continuous (see e.g. [23]). For T ⊂ [0, 1] and δ > 0, let

w_{x}(T) = sup_{t_{1},t_{2}∈T} |x(t_{2}) − x(t_{1})|,
w′′_{x}(δ) = sup_{t,t_{1},t_{2}}{|x(t) − x(t_{1})| ∧ |x(t_{2}) − x(t)|},

where the supremum in the definition of w′′_{x}(δ) is over all (t, t_{1}, t_{2}) satisfying 0 ⩽ t_{1} ⩽ t ⩽ t_{2} ⩽ 1 and t_{2} − t_{1} ⩽ δ.

Theorem 4.5. A set M ⊂ M_{0}(D) is relatively compact if and only if for each r > 0 and ε > 0

(4.7)  sup_{μ∈M} μ(S ∖ B_{0,r}) < ∞,
(4.8)  lim_{R→∞} sup_{μ∈M} μ(x : |x|_{∞} > R) = 0,
(4.9)  lim_{δ→0} sup_{μ∈M} μ(x : w′′_{x}(δ) ⩾ ε) = 0,
(4.10)  lim_{δ→0} sup_{μ∈M} μ(x : w_{x}([0, δ)) ⩾ ε) = 0,
(4.11)  lim_{δ→0} sup_{μ∈M} μ(x : w_{x}([1 − δ, 1)) ⩾ ε) = 0.

Similar to the space C, regular variation for measures on D is typically proved by showing relative compactness in M_{0}(D) and convergence of finite dimensional projections (see [13, Theorem 10]).

Theorem 4.6. Let ν, μ ∈ M_{0}(D) be nonzero and let {c_{n}} be a regularly varying sequence of positive numbers. Then c_{n}ν(n·) → μ(·) in M_{0}(D) as n → ∞ if and only if there exists T ⊂ [0, 1] containing 0, 1 and all but at most countably many points of [0, 1] such that

c_{n}νπ_{t_{1},...,t_{k}}^{−1}(n·) → μπ_{t_{1},...,t_{k}}^{−1}(·) (4.12)

in M_{0}(R^{dk}) as n → ∞ whenever t_{1}, …, t_{k} ∈ T, and for each ε > 0

(4.13)  lim_{δ→0} lim sup_{n} c_{n}ν(w′′_{x}(δ) ⩾ nε) = 0
(4.14)  lim_{δ→0} lim sup_{n} c_{n}ν(w_{x}([0, δ)) ⩾ nε) = 0
(4.15)  lim_{δ→0} lim sup_{n} c_{n}ν(w_{x}([1 − δ, 1)) ⩾ nε) = 0.

Notice that the regular variation statement c_{n}ν(n·) → μ(·) can be replaced by any of the equivalent statements in Theorem 3.1. Notice also that the set T in Theorem 4.6 appears because for t ∈ (0, 1) the map π_{t}(x) = x(t) is continuous at x ∈ D if and only if x is continuous at t. Regular variation on D can be used for studying extremal properties of stochastic processes, see [13] and [15]. In particular, the mapping theorem can be used to determine the tail behavior of functionals of a heavy-tailed stochastic process with càdlàg sample paths. Regular variation is also closely connected to max-stable distributions on D, see [12].

Regularly varying sequences of measures in M_{0}(D) appear for instance when studying large deviations. Consider for instance the stochastic process X_{t}^{(n)} = Z_{1} + ⋯ + Z_{[nt]}, t ∈ [0, 1], where {Z_{k}} is a sequence of independent and identically distributed R^{d}-valued random variables whose common probability distribution P (Z_{1} ∈ ·) is regularly varying on R^{d} with α > 1. Then the sequence ν_{n}(·) = P (n^{−1}X^{(n)} ∈ ·) is regularly varying according to Definition 3.1 with c_{n} = [nP (|Z_{1}| > n)]^{−1}, see [14, Theorem 2.1].

5. Proofs

Proof of Theorem 2.1. Suppose μ_{n} → μ in M_{0} and take f ∈ 𝒞_{0}. Given ε > 0 consider the neighborhood N_{ε,f}(μ) = {ν : | ∫ f dν − ∫ f dμ| < ε}. By assumption there exists n_{0} such that n ⩾ n_{0} implies μ_{n} ∈ N_{ε,f}(μ), i.e. | ∫ f dμ_{n} − ∫ f dμ| < ε. Hence ∫ f dμ_{n} → ∫ f dμ.

Conversely, suppose that ∫ f dμ_{n} → ∫ f dμ for each f ∈ 𝒞_{0}. Take ε > 0 and a neighborhood N_{ε,f_{1},...,f_{k}}(μ) = {ν : | ∫ f_{i} dν − ∫ f_{i} dμ| < ε, i = 1, …, k}. Let n_{i} be an integer such that n ⩾ n_{i} implies | ∫ f_{i} dμ_{n} − ∫ f_{i} dμ| < ε. Hence, n ⩾ max(n_{1}, …, n_{k}) implies μ_{n} ∈ N_{ε,f_{1},...,f_{k}}(μ). It follows that μ_{n} → μ in M_{0}. □

Lemma 5.1. Let μ ∈ M_{0}, let r > 0 be such that μ(∂B_{0,r}) = 0 and let f ∈ 𝒞_{b}(S ∖ B_{0,r}) be nonnegative. For each ε > 0 there exist nonnegative f_{1}, f_{2} ∈ 𝒞_{0} such that f_{1} ⩽ f ⩽ f_{2} on S ∖ B_{0,r} and | ∫_{S} f_{2} dμ − ∫_{S} f_{1} dμ| ⩽ ε.

