Integrability of Seminorms

We study integrability and equivalence of L^p-norms of polynomial chaos elements. Relying on known results for Banach space valued polynomials, a simple technique is presented to obtain integrability results for random elements that are not necessarily limits of Banach space valued polynomials. This enables us to prove integrability results for a large class of seminorms of stochastic processes and to answer, partially, a question raised by C. Borell (1979, Seminaire de Probabilites, XIII, 1-3).


Introduction
The purpose of the present paper is to unify and extend results on integrability of seminorms of polynomial chaos elements taking values in a topological vector space. The chaos are understood in the weak sense, in the spirit of Ledoux and Talagrand (1991). The motivation for this comes from stochastic processes. For example, in order to study U := sup t∈T |X t |, where T is a countable set, we may think of X as a map from Ω into l ∞ (T ). However, the non-separability of l ∞ (T ) causes many problems, e.g. with measurability of X . The approach in this paper is instead to view X as a random element in the separable topological space R T . Then U = N (X ), where N ( f ) = sup t∈T | f (t)| is a lower semicontinuous seminorm on R T (taking values in [0, ∞]). When X is a weak chaos process, Theorem 2.2 provides conditions under which U is integrable.
Weak chaos processes appear in the context of multiple integral processes; see e.g. Krakowiak and Szulga (1988) for the α-stable case. Rademacher chaos processes are applied repeatedly when studying U-statistics; see de la Peña and Giné (1999). They are also used to study infinitely divisible chaos processes; see Basse and Pedersen (2009), Marcus and Rosiński (2003) and Rosiński and Samorodnitsky (1996). Using the results of the present paper, Basse-O'Connor and Graversen (2010) extend some results on Gaussian semimartingales (e.g. Jain and Monrad (1982) and Stricker (1983)) to a large class of chaos processes.
Let N be a measurable seminorm on R T . For X Gaussian, Fernique (1970) shows that e N (X ) 2 is integrable for some > 0. This result is extended to Gaussian chaos processes by Borell (1978), Theorem 4.1. Moreover, if X is α-stable for some α ∈ (0, 2), de Acosta (1975), Theorem 3.2, shows that N (X ) p is integrable for all p < α. When X is infinitely divisible, Rosiński and Samorodnitsky (1993) provide conditions on the Lévy measure ensuring integrability of N (X ). See also Hoffmann-Jørgensen (1977) for further results.
Given a sequence (Z n ) n∈N of independent random variables, Borell (1984) studies, under the condition sup n≥1 Z n − EZ n q Z n − EZ n 2 < ∞, q ∈ (2, ∞], (1.1) integrability of Banach space valued random elements which are limits in probability of tetrahedral polynomials associated with (Z n ) n∈N . As shown in Borell (1984), (1.1) implies equivalence of L p -norms for Hilbert space valued tetrahedral polynomials for p ≤ q, but not for Banach space valued tetrahedral polynomials except in the case q = ∞. We impose the stronger condition C q on (Z n ) n∈N , see (1.2)-(1.3), which in the case q = ∞ equals (1.1). Under C q with q < ∞, Kwapień and Woyczyński (1992), Theorem 6.6.2, show equivalence of L p -norms of Banach space valued tetrahedral polynomials. We extend and unify Borell (1984), Kwapień and Woyczyński (1992) and and others, by considering random elements which are not necessarily limits of tetrahedral polynomials. Moreover, for lower semicontinuous seminorms Borell (1978), de Acosta (1975 and Fernique (1970) are special cases of Theorem 2.1.

