https://doi.org/10.1007/s10959-021-01085-6
On Decoupling in Banach Spaces
Sonja Cox1·Stefan Geiss2
Received: 19 June 2018 / Revised: 8 February 2021 / Accepted: 17 February 2021 / Published online: 14 March 2021
© The Author(s) 2021
Abstract
We consider decoupling inequalities for random variables taking values in a Banach space X. We restrict the class of distributions that appear as conditional distribu- tions while decoupling and show that each adapted process can be approximated by a Haar-type expansion in which only the pre-specified conditional distributions appear.
Moreover, we show that in our framework a progressive enlargement of the underlying filtration does not affect the decoupling properties (in particular, it does not affect the constants involved). As a special case, we deal with one-sided moment inequalities for decoupled dyadic (i.e., Paley–Walsh) martingales and show that Burkholder–Davis–
Gundy-type inequalities for stochastic integrals ofX-valued processes can be obtained from decoupling inequalities forX-valued dyadic martingales.
Keywords Decoupling in Banach spaces·Regular conditional probabilities·Dyadic martingales·Stochastic integration
Mathematics Subject Classification 60E15·60H05·46B09
1 Introduction
The UMD property is crucial in harmonic and stochastic analysis in Banach spaces, see, e.g., [17,18]. A Banach space X is said to satisfy the UMD property if there exists a constantc(1)≥1 such that for everyX-valued martingale difference sequence
B
Stefan Geiss stefan.geiss@jyu.fi Sonja Cox s.g.cox@uva.nl1 Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Postbus 94248, 1090 GE Amsterdam, Netherlands
2 Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, 40014 Jyvaskyla, Finland
(dn)Nn=1one has that
N n=1
dn
L2(P;X)
≤c(1)
N n=1
θndn
L2(P;X)
(1)
for all signs θn ∈ {−1,1}, i.e., one has Unconditional Martingale Differences. It follows from Maurey [27], that, in order to verify the UMD property of the Banach space X, it is sufficient to consider X-valued Haar- or dyadic martingales (dyadic martingales are also known asPaley–Walsh martingales).
On the other hand, McConnell [28, Theorem 2.2] (see also Hitczenko [15]) proved that the UMD property is equivalent to the existence of ac(2)≥1 such that
N n=1
dn
L2(P;X)
≤c(2)
N n=1
en
L2(P;X)
(2)
for all N ∈ Nand all X-valued(Fn)nN=1-martingale difference sequences(dn)Nn=1 and(en)nN=1 such that L(dn|Fn−1) = L(en|Fn−1), i.e.,(en)nN=1 and(dn)nN=1 are tangent. Imposing additional assumptions on either(en)Nn=1or(dn)nN=1in (2) results in an (a priori) weaker Banach space property, e.g., imposing that(en)nN=1in (2) is thedecoupled tangent sequence of(dn)Nn=1(see Definition2.5) results in thelower decoupling property for tangent martingales. The Banach spaceL1satisfies the lower decoupling property for tangent martingales (see Cox and Veraar [9, Example 4.7]), but fails to have the UMD property (see, e.g., [17, Example 4.2.20]). The notion of decoupled tangent sequences was introduced by Kwapie´n and Woyczy´nski [23,24].
The decoupled tangent sequence(en)nN=1of a sequence(dn)n∈N(adapted to a filtration (Fn)n∈N) is unique in distribution and replaces parts of the dependence structure of (dn)Nn=1by a sequence of conditionally independent random variables. Although the definition of decoupling might not be explicit, there are canonical representations of a decoupled tangent sequence, see Kwapie´n and Woyczy´nski [24] and Montgomery- Smith [29].
Inequalities (1) and (2) describetwo-sided decoupling propertiesdue to the sym- metry between the left- and right-hand side. The lower decoupling property for tangent martingales however is an example of aone-sided decoupling property. An a priori different one-sided decoupling property is obtained by considering (1) where(θn)Nn=1 is replaced by a Rademacher sequence that is independent of(dn)nN=1. This one-sided decoupling property was first studied explicitly in [13] and is also satisfied byL1. The goal of this article is to gain insight into the relation between these different kinds of one-sided decoupling properties. First, however, let us discuss some instances in the literature where decoupling inequalities play a crucial role.
The proofs by Burkholder [4] and Bourgain [2] of the equivalence of the UMD property of a Banach space X and the continuity of the X-valued Hilbert transform use thatXhas the UMD property if and only if it has both a lower- and an upper decou- pling property. For certain applications, only a single one-sided decoupling property is
needed. For example, the lower (resp. upper) decoupling property for martingales and the type (resp. cotype) property imply martingale type (resp. cotype) and therefore by Pisier [31] an equivalent re-norming of the Banach space with a norm having a certain modulus of continuity (resp. convexity). A classical case of decoupling, studied on its own, concerns randomly stopped sums of independent random variables, see for example the results of Klass [21,22].