Proof. For any r′ > r let f_{1,r′} be a function on S_{0} given by

f_{1,r′} = \{ \begin{matrix}g_{r,r′}f & on S ∖ B_{0,r}, \\ 0 & on B_{0,r} ∖ \{0\}, \end{matrix}

where g_{r,r′}(x) = max{min{d(x, s_{0}), r′} − r, 0}/(r′ − r) for x ∈ S_{0}. Then f_{1,r′} is continuous, f_{1,r′} ⩽ f on S∖B_{0,r} and f_{1,r′}(x) ↑ f(x) pointwise on S∖B_{0,r}^{−} as r′ ↓ r. We now consider an upper bound. By the Tietze extension theorem (Theorem 3.6.3 in [9]) there exists a nonnegative, bounded continuous extension F of f to S_{0} such that F = f on S ∖ B_{0,r} and sup |F | = sup |f|. For r′′ < r let f_{2,r′′} be a function on S_{0} given by f_{2,r′′} = g_{r′′,r}F on S ∖ B_{0,r} and 0 otherwise. Then f_{2,r′′} is continuous, f_{2,r′′} ⩾ f on S ∖ B_{0,r} and f_{2,r′′}(x) ↓ f(x) pointwise on S ∖ B_{0,r} as r′′ ↑ r. In particular,

| ∫_{S} f_{2,r′′} dμ − ∫_{S} f_{1,r′} dμ| ⩽ sup |f| μ(B_{0,r′} ∖ B_{0,r′′}) → sup |f| μ(∂B_{0,r}) = 0

as r′ ↓ r and r′′ ↑ r. Hence, for r′ and r′′ sufficiently close to r we may take f_{1} = f_{1,r′} and f_{2} = f_{2,r′′}. □

Proof of Theorem 2.2. (i) Let R_{μ} = {r ∈ (0, ∞) : μ(∂B_{0,r}) = 0} and notice that (0, ∞) ∖ R_{μ} is at most countable. Take r ∈ R_{μ} and, without loss of generality, a nonnegative f ∈ 𝒞_{b}(S ∖ B_{0,r}). Given ε > 0 there exist, by Lemma 5.1, nonnegative f_{1}, f_{2} ∈ 𝒞_{0}(S) with f_{1} ⩽ f ⩽ f_{2} on S ∖ B_{0,r} such that | ∫ f_{2} dμ − ∫ f_{1} dμ| ⩽ ε. Hence, ∫ f_{1} dμ_{n} ⩽ ∫ f dμ_{n} ⩽ ∫ f_{2} dμ_{n} and by Theorem 2.1 μ_{n} → μ in M_{0} implies that

∫ f_{1} dμ ⩽ lim inf_{n} ∫ f dμ_{n} ⩽ lim sup_{n} ∫ f dμ_{n} ⩽ ∫ f_{2} dμ.

Since ε > 0 was arbitrary it follows that ∫ f dμ_{n} → ∫ f dμ.

(ii) Take f ∈ 𝒞_{0}(S); without loss of generality f can be chosen nonnegative. The support of f is contained in S ∖ B_{0,r_{i}} for some r_{i} > 0 such that μ_{n}^{(r_{i})} → μ^{(r_{i})} in M_{b}(S ∖ B_{0,r_{i}}). Hence f ∈ 𝒞_{b}(S ∖ B_{0,r_{i}}) and ∫ f dμ_{n} = ∫ f dμ_{n}^{(r_{i})} → ∫ f dμ^{(r_{i})} = ∫ f dμ. □

Proof of Theorem 2.3. The proof consists of minor modifications of arguments that can be found in [7, pp. 628–630]. Here we change from r to 1/r. For the sake of completeness we have included a full proof.

We show that (i) μ_{n} → μ in M_{0} if and only if d_{M_{0}}(μ_{n}, μ) → 0, and (ii) (M_{0}, d_{M_{0}}) is complete and separable.

(i) Suppose that d_{M_{0}}(μ_{n}, μ) → 0. The integral expression in (2.1) can be written d_{M_{0}}(μ_{n}, μ) = ∫_{0}^{∞} e^{−r}g_{n}(r) dr, so that for each n, g_{n}(r) decreases with r and is bounded by 1. Helly’s selection theorem (p. 336 in [4]), applied to 1 − g_{n}, implies that there exists a subsequence {n′} and a non-increasing function g such that g_{n′}(r) → g(r) for all continuity points of g. By dominated convergence, ∫_{0}^{∞} e^{−r}g(r) dr = 0 and since g is monotone this implies that g(r) = 0 for all finite r > 0. Since this holds for all convergent subsequences {g_{n′}(r)}, it follows that g_{n}(r) → 0 for all continuity points r of g, and hence, for such r, p_{r}(μ_{n}^{(r)}, μ^{(r)}) → 0 as n → ∞. By Theorem 2.2, μ_{n} → μ in M_{0}.

Suppose that μ_{n} → μ in M_{0}. Then theorem 2.2 implies that μ_{n}^{(r)} → μ^{(r)} in M_{b}(S ∖ B_{0,r}) for all but at most countably many r > 0. Hence, for such r, p_{r}(μ_{n}^{(r)}, μ^{(r)})[1 + p_{r}(μ_{n}^{(r)}, μ^{(r)})]^{−1} → 0, which by the dominated convergence theorem implies that d_{M_{0}}(μ_{n}, μ) → 0.

(ii) Separability: For r > 0 let D_{r} be a countable dense set in M_{b}(S ∖ B_{0,r}) with the weak topology. Let D be the union of D_{r} for rational r > 0. Then D is countable. Let us show D is dense in M_{0}. Given ε > 0 and μ ∈ M_{0} pick r′ > 0 such that ∫_{0}^{r′} e^{−r} dr < ε/2. Take μ_{r′} ∈ D_{r′} such that p_{r′}(μ_{r′}, μ^{(r′)}) < ε/2. Then p_{r}(μ_{r′}^{(r)}, μ^{(r)}) < ε/2 for all r > r′. In particular, d_{M_{0}}(μ_{r′}, μ) < ε.