Chaos Processes and Condition C q
Let (Ω, , P) denote a probability space. When F is a topological space, a Borel measurable map X : Ω → F is called an F -valued random element, however when F = R, X is, as usual, called a random variable. For each p > 0 and random variable X we let X p := E[|X | p ] 1/p , which defines a norm when p ≥ 1; moreover, let X ∞ := inf{t ≥ 0 : P(|X | ≤ t) = 1}. When F is a Banach space, L p (P; F ) denotes the space of all F -valued random elements, X , satisfying X L p (P;F ) = E[ X p ] 1/p < ∞. Throughout the paper I denotes a set and for all ξ ∈ I, ξ is a family of independent random variables. Set = { ξ : ξ ∈ I}. Furthermore, d ≥ 1 is a natural number and F is a locally convex Hausdorff topological vector space (l.c.TVS) with dual space F * , see Rudin (1991). Following Fernique (1997), a map N from F into [0, ∞] is called a pseudoseminorm if for all x, y ∈ F and λ ∈ R, we have and For ξ ∈ I let d ( ξ ; F ) denote the set of p(Z 1 , . . . , Z n ) where n ∈ N, Z 1 , . . . , Z n are different elements in ξ and p is an F -valued tetrahedral polynomial of order d. Recall that p : R n → F is called is the set of all F -valued random elements X for which there exists a sequence (X k ) k∈N ⊆ ∪ ξ∈I d ( ξ ; F ) converging weakly to X . In the spirit of Ledoux and Talagrand (1991) we introduce the following: Definition 1.1. An F -valued random element X is said to be a weak chaos element of order d associated with if for all n ∈ N and (x * i ) n i=1 ⊆ F * we have (x * 1 (X ), . . . , x * n (X )) ∈ d ( ; R n ), and in this case we write X ∈ weakd ( ; F ). Similarly, a real-valued stochastic process (X t ) t∈T is said to be a weak chaos process of order d associated with if for all n ∈ N and (t i ) n i=1 ⊆ T we have In what follows we shall need the next conditions: Condition C q . For q ∈ (0, ∞), is said to satisfy C q if there exists β 1 , β 2 > 0 such that for all Z ∈ ∪ ξ∈I ξ there exists c Z > 0 with P(|Z| ≥ c Z ) ≥ β 1 and Let us start by noticing that C q implies equivalence of moments, that is, if satisfies C q with q ∈ (0, ∞) then for all p ∈ (0, q) we have (a):

Main results
Recall that an F -valued random element X is said to be a.s. separably valued if P(X ∈ A) = 1 for some separable closed subset A of F , and a map f : where k p,r,d,β depends only on p, q, d and the β's from C q . Furthermore, in the case q = ∞ we have For q = ∞, Theorem 2.1 answers in the case where the pseudo-seminorm is lower semicontinuous a question raised by Borell (1979) concerning integrability of pseudo-seminorms of Rademacher chaos elements. This additional assumption is satisfied in most examples, in particular the one considered in the Introduction. We prove Theorem 2.1 by representing N on the form N (x) = sup n∈N |x * n (x)| where (x * n ) n∈N ⊆ F * , which enables us to obtain the result by a suitable application of Kwapień and Woyczyński (1992) when q < ∞ and Borell (1984) where k p,r,d,β depends only on p, q, d and the β's from C q . If q = ∞ and p ≥ 2 we may choose Kwapień and Woyczyński (1992), Equation (2.2.4). Furthermore, for q ∈ (1, ∞) and d ≥ 2 it is taken from the proof of Kwapień and Woyczyński (1992), Theorem 6.6.2, and using Kwapień and Woyczyński (1992), Remark 6.9, the result is seen to hold also for q ∈ (0, 1]. For q = ∞, (2.1) is a consequence of Borell (1984), Theorem 4.1. In Borell (1984) the result is only stated for 2 ≤ p < r, however, a standard application of Hölder's inequality shows that it is valid for all 0 < p < r; see e.g. Pisier (1978), Lemme 1.1. Finally, in Borell (1984) there is no explicit expression for A d ; this can, however, be obtained by applying Lemma A.1 from the Appendix, in the proof of Borell (1984), Theorem 4.1, top of page 198.