Another application for decoupling is stochastic integration. Indeed, only the lower decoupling property is needed to obtain sufficient conditions for the existence of stochastic integrals. This can be inferred from [12, the proof of Theorem 2], where stochastic integration of UMD Banach space-valued processes with respect to a Brow- nian motion is considered. One-sided decoupling is used more explicitly in [24, Section 6], where the existence of decoupled tangent processes for left quasi-continuous pro- cesses in the Skorokhod space is studied. In [34] and [9, Section 5], decoupling inequalities were used to give sufficient conditions for the existence of a Banach space- valued stochastic process with respect to a cylindrical Brownian motion. Very recently, Kallenberg [20] proved the existence of decoupled tangent semi-martingales and two-sided decoupling inequalities, and considered applications to multiple stochastic integrals. Moreover, quasi-Banach spaces fail to satisfy the UMD property, but may satisfy decoupling inequalities, see [7, Section 5.1] and e.g. [9, Example 4.7]. Sec- tion5contains our contribution to this topic, see also Theorem1.7and Remark5.6.
Before discussing this contribution, let us turn to the open problem that motivated our research:
Open Problem 1.1 If a Banach space X has the lower decoupling property for tangent dyadic martingales, does it also have the lower decoupling property for general tangent martingales?
We were not able to answer this question in this generality. However, our main result (Theorem1.4) provides a reduction of this problem to simple Haar-type series and gives a partial answer (see Corollary1.6for a special case). The proof of Theorem1.4 is inspired by the aforementioned work by Maurey [27]. Open Problem1.1can be split into two subproblems; consequently, our proof of Theorem1.4consists of two parts [completely solving subproblem (A) and partially solving subproblem (B)]:
(A) If a Banach space X has a lower (upper) decoupling property for X-valued sequences of random variables adapted to a (in a certain sense) natural minimal filtration(Fn)∞n=1and with conditional distributions in a set of measuresP, does Xalso have a lower (upper) decoupling property forX-valued sequences adapted toanyfiltration(Fn)∞n=1and with conditional distributions inP?
(B) Given thatXhas a lower (resp. upper) decoupling property forX-valued sequences with conditional distributions in a certainP, doesXalso have a lower (resp. upper) decoupling property for generalX-valued sequences?
Problem (A) is of fundamental importance in stochastic integration theory as, given the driving process, the underlying filtration determines the set of integrands we may use. We now describe the content of the article in more detail:
Section 3: Theorem 3.1 provides a factorization of a random variable along regular conditional probabilities. With this result, we contribute to the results of
Montgomery-Smith [29] (see also Kallenberg [19, Lemma 3.22]). This result is the key to approximate our adapted processes in terms of Haar-like series.
Section4: this section is devoted to our main results, Theorem4.3and its corollary, Theorem1.4. It contains the key ingredients for the proofs and some examples.
To formulate Theorem1.4, we recall some definitions. For a separable Banach space X, we denote byB(X)theσ-algebra generated by the norm-open sets. ByP(X), we denote the set of all probability measures on(X,B(X)), forp∈(0,∞)we let
Pp(X):=
μ∈P(X):
X
xpXdμ(x) <∞
,
andδx ∈P(X)stands for the Dirac measure atx ∈ X. The next definition concerns a set of admissible adapted processes characterized by an assumption on the regular versions of the—in a sense—predictable projections:
Definition 1.2 LetXbe a separable Banach space,p∈(0,∞),∅ =P ⊆Pp(X), and (,F,P, (Fn)∞n=0)be a stochastic basis. We denote byAp(, (Fn)∞n=0;X,P)the set of(Fn)∞n=1-adapted sequences(dn)∞n=1inLp(P;X)with the property that for all n ≥1 there exists ann−1∈F satisfyingP(n−1)=1 andκn−1[ω,·] ∈P for all ω∈n−1, whereκn−1is a regular conditional probability kernel forL(dn|Fn−1).
The concept of regular conditional probability kernels is recalled in Sect.2.2. Next, we introduce an extension of a given set of probability measures that is natural in our context:
Definition 1.3 For a separable Banach spaceX,p∈(0,∞)and∅ =P ⊆Pp(X)we let
Pp-ext :=
μ∈Pp(X): ∀j≥1∃Kj ≥1 andμj,1, . . . , μj,Kj ∈P such thatμj,1∗ · · · ∗μj,Kj
w∗
→μas j → ∞ and
μj,1∗ · · · ∗μj,Kj j∈N is uniformlyLp-integrable .
The convergence of the convolutionsμj,1∗ · · · ∗μj,Kj towardμin Definition1.3 is known to be the convergence in the p-Wasserstein distance if p ∈ [1,∞)(cf. [6, Theorem 5.5, p. 358]).
Theorem 1.4 Let X,Y,Z be Banach spaces, where X is separable, let S : X → Y and T : X →Z be linear and bounded, p∈(0,∞),an index set, let : [0,∞)→ [0,∞)be upper semi-continuous, and let λ: [0,∞)→ [0,∞),λ∈, be a family of lower semi-continuous functions such that
sup
ξ∈(0,∞)(1+ |ξ|)−p (ξ) <∞ and sup
ξ∈(0,∞)(1+ |ξ|)−p λ(ξ) <∞ (3) for allλ∈ . Then, for a setP ⊆Pp(X)withδ0 ∈P, the following assertions are equivalent:
(i) For every stochastic basis(,F,P, (Fn)∞n=1)and finitely supported1 (dn)∞n=1
∈Ap(,F;X,Pp-ext), it holds that
λ∈supE λ
∞ n=1
Sdn
Y
≤E
∞ n=1
T en
Z
, (4)
whenever(en)n∈Nis anF-decoupled tangent sequence of(dn)n∈N.