Completeness: Let {μ_{n}} be a Cauchy sequence for d_{M_{0}}. Then {μ_{n}^{(r)}} is a Cauchy sequence for p_{r} for all but at most countably many r > 0 and by completeness of M_{b}(S ∖ B_{0,r}) it has a limit μ_{r}. These limits are consistent in the sense that μ_{r′}^{(r)} = μ_{r} for r′ < r. On 𝒮_{0} put μ(A) = lim_{r→0} μ_{r}(A ∩ S ∖ B_{0,r}). Then μ is a measure. Clearly, μ ⩾ 0 and μ(∅) = 0. Moreover, μ is countably additive: for disjoint A_{n} ∈ 𝒮_{0} the monotone convergence theorem implies that

μ( ⋃_{n} A_{n}) = lim_{r→0} μ_{r}( ⋃_{n} A_{n} ∩ [S ∖ B_{0,r}])
= lim_{r→0} ∑_{n} μ_{r}(A_{n} ∩ [S ∖ B_{0,r}]) = ∑_{n} μ(A_{n}).  □

Proof of Theorem 2.4. We show that (i) ⇔ (ii) and (ii) ⇔ (iii).

Suppose that (i) holds and take A ∈ 𝒮 with s_{0} ∉ A^{−} and μ(∂A) = 0. Since s_{0} ∉ A^{−} there exists r > 0 with μ(∂B_{0,r}) = 0 such that A ⊂ S ∖ B_{0,r}. By Theorem 2.2, μ_{n}^{(r)} → μ^{(r)} in M_{b}(S ∖ B_{0,r}). The Portmanteau theorem for weak convergence implies (ii).

Suppose that (ii) holds. The Portmanteau theorem for weak convergence implies that μ_{n}^{(r)} → μ^{(r)} in M_{b}(S ∖ B_{0,r}) for all r > 0 for which μ(∂B_{0,r}) = 0. Since μ(∂B_{0,r}) = 0 for all but at most countably many r > 0, Theorem 2.2 implies μ_{n} → μ in M_{0}.

Suppose that (iii) holds and take A ∈ 𝒮 with s_{0} ∉ A^{−} and μ(∂A) = 0. Then,

lim sup_{n→∞} μ_{n}(A) ⩽ lim sup_{n→∞} μ_{n}(A^{−}) ⩽ μ(A^{−})

= μ(A^{∘}) ⩽ lim inf_{n→∞} μ_{n}(A^{∘}) ⩽ lim inf_{n→∞} μ_{n}(A).

Hence, lim_{n→∞} μ_{n}(A) = μ(A), so that (ii) holds.

Suppose that (ii) holds and take a closed F ∈ 𝒮 with s_{0} ∉ F. Notice that for F_{ε} = {s : d(s, F) ⩽ ε} it holds that for small ε, say ε ∈ (0, c), ∂F_{ε} = {s : d(s, F) = ε}, s_{0} ∉ F_{ε} and μ(∂F_{ε}) = 0 for all but at most countably many ε ∈ (0, c). Hence, lim sup_{n→∞} μ_{n}(F) ⩽ μ(F_{ε}). Since F_{ε} ↓ F as ε ↓ 0 and since F is closed, μ(F_{ε}) ↓ μ(F) as ε ↓ 0. Hence, lim sup_{n→∞} μ_{n}(F) ⩽ μ(F). Now take an open G ∈ 𝒮 with s_{0} ∉ G^{−} and an r > 0 such that μ(∂B_{0,r}) = 0 and G ⊂ S ∖ B_{0,r}. Set A = S ∖ B_{0,r}^{−} and so that F = A ∖ G is closed. Then

lim inf_{n→∞} μ_{n}(G) = lim inf_{n→∞}(μ_{n}(A) − μ_{n}(F)) ⩾ μ(A) − μ(F) = μ(G).

Hence, (ii) holds. □

Proof of Lemma 2.1. (ii) ⇔ (iii): Notice that s′_{0} ∉ A^{−} if and only if there exists ε > 0 such that A ⊂ S′ ∖ B_{0′,ε}, and that s_{0} ∉ h^{−1}(A)^{−} if and only if there exists δ > 0 such that h^{−1}(A) ⊂ S ∖ B_{0,δ}. Hence, (ii) holds if and only if for every ε > 0 there exists δ > 0 such that h^{−1}(S′ ∖ B_{0′,ε}) ⊂ S ∖ B_{0,δ}. Taking complements shows that h^{−1}(S′ ∖ B_{0′,ε}) ⊂ S ∖ B_{0,δ} if and only if B_{0,δ} ⊂ h^{−1}(B_{0′,ε}).

(iii) ⇒ (i):

(iii) implies that h(B_{0,δ}) ⊂ h(h^{−1}(B_{0′,ε})). Since h(h^{−1}(B_{0′,ε})) ⊂ B_{0′,ε} holds for any h, it follows that (i) holds.

(i) ⇒ (iii): (i) implies that for every ε > 0 there exists δ > 0 such that h(B_{0,δ}) ⊂ B_{0′,ε}, which implies that h^{−1}(h(B_{0,δ})) ⊂ h^{−1}(B_{0′,ε}). Since B_{0,δ} ⊂ h^{−1}(h(B_{0,δ})) holds for any h, it follows that (iii) holds. □

Proof of Theorem 2.5. Take A ∈ 𝒮′ with s′_{0} ∉ A^{−} and μh^{−1}(∂A) = 0. Since ∂h^{−1}(A) ⊂ h^{−1}(∂A) ∪ D_{h} (see e.g. (A2.3.2) in [7]), we have μ(∂h^{−1}(A)) ⩽ μh^{−1}(∂A) + μ(D_{h}) = 0. Since μ_{n} → μ in M_{0}(S), μ(∂h^{−1}(A)) = 0 and, by Lemma 2.1, s_{0} ∉ h^{−1}(A)^{−}, it follows by Theorem 2.4 (ii) that μ_{n}h^{−1}(A) → μh^{−1}(A). Hence, μ_{n}h^{−1} → μh^{−1} in M_{0}(S′). □

Proof of Theorem 2.6. Suppose M ⊂ M_{0}(S) is relatively compact. Let {μ_{n}} be a subsequence in M. Then there exists a convergent subsequence μ_{n_{k}} → μ for some μ ∈ M^{−}. By Theorem 2.2, there exists a sequence {r_{i}} with r_{i} ↓ 0 such that μ_{n_{k}}^{(r_{i})} → μ^{(r_{i})} in M_{b}(S ∖ B_{0,r_{i}}). Hence, M^{(r_{i})} is relatively compact in M_{b}(S ∖ B_{0,r_{i}}) for each such r_{i}.