Let l n ∞ be R n equipped with the sup norm. Fix finite p, r with 0 < p < r ≤ r and let C := k p,r,d,β . Let us show that for all n ∈ N and Y ∈ d ( ; R n ) we have (2.3) and Krakowiak and Szulga (1986) Arguing as in Fernique (1997), Lemme 1.2.2, we will show that there exists ( Then A is convex and balanced since N is a pseudoseminorm and closed since N is lower semicontinuous. Thus by the Hahn-Banach theorem, see Rudin (1991), Theorem 3.7, for all Since X is a.s. separably valued we may and will assume that F is separable and hence strongly Lindelöf since it is metrizable by assumption, see Gemignani (1990). Thus, since (2.5) is an open For all finite 0 < p < r ≤ q, (2.2) shows that U n q ≤ C U n p < ∞ for all n ∈ N. This implies that {U p n : n ∈ N} is uniformly integrable and hence To prove the last statement of the theorem let which completes the proof.
Let T denote a countable set and F := R T be equipped with the product topology. Then F is a separable and locally convex Fréchet space and all x * ∈ F * are of the form for some n ∈ N, t 1 , . . . , t n ∈ T and α 1 , . . . , α n ∈ R. Thus for X = (X t ) t∈T we have that X ∈ weakd ( ; F ) if and only if X is a weak chaos process of order d. Rewriting Theorem 2.1 in the case F = R T we obtain the following result: Theorem 2.2. Assume satisfies C q for some q ∈ (0, ∞] and if q < ∞ and d ≥ 2 that all elements in ∪ ξ∈I ξ are symmetric. Let T denote a countable set, (X t ) t∈T be a weak chaos process of order d and N be a lower semicontinuous pseudo-seminorm on R T such that N (X ) < ∞ a.s. Then for all finite Let denote a vector space of Gaussian random variables and Π d ( ; R) be the closure in probability of the random variables p(Z 1 , . . . , Z n ), where n ∈ N, Z 1 , . . . , Z n ∈ and p : R n → R is a polynomial of degree at most d (not necessary tetrahedral). Recall that a sequence of independent, identically distributed random variables (Z n ) n∈N such that P(Z 1 = ±1) = 1/2 is called a Rademacher sequence.

Proposition 2.3. Suppose F is a l.c.TVS and X is an F -valued random element such that x
Proof. Let n ∈ N, x * 1 , . . . , x * n ∈ F * and W = (x * 1 (X ), . . . , x * n (X )). We need to show that W ∈ d ( ; R n ). For all k ≥ 1 we may choose polynomials p k : R k → R n of degree at most d and Y 1,k , . . . , Y k,k independent standard normal random variables such that with Y k = (Y 1,k , . . . , Y k,k ) we have lim k p k (Y k ) = W in probability. Hence it suffices to show p k (Y k ) ∈ d ( ; R n ) for all k ∈ N. Fix k ∈ N and let us write p and Y for p k and Y k . Reenumerate 0 as k independent Rademacher sequences (Z i,m ) i≥1 with m = 1, . . . , k and set By applying Theorem 2.1, the conclusion follows since satisfies C ∞ with β 3 = 1.
The integrability of e N (X ) 2/d , in Proposition 2.3, is a consequence of the seminal work Borell (1978), Theorem 4.1. However, Proposition 2.3 provides a simple proof of this result and also provides equivalence of L p -norms and explicit constants. When F = R T for some countable set T , Proposition 2.3 covers processes X = (X t ) t∈T , where all time variables, X t , have the following representation in terms of multiple Wiener-Itô integrals with respect to a Brownian motion W , For basic fact about multiple integrals see Nualart (2006).
The next result is known from Arcones and Giné (1993), Theorem 3.1, for general Gaussian polynomials.
Proposition 2.4. Assume that = { 0 } satisfies C q for some q ∈ [2, ∞] and 0 consists of symmetric random variables. Let F denote a Banach space and X an a.s. separably valued random element in F with x * (X ) ∈ d ( ; R) for all x * ∈ F * . Then there exist x 0 , x i 1 ,...,i k ∈ F and {Z n : n ≥ 1} ⊆ 0 such that for all finite p ≤ q s. and in L p (P; F ).
Proof. We follow Arcones and Giné (1993), Lemma 3.4. Since X is a.s. separably valued we may and do assume F that is separable, which implies that F * 1 := {x * ∈ F * : x * ≤ 1} is metrizable and compact in the weak*-topology by the Banach-Alaoglu theorem; see Rudin (1991), Theorem 3.15+3.16. Moreover, the map x * → x * (X ) from F * 1 into L 0 (P) is trivially weak*-continuous and thus a weak*continuous map into L 2 (P) by a combination of the equivalence of norms from Theorem 2.1 and Krakowiak and Szulga (1986), Corollary 1.4. This shows that {x * (X ) : x * ∈ F * 1 } is compact in L 2 (P) and hence separable. By definition of d ( ; R), this implies that there exists a countable set {Z n : n ∈ N} ⊆ 0 such that for some a(A, x * ) ∈ R, where N d = {A ⊆ N : |A| ≤ d} and Z A = i∈A Z i for A ∈ N d . For A ∈ N d , the map x * → a(A, x * ) from F * into R is linear and weak*-continuous and hence there exists Since F is separable, (2.6) and Kwapień and Woyczyński (1992), Theorem 6.6.1, show that lim n→∞ A∈N n d x A Z A = X a.s.
As above it follows that the convergence also takes place in L p (P; F ) for all finite p ≤ q, which completes the proof.
The above proposition gives rise to the following corollary: Corollary 2.5. Assume that = { 0 } satisfies C q for some q ∈ [2, ∞] and 0 consists of symmetric random variables. Let T denote a set, V (T ) ⊆ R T a separable Banach space where the maps f → f (t) from V (T ) into R is continuous for all t ∈ T , and X = (X t ) t∈T a stochastic process with sample paths in V (T ) satisfying X t ∈ d ( ; R) for all t ∈ T . Then there exists x 0 , x i 1 ,...,i k ∈ V (T ) and Rosiński (1986), page 287. By (i) we may regard X as a random element in V (T ) and by Hence the result is a consequence of Proposition 2.4. Borell (1984), Theorem 5.1, shows Corollary 2.5 assuming (1.1), T is a compact metric space, V (T ) = C(T ) and X ∈ L q (P; V (T )). By assuming C q instead of the weaker condition (1.1) we can omit the assumption X ∈ L q (P; V (T )). Note also that by Theorem 2.2 the last assumption is satisfied under C q . When 0 consists of symmetric α-stable random variables and d = 1, Corollary 2.5 is known from Rosiński (1986), Corollary 5.2. The separability assumption on V (T ) in Corollary 2.5 is crucial. Indeed, for all p > 1, Jain and Monrad (1983), Proposition 4.5, construct a separable centered Gaussian process X = (X t ) t∈ [0,1] with sample paths in the non-separable Banach space B p of functions of finite p-variation on [0, 1] such that the range of X is a non-separable subset of B p and hence the conclusion in Corollary 2.5 can not be true. However, for the non-separable Banach space B 1 a result similar to Corollary 2.5 is shown in Jain and Monrad (1982) for Gaussian processes, and extended to weak chaos processes in Basse-O'Connor and Graversen (2010).

A class of infinitely divisible processes
An important example of a weak chaos process of order one is (X t ) t∈T of the form where Λ is an independently scattered infinitely divisible random measure (or random measure for short) on some non-empty space S equipped with a δ-ring , and s → f (t, s) are Λ-integrable deterministic functions in the sense of Rajput and Rosiński (1989). To obtain the associated let I be the set of all ξ given by ξ = {A 1 , . . . , A n } for some n ∈ N and disjoint sets A 1 , . . . , A n in , and let Then, by definition of the stochastic integral (3.1) as the limit of integrals of simple functions, (X t ) t∈T is a weak chaos process of order one associated with .