Conversely, suppose there exists a sequence {r_{i}} with r_{i} ↓ 0 such that M^{(r_{i})} ⊂ M_{b}(S∖B_{0,r_{i}}) is relatively compact for each i, and let {μ_{n}} be a sequence of elements in M. We use a diagonal argument to find a convergent subsequence. Since M^{(r_{1})} is relatively compact there exists a subsequence {μ_{n_{1}(k)}} of {μ_{n}} such that μ_{n_{1}(k)}^{(r_{1})} converges to some μ_{r_{1}} in M_{b}(S∖B_{0,r_{1}}). Similarly since M^{(r_{2})} is relatively compact and {μ_{n_{1}(k)}} ⊂ M there exists a subsequence {μ_{n_{2}(k)}} of {μ_{n_{1}(k)}} such that μ_{n_{2}(k)}^{(r_{2})} converges to some μ_{r_{2}} in M_{b}(S∖B_{0,r_{2}}). Continuing like this; for each i ⩾ 3 let n_{i}(k) be a subsequence of n_{i−1}(k) such that μ_{n_{i}(k)}^{(r_{i})} converges to some μ_{r_{i}} in M_{b}(S∖B_{0,r_{i}}).

Then the diagonal sequence {μ_{n_{k}(k)}} satisfies μ_{n_{k}(k)}^{(r_{i})} → μ_{r_{i}} in M_{b}(S ∖ B_{0,r_{i}}) for each i ⩾ 1. Take f ∈ 𝒞_{0}(S). There exists some i_{0} ⩾ 1 such thatf vanishes on B_{0,r_{i}} for each i ⩾ i_{0}. In particular f ∈ 𝒞_{b}(S ∖ B_{0,r_{i}}) for each i ⩾ i_{0} and

∫ f dμ_{r_{i}} = lim_{k} ∫ f dμ_{n_{k}(k)}^{(r_{i})} = lim_{k} ∫ f dμ_{n_{k}(k)}^{(r_{i_{0}} )} = ∫ f dμ_{r_{i_{0}}} .

Hence, we can define μ′ : 𝒞_{0}(S) → [0, ∞] by μ′(f) = lim_{i→∞} ∫ f dμ_{r_{i}}. This μ′ induces a measure μ in M_{0}. Indeed, for A ∈ 𝒮_{0} we can find a sequence f_{n} ∈ 𝒞_{0}(S) such that 0 ⩽ f_{n} ↑ I_{A} and put μ(A) = lim_{n} μ′(f_{n}). If A ∩ B_{0,r} = ∅ for some r > 0, then exists f_{n} ∈ 𝒞_{0}(S) such that f_{n} ↓ I_{A} and hence μ(A) ⩽ μ′(f_{n}) < ∞. Thus, μ is finite on sets A with s_{0} ∉ A^{−}. To show μ is countably additive, let A_{1}, A_{2}, … be disjoint sets in 𝒮_{0} and 0 ⩽ f_{nk} ↑ I_{A_{k}} for each k. Then ∑_{k} f_{nk} ↑ I_{∪_{k}A_{k}} and we have by Fubini’s theorem and the monotone convergence theorem that

μ(∪_{k}A_{k}) = lim_{n} μ′( ∑_{k} f_{nk}) = ∑_{k} lim_{n} μ′(f_{nk}) = ∑_{k} μ(A_{k}).

By construction ∫ f dμ = μ′(f) for each f ∈ 𝒞_{0}(S). Hence, ∫ f dμ_{n_{k}(k)} → ∫ f dμ for each f ∈ 𝒞_{0}(S) and we conclude that M is relatively compact in M_{0}. □

Proof of Theorem 2.7. Suppose M ⊂ M_{0} is relatively compact. By Theorem 2.6, there exists a sequence {r_{i}} with r_{i} ↓ 0 such that M^{(r_{i})} ⊂ M_{b}(S∖B_{0,r_{i}}) is relatively compact for each r_{i}. Prohorov’s theorem (Theorem A2.4.1 in [7]) implies that (2.2) and (2.3) hold.

Conversely, suppose there exists a sequence {r_{i}} with r_{i} ↓ 0 such that (2.2) and (2.3) hold. Then, by Prohorov’s theorem, M^{(r_{i})} ⊂ M_{b}(S ∖ B_{0,r_{i}}) is relatively compact for each i. By Theorem 2.6, M ⊂ M_{0} is relatively compact. □

Proof of Theorem 2.8. Take an open set G ∈ 𝒮_{0}. For each x ∈ G there exists a set A_{x} ∈ 𝒜 such that x ∈ A_{x}^{∘} ⊂ A_{x} ⊂ G. Since S is separable, there exists a countable subcollection {A_{x_{i}}^{∘}} of {A_{x}^{∘}, x ∈ G} that covers G. Hence G ⊂ ∪_{i}A_{x_{i}}^{∘}. Since A_{x} ⊂ G we also have ∪_{i}A_{x_{i}} ⊂ G. Hence G = ∪_{i}A_{x_{i}}. It follows that the πsystem 𝒜 generates the open sets and hence the Borel σ-algebra. By the uniqueness theorem μ = ν on 𝒮_{0}. □

For the proof of Theorem 3.1 the following simple observation will be helpful.

Lemma 5.2. Let μ ∈ M_{0}. Let E_{1} = S ∖ B_{0,1}, rE_{1} = {rx : x ∈ E_{1}} for r > 0, and R_{μ} = {r > 0 : μ(∂rE_{1}) = 0}. Then (0, ∞) ∖ R_{μ} is at most countable and for some r_{0} ∈ R_{μ} it holds that μ(rE_{1}) > 0 for r ∈ (0, r_{0}).

Proof. Since μ ∈ M_{0} it holds that μ(rE_{1}) < ∞ for all r > 0. In addition ∂[rE_{1}] ∩ ∂[r′E_{1}] = ∅ for r ≠ r′. Hence, μ(∂[rE_{1}]) = 0 for all but at most countably many r > 0. Since μ is nonzero μ(rE_{1}) > 0 for r ∈ (0, r_{0}) for some r_{0} > 0 and we may choose r_{0} ∈ R_{μ}. □

Proof of Theorem 3.1(a). We show this statement under the assumption that (iii) holds. This is sufficient since in the proof of statement (b), below, it is shown that the limiting measures are the same up to a constant factor.

Suppose that (iii) holds and set E_{1} = S ∖ B_{0,1}. By Lemma 5.2, μ(∂[rE_{1}]) = 0 for all r ∈ R_{μ} ⊂ (0, ∞), where (0, ∞) ∖ R_{μ} is at most countable. Moreover, μ(rE_{1}) > 0 for r ∈ (0, r_{0}) for some r_{0} ∈ R_{μ}. Hence, for λ in a set of positive measure, e.g. (1/2, 1),

\frac{ν(tλ[r_{0}E_{1}])}{ν(t[r_{0}E_{1}])} = \frac{ν(tB)}{ν(t[r_{0}E_{1}])} \frac{ν(tλ[r_{0}E_{1}])}{ν(tB)} → \frac{μ(λ[r_{0}E_{1}])}{μ(r_{0}E_{1})} ∈ (0, ∞)

as t → ∞. Hence, by Theorem 1.4.1 in [6], t ↦ ν(t[r_{0}E_{1}]) is regularly varying and

lim_{t→∞} \frac{ν(tλE_{1})}{ν(tE_{1})} = lim_{t→∞} \frac{ν(tλ[r_{0}E_{1}])}{ν(t[r_{0}E_{1}])} = λ^{−α}

for some α > 0. In particular, μ(∂[λE_{1}]) = 0 for all λ > 0 and ν(tE_{1})^{−1}ν(t·) → μ(E_{1})^{−1}μ(·) in M_{0} as t → ∞. Moreover, if A ∈ 𝒮 with 0 ∉ A^{−} and μ(∂A) = 0, then for any λ > 0,

\frac{ν(tλA)}{ν(tE_{1})} = \frac{ν(tλA)}{ν(tλE_{1})} \frac{ν(tλE_{1})}{ν(tE_{1})} → λ^{−α} \frac{μ(A)}{μ(E_{1})}

as t → ∞. Hence, μ(λA) = λ^{−α}μ(A) for such set A and all λ > 0. Since these sets A form a π-system that generate the σ-algebra 𝒮_{0}, μ(λA) = λ^{−α}μ(A) for all A ∈ 𝒮_{0} and λ > 0. □

In the proof of Theorem 3.1(b) we will use a particular determining class. For A ⊂ S, let S(A) = {sx : s ⩾ 1, x ∈ A} and let 𝒜̃= {A ∈ 𝒮 : A = S(A), 0 ∉ A^{−}}. Given a measure μ ∈ M_{0} we write 𝒜̃_{μ} for the class of sets Ã ∈ 𝒜̃with μ(∂Ã) = 0.

Lemma 5.3. Take A ⊂ S with 0 ∉ A^{−}. If A is open (closed), then S(A) is open (closed).

Proof. Assume that A is open. Since (λ, x) ↦ λx is continuous by assumption, the map f_{λ} given by f_{λ}(x) = λx is continuous. Moreover, S(A) = ∪_{λ⩾1}λA = ∪_{λ⩾1}f_{1/λ}^{−1}(A). Hence, S(A) is open.

Assume that A is closed. Take y_{n} ∈ S(A) with y_{n} → y for some y ∈ S. Write y_{n} = s_{n}x_{n}, where s_{n} ⩾ 1 and x_{n} ∈ A. If s_{n} → ∞, then x_{n} ∉ A for n sufficiently large (recall that 0 ∉ A). Hence, {s_{n}} has an accumulation point s ∈ [1, ∞) so that s_{n′} → s for some subsequence {s_{n′}}. Hence, x_{n′} → x for some x ∈ A and it follows that y_{n′} → y′ for some y′ ∈ S(A). Since y_{n} → y we must have y = y′. □

Lemma 5.4. Let μ, ν ∈ M_{0}. If μ(Ã) = ν(Ã) for each Ã ∈ 𝒜̃_{μ}, then μ = ν.

Proof. If μ and ν coincide on 𝒜̃_{μ} then they coincide on the π-system of finite differences of sets in 𝒜̃_{μ}. By Theorem 2.8 it is sufficient to show that for each x ∈ S_{0} and ε > 0 there exists a set D of the form D = Ã_{1} ∖ Ã_{2} with Ã_{1}, Ã_{2} ∈ 𝒜̃_{μ} such that x ∈ D^{∘} ⊂ D ⊂ B_{x,ε}. Take δ ∈ (0, ε] such that 0 ∉ B_{x,δ}^{−}, μ(∂B_{x,δ}) = 0, and μ(∂S(B_{x,δ})) = 0. Then S(B_{x,δ}) ∖ B_{x,δ} ∈ 𝒜̃_{μ} and hence we may choose Ã_{1} = B̃_{x,δ} and Ã_{2} = S(B_{x,δ}) ∖ B_{x,δ}. □

Proof of Theorem 3.1(b). We show that (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) ⇒ (i) and that (iv) ⇔ (v).