As we saw in Section 2, C q is crucial in order to obtain integrability results and equivalence of L pnorms, so let us consider some cases where the important example (3.1) does or does not satisfy C q . For this purpose let us introduce the following distributions: The inverse Gaussian distribution IG(µ, λ) with µ, λ > 0 is the distribution on R + with density Moreover, the normal inverse Gaussian distribution NIG(α, β, µ, δ) with µ ∈ R, δ ≥ 0, and 0 ≤ β ≤ α, is symmetric if and only if β = µ = 0, and in this case it has the following density where K 1 is the modified Bessel function of the third kind and index 1 given by K 1 (z) = 1 2 ∞ 0 e −z( y+ y −1 )/2 d y for z > 0. For each finite number t 0 > 0, a random measure Λ is said to be induced by a Lévy process Y = By the scaling property it is not difficult to show that if Λ is a symmetric α-stable random measure with α ∈ (0, 2), then satisfies C q if and only if q < α. The next result studies C q in some non-trivial cases. (i) If Y 1 has an IG-distribution, then satisfies C q if and only if q ∈ (0, 1 2 ).
(ii) If Y 1 has a symmetric NIG-distribution, then satisfies C q if and only if q ∈ (0, 1).
(iii) If Y is non-deterministic and has no Gaussian component, then does not satisfy C q for any q ≥ 2. In fact, all integrable non-deterministic Lévy processes Y satisfies lim t→0 ( Y t 2 / Y t 1 ) = ∞.
Proof. Assume that Λ is a random measure induced by a Lévy process Y = (Y t ) t∈ [0,T ] . For arbitrary To prove the if -implication of (i) let q ∈ (0, 1 2 ) and assume that where m is the Lebesgue measure, and hence with c Z = m(A) 2 λ we have that Z/c Z = d IG(µ/(λm(A)), 1), which has a density which on [1, ∞) is bounded from below and above by constants (not depending on x) times g Z (x), where Thus there exists a constant c > 0, not depending on A or s, such that Using e.g. l'Hôpital's rule it is easily seen that (3.4) is finite, showing (1.2). Therefore C q follows by the inequality To show the only if -implication of (i) note that n 2 Y 1/n → d X as n → ∞, where X follows a 1 2 -stable distribution on R + . Assume that satisfies C q for some q ≥ 1/2. Then, by (1.4) there exists c > 0 such that Y t 1/2 ≤ c Y t 1/4 for all t ∈ [0, 1], and since {n 2 Y 1/n : n ≥ 1} is bounded in L 0 it is also bounded in L 1/2 . But this contradicts ∞ = X 1/2 ≤ lim inf n→∞ n 2 Y 1/n 1/2 , and shows that does not satisfy C q .
To show the if -implication of (ii) assume that Y 1 = d NIG ( Using the above (i) on U Z and q/2, there exists a constant c 1 > 0 such that Furthermore, it is well known that there exists a constant c 2 > 0 such that for all s ≥ 1 Since U Z has a density given by (3.3) it is easily seen that Moreover, using that Z/c Z = d NIG(m(A)αδ, 0, 0, 1) and that K 1 (z) ≥ e −z /z for all z > 0, it is not difficult to show that there exists a constant c 3 , not depending on s and A, such that ∞ s e −x 2 /2 d x ≤ c 3 P(|Z/c Z | > s), for all s ≥ 1.
(3.5) By combining the above we obtain (1.2) and by (3.5) applied on s = 1, C q follows. The only ifimplication of (ii) follows similar to the one of (i), now using that (n −1 Y 1/n ) n≥1 converge weakly to a symmetric 1-stable distribution.
To show (iii) it is enough to prove that for all non-deterministic and square-integrable Lévy processes, Y , with no Gaussian component we have as t → 0. The latter statement follows by the equality To show that Y t 1 = o(t 1/2 ) as t → 0 we may assume that Y is symmetric.
Now let x 0 , . . . , x d ∈ V . Since N is a seminorm, the Hahn-Banach theorem (see Rudin (1991), Theorem 3.2) shows that there exists a family Λ of linear functionals on V such that Assuming that the left-hand side of (A.1) is satisfied we have This completes the proof.