Suppose that (i) holds and set c(t) = c_{[t]}. For each Ã ∈ 𝒜̃(recall the definition of 𝒜̃before Lemma 5.4) and t ⩾ 1 it holds that

\frac{c_{[t]}}{c_{[t]+1}} c_{[t]+1}ν(([t] + 1)Ã) ⩽ c(t)ν(tÃ) ⩽ c_{[t]}ν([t]Ã). (5.1)

Since {c_{n}}_{n⩾1} is regularly varying it holds that lim_{n→∞} c_{n}/c_{n+1} = 1. Hence, lim_{t→∞} c(t)ν(tÃ) = μ(Ã) for all Ã ∈ 𝒜̃_{μ}. By Lemma 5.4 statement (ii) follows if we show that {c(t)ν(t·) : t > 0} is relatively compact in M_{0}. Indeed, Lemma 5.4 implies that all subsequential limits coincide. We know from (i) that {c_{n}ν(n·) : n ⩾ 1} is relatively compact in M_{0} and by Theorem 2.7 there is a sequence r_{i} ↓ 0 such that for each i, sup_{n} c(n)ν(n[S ∖ B_{0,r_{i}}]) < ∞ and given η > 0 there is a compact set C_{i} ⊂ S ∖ B_{0,r_{i}} such that

sup_{n} c(n)ν(n[S ∖ (B_{0,r_{i}} ∪ C_{i})]) ⩽ η. (5.2)

Since S ∖ B_{0,r} ∈ 𝒜̃, (5.1) implies that sup_{t} c(t)ν(t[S ∖ B_{0,r_{i}}]) < ∞. Moreover, (5.2) holds for any compact set C′_{i} with C_{i} ⊂ C′_{i} ⊂ S ∖ B_{0,r_{i}}. We claim that we can choose this C′_{i} such that S ∖ (B_{0,r_{i}} ∪ C′_{i}) ∈ 𝒜̃. Then (5.1) implies that sup_{t} c(t)ν(t[S∖(B_{0,r} ∪C′_{i})]) ⩽ η and by Theorem 2.7 {c(t)ν(t·) : t > 0} is relatively compact. We now show that it is possible to choose such a set C′_{i}. Take C′_{i} = {sx : s ∈ [0, 1], x ∈ C_{i}} ∩ (S ∖ B_{0,r_{i}}). The set {sx : s ∈ [0, 1], x ∈ C_{i}} is compact because it is the image of the compact set [0, 1] × C_{i} ⊂ R_{+} × S under the continuous map from R_{+} × S to S given by (λ, x) ↦ λx. Since C′_{i} is the intersection of a compact set and a closed set it is compact. Clearly, C_{i} ⊂ C′_{i} and S ∖ (B_{0,r_{i}} ∪ C′_{i}) ∈ 𝒜̃. This completes the proof of (ii).

Suppose that (ii) holds. By Lemma 5.2 there exists r > 0 such that μ(∂[rE_{1}]) = 0 and μ(rE_{1}) > 0. For t > 0 and A ∈ 𝒮 with 0 ∉ A^{−} and μ(∂A) = 0,

\frac{ν(tA)}{ν(t[rE_{1}])} = \frac{c(t)ν(tA)}{c(t)ν(t[rE_{1}])} → \frac{μ(A)}{μ(rE_{1})}

as t → ∞. Hence, by Theorem 2.4 (ii), (iii) holds.

The implication (iii) ⇒ (iv) is shown above (proof of statement (a)).

Suppose that (iv) holds. Lemma 5.2 implies that for r in a subset of (0, ∞) of positive measure, lim_{t→∞} ν(trE_{1})/ν(tE_{1}) exists and is positive. Hence, by Theorem 1.4.1 in [6], c(t) = ν(tE_{1})^{−1} is regularly varying with some index α > 0. Hence, the sequence c_{n} = ν(nE_{1})^{−1} is regularly varying with index α > 0 and c_{n}ν(n·) → μ(·) in M_{0}, i.e. (i) holds.

Suppose that (iv) holds. As above, this implies that t ↦ ν(tE_{1})^{−1} is regularly varying with some index α > 0. By Theorem 1.5.12 in [6] there exists a sequence {a_{n}}, which is regularly varying with index 1/α, such that lim_{n→∞} n/ν(a_{n}E_{1}) = 1. Hence, (v) holds.

Suppose that (v) holds. Take r > 0 such that μ(∂[rE_{1}]) = 0 and μ(rE_{1}) > 0. For t > a_{1}, let k = k(t) be the largest integer with a_{k} ⩽ t. Then a_{k} ⩽ t < a_{k+1} and k → ∞ as t → ∞. Hence, for Ã ∈ 𝒜̃,

\frac{k}{k + 1} \frac{(k + 1)ν(a_{k+1}Ã)}{kν(a_{k}[rE_{1}])} ⩽ \frac{ν(tÃ)}{ν(t[rE_{1}])} ⩽ \frac{k + 1}{k} \frac{kν(a_{k}Ã)}{(k + 1)ν(a_{k+1}[rE_{1}])}

from which it follows that lim_{t→∞} ν(tÃ)/ν(t[rE_{1}]) = μ(Ã)/μ(rE_{1}). To show that {ν(t·)/ν(t[rE_{1}]) : t > 0} is relatively compact we can apply the same argument as in the proof of the implication (i) ⇒ (ii). Hence, (iii) holds. □

Proof of Theorem 4.1. By Theorem 3.1(a), the measure μ has the scaling property μ(λA) = λ^{−α}μ(A) for λ > 0 and A ∈ 𝒮_{0}. It follows that μ(∂[R^{d}∖B_{0,u}]) = 0 for each u > 0 and therefore lim_{n→∞} c_{n}ν(n[R^{d} ∖ B_{0,u}]) = μ(R^{d} ∖ B_{0,u}) for each u > 0. Hence, {c_{n}ν(n·)} is relatively compact in M_{0}. By assumption all subsequential limits agree on sets V_{u,S} with μ(∂V_{u,S}) = 0, and by Theorem 2.8 the finite differences of such sets form a determining class. Hence, all subsequential limits coincide and the proof is complete. □

Proof of Theorem 4.2. Similar to the proof of Theorem 4.1. □

Proof of Theorem 4.3. Let r_{i} ↓ 0. If (4.1)–(4.3) hold for M then they hold for each M^{(r_{i})}. By [5, Theorem 7.3 p. 82] M^{(r_{i})} is relatively compact in M_{b}(C ∖ B_{0,r_{i}}) which by Theorem 2.6 implies that M is relatively compact.

Conversely, suppose that M is relatively compact in M_{0}(C). By Theorem 2.7 there exists a sequence r_{i} ↓ 0 such that (4.1) holds for r = r_{i} and given η > 0 there exists a compact set C_{i} ⊂ C ∖ B_{0,r_{i}} such that sup_{μ∈M} μ(C ∖ (C_{i} ∪ B_{0,r_{i}})) < η. Choose such C_{i}. Given ε > 0, by the Arzelà–Ascoli theorem there exist R_{i} and δ_{i} such that C_{i} ⊂ {x : |x(0)| ⩽ R_{i}} and C_{i} ⊂ {x : w_{x}(δ_{i}) ⩽ ε}. We may assume that R_{1} > r_{1} (otherwise C_{1} = ∅) and since {x : |x(0)| > R_{1}} ⊂ {x : |x|_{∞} > r_{1}} we have

sup_{μ∈M} μ(x : |x(0)| > R_{1}) = sup_{μ∈M} μ(x : |x|_{∞} > r_{1}, |x(0)| > R_{1}) < η.

Since η was arbitrary(4.2) follows. Finally, let i_{0} be an integer such that r_{i_{0}} < ε/2. Then {x : w_{x}(δ_{i_{0}}) > ε} ⊂ {x : |x|_{∞} > r_{i_{0}}} and hence

sup_{μ∈M} μ(x : w_{x}(δ_{i_{0}}) > ε) = sup_{μ∈M} μ(x : w_{x}(δ_{i_{0}}) > ε, |x|_{∞} > r_{i_{0}}) < η.

Since η was arbitrary (4.3) holds. □

Proof of Theorem 4.4. By (4.4) it follows in particular that c_{n}νπ_{0}^{−1}(n·) → μπ_{0}^{−1}(·) in M_{0}(R^{d}). Hence {c_{n}νπ_{0}^{−1}(n·)} is relatively compact and (4.2) holds. In addition, the conditions (4.5)–(4.6) imply by Theorem 4.3 that {c_{n}ν(n·)} is relatively compact. Let μ and μ′ be subsequential limits. By (4.4) μ = μ′ on the π-system of sets π_{t_{1},...,t_{k}}^{−1}(H), for Borel sets H ⊂ R^{dk}. Hence, μ = μ′ and c_{n}ν(n·) → μ(·) in M_{0}.

Conversely, if c_{n}ν(n·) → μ(·) in M_{0}, then (4.4) follows from the Mapping Theorem (Theorem 2.5). Since {c_{n}ν(n·)} is relatively compact (4.5)–(4.6) hold by Theorem 4.3. □

Proof of Theorem 4.5. Let r_{i} ↓ 0. If (4.7)–(4.11) hold then they hold they hold for each M^{(r_{i})}. By [5, Theorem 13.2 p. 139 and (13.8) p. 141] M^{(r_{i})} is relatively compact in M_{b}(D ∖ B_{0,r_{i}}) which by Theorem 2.6 implies that M is relatively compact.

Conversely, suppose that M is relatively compact in M_{0}(D). By Theorem 2.7 there exists a sequence r_{i} ↓ 0 such that (4.7) holds and given η > 0 there exists a compact set C_{i} ⊂ D ∖ B_{0,r_{i}} such that sup_{μ∈M} μ(D ∖ (C_{i} ∪ B_{0,r_{i}})) < η. Choose such C_{i}. As in the proof of Theorem 4.3 it follows that (4.8) holds. Given ε > 0, by Theorem 13.2 in [5, p. 139 and (13.8) p. 141] there exist δ_{i} such that C_{i} ⊂ {x : w′′_{x}(δ_{i}) ⩽ ε}, C_{i} ⊂ {x : w_{x}([0, δ_{i})) ⩽ ε}, and C_{i} ⊂ {x : w_{x}([1 − δ_{i}, 1)) ⩽ ε}. Let i_{0} be an integer such that r_{i_{0}} < ε/2. Then

{x : w′′_{x}(δ_{i_{0}}) > ε} ⊂ {x : |x|_{∞} > r_{i_{0}}}
{x : w_{x}([0, δ_{i_{0}})) > ε} ⊂ {x : |x|_{∞} > r_{i_{0}}}
{x : w_{x}([1 − δ_{i}, 1)) > ε} ⊂ {x : |x|_{∞} > r_{i_{0}}}.

and hence

sup_{μ∈M} μ(x : w′′_{x}(δ_{i_{0}}) > ε) = sup_{μ∈M} μ(x : w′′_{x}(δ_{i_{0}}) > ε, |x|_{∞} > r_{i_{0}}) < η,
sup_{μ∈M} μ(x : w_{x}([0, δ_{i_{0}})) > ε) = sup_{μ∈M} μ(x : w_{x}([0, δ_{i_{0}})) > ε, |x|_{∞} > r_{i_{0}}) < η,
sup_{μ∈M} μ(x : w_{x}((1 − δ_{i_{0}}, 1)) > ε) = sup_{μ∈M} μ(x : w_{x}((1 − δ_{i_{0}}, 1)) > ε, |x|_{∞} > r_{i_{0}}) < η.

Since η was arbitrary (4.9)–(4.11) hold. □

Proof of Theorem 4.6. Suppose ν satisfies (4.12)–(4.15). We will use Theorem 4.5 to show that {c_{n}ν(n·)} is relatively compact in M_{0}(D). For this we only need to check (4.8). Take η > 0. For any R > 0 and 0 = t_{0} < t_{1} < ⋯ < t_{k} = 1 in T such that t_{i} − t_{i−1} < δ where δ > 0

c_{n}ν(x : |x|_{∞} > nR) ⩽ c_{n}ν(x : max_{1⩽i⩽k} |x(t_{i})| > nR/2)

+ c_{n}ν(x : max_{1⩽i⩽k} |x(t_{i})| ⩽ nR/2, max_{1⩽i⩽k} sup_{t_{i−1}⩽t⩽t_{i}} |x(t)| > R)

⩽ c_{n}ν(x : max_{1⩽i⩽k} |x(t_{i})| > nR/2) + c_{n}ν(x : w′′(x, δ) > nR/2)

By (4.12) {c_{n}νπ_{t_{1},...,t_{k}}^{−1}(n·)} is relatively compact and hence we may choose R big enough such that sup_{n} c_{n}ν(x : max_{1⩽i⩽k} |x(t_{i})| > nR/2) < η/2. By (4.13) we may choose δ > 0 such that sup_{n} c_{n}ν(x : w′′(x, δ) > nR/2) < η/2. Hence (4.8) holds. It follows that {c_{n}ν(n·)} is relatively compact in M_{0}.

Let μ and μ′ be subsequential limits. We will show μ = μ′. Let T_{μ} (T_{μ′}) consist of the points where the projection π_{t} is continuous except at a set of μ-measure (μ′-measure) zero (see [5, p. 138]). By (4.12) we have μπ_{t_{1},...,t_{k}}^{−1} = μ′π_{t_{1},...,t_{k}}^{−1} for t_{1}, …, t_{k} ∈ T ∩ T_{μ} ∩ T_{μ′}. Since 0, 1 ∈ T ∩ T_{μ} ∩ T_{μ′} and this set is dense in [0, 1] it follows by Theorem 13.1 in [5] that μ = μ′.

Conversely, suppose that c_{n}ν(n·) → μ(·) in M_{0}. By the mapping theorem (Theorem 2.5) (4.12) holds for t_{1}, …, t_{k} ∈ T_{μ} and since {c_{n}ν(n·)} is relatively compact (4.13)–(4.15) hold. □

Acknowledgment

The authors are grateful for the detailed and enlightening comments made by an anonymous referee which significantly helped to improve the paper.

References

[1] A. Araujo and E. Giné, The Central Limit Theorem for Real and Banach Valued Random Variables, Wiley, New York, 1980.

[2] M. Barczy and G. Pap, Portmanteau theorem for unbounded measures, Statist. Probab. Lett. 76 (2006), 1831–1835.

[3] B. Basrak, The Sample Autocorrelation Function of Non-Linear Time Series, PhD Thesis, University of Groningen, Department of Mathematics, 2000.

[4] P. Billingsley, Probability and Measures, 3rd edition, Wiley, New York, 1995.

[5] P. Billingsley, Convergence of Probability Measures, 2nd edition, Wiley, New York, 1999.

[6] N. Bingham, C. Goldie and J. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987.

[7] D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, Springer-Verlag, New York, 1988.

[8] W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edition, Wiley, New York, 1971.

[9] A. Friedman, Foundations of Modern Analysis, Dover Pulications, New York, 1982.

[10] E. Giné, Central limit theorems in Banach spaces: A survey, in Lecture Notes in Math. 860, 1981, 138–152.

[11] E. Giné, M. G. Hahn and P. Vatan, Max-infinitely divisible and max-stable sample continuous processes, Probab. Theory Related Fields. 87 (1990), 139–165.

[12] L. de Haan and T. Lin, On convergence toward an extreme value distribution in C[0, 1], Ann. Probab. 29 (2001), 467–483.

[13] H. Hult and F. Lindskog, Extremal behavior of regularly varying stochastic processes, Stochast. Process. Appl. 115 (2005), 249–274.

[14] H. Hult, F. Lindskog, T. Mikosch and G. Samorodnitsky, Functional large deviations for multivariate regularly varying random walks, Ann. Appl. Probab. 15 (2005), 2651–2680.

[15] H. Hult and F. Lindskog, Extremal behavior of stochastic integrals driven by regularly varying Lévy processes, to appear in Ann. Probab.

[16] H. Kesten, Random difference equations and renewal theory for products of random matrices, Acta Math. 131 (1973), 207–248.

[17] O. Kallenberg, Random Measures, 3rd edition, Akademie-Verlag, Berlin, 1983.

[18] M. M. Meerschaert, Regular variation and domains of attraction in R^{k}, Stat. Probab. Lett. 4 (1986), 43–45.

[19] M. M. Meerschaert and H.-P. Scheffler, Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice, Wiley, New York, 2001.

[20] S. Resnick, Extreme Values, Regular Variation, and Point Processes, Springer-Verlag, New York, 1987.

[21] S. Resnick, On the foundations of multivariate heavy-tail analysis, J. Appl. Probab. 41A (2004), 191–212.

[22] E. L. Rvačeva, On domains of attraction of multi-dimensional distributions, Sel. Transl. Math. Stat. Probab. 2 (1962), 183–205.

[23] W. Whitt, Some useful functions for functional limit theorems, Math. Oper. Res. 5 (1980), 67–85.

Division of Applied Mathematics (Received 06 07 2006) Brown University (Revised 31 10 2006) Providence USA

henrik hult@brown.edu

Department of Mathematics KTH Stockholm Sweden

lindskog@kth.se