(ii) For every sequence of independent random variables(ϕn)nN=1⊂Lp(P;X), N ≥ 1, satisfying L(ϕn) ∈ P, and every A0 ∈ {∅, }, An ∈ σ (ϕ1, . . . , ϕn), n ∈ {1, . . . ,N}, it holds that
λ∈supE λ
N n=1
1An−1Sϕn
Y
≤E
N n=1
1An−1Tϕn
Z
, (5)
where(ϕn)Nn=1is an independent copy of(ϕn)nN=1. Some remarks concerning Theorem1.4are at place:
(1) The condition that δ0 ∈ P ensures that finitely supported sequences fit in our setting and is used at several instances in the proof.
(2) The condition thatX is separable is mainly to simplify our presentation: after all, we can apply Theorem1.4whenever(dn)n∈Nis a sequence of random variables taking values in a separable subspaceX of some non-separable spaceX.˜ (3) The table below provides some typical choices for, , and ( λ)λ∈ given
p∈(0,∞)(hereC∈(0,∞)and f,gareR-valued random variables):
λ(ξ) (ξ) p
supλ∈E λ(f)≤√p E (g)
card()=1 ξp Cpξp fLp(P)≤CgLp(P)
card()=1 1{ξ>μ}, μ≥0 Cpξp √p
P(f > μ)≤CgLp(P) (0,∞) λp1{ξ>λ} Cpξp fLp,∞(P)≤CgLp(P)
(4) For relevant choices forP, see Examples4.5–4.8; Corollary1.6uses Example4.8.
(5) Theorem1.4 remains valid if one exchanges (dn)nN=1 with(en)Nn=1 in (4) and (ϕn)nN=1with(ϕn)nN=1in (5), respectively.
Section 5: we use Theorem 1.4to obtain relevant upper bounds for stochastic integrals, see Theorem1.7. In order to formulate that theorem, we need the following definition:
1 There are only finitely manynfor whichdn≡0.
Definition 1.5 If p ∈ (0,∞)and if X is a separable Banach space X, then we let Dp(X):=infc, where the infimum is taken over allc∈ [0,∞]such that
N n=1
rnvn−1
Lp(P;X)
≤c
N n=1
rnvn−1
Lp(P;X)
(6)
for all N ≥ 2, v0 ∈ X andvn := hn(r1, . . . ,rn)withhn: {−1,1}n → X,n ∈ {1, . . . ,N−1}, where the(rn)Nn=1are independent and take the values−1 and 1 with probability 1/2, and(rn)nN=1is an independent copy of(rn)nN=1.
The process (n
k=1rkvk−1)Nn=1 in Definition 1.5is a dyadic martingale. Theo- rem 1.4with X = Y = Z, S = T = Id, = {λ}, and (ξ) = λ(ξ) = ξp implies:
Corollary 1.6 Let X be a separable Banach space, let p ∈ (0,∞) be such that Dp(X) <∞, and letP:= {12(δ−x+δ−x): x∈X}. Then,
N n=1
dn
Lp(P;X)
≤Dp(X)
N n=1
en
Lp(P;X)
(7)
for all N ≥1, every stochastic basis(,F,P, (Fn)Nn=0), every strongly p-integrable and (Fn)nN=1 adapted sequence of random variables(dn)nN=1such that, on a set of measure one, theFn−1-conditional laws of all dnbelong toPp-ext, and every decoupled tangent sequence(en)nN=1of(dn)nN=1.
Cox and Geiss [8, Section 5] contains a characterization of Pp-ext when P :=
{12(δ−x +δ−x): x ∈ R}. Corollary1.6combined with the Central Limit Theorem results in Theorem 1.7. For details, see the proof of Part (ii) of Theorem5.2. The- orem1.7extends both [9, Theorem 5.4] and [12, Theorem 2]: see Remark 5.6for details.
Theorem 1.7 For a separable Banach space X and p,q ∈ (0,∞), the following assertions are equivalent:
(i) Dp(X) <∞.
(ii) For every stochastic basis (,F,P,F = (Ft)t∈[0,∞)), every F-Brownian motion W = (W(t))t∈[0,∞), and every simple F-predictable X -valued process (H(t))t∈[0,∞)it holds that
∞
0
H(t)dW(t)
Lq(P;X)≤ Kp,2Dp(X)S(H)Lq(P) with the square function S(H)(ω) := f → ∞
0 f(t)H(t, ω)dtγ (L2((0,∞);X)
and Kp,2the constant in theLp-to-L2Kahane–Khintchine inequality(see Sect.5.1 for details on theγ-radonifying normγ (L2((0,∞);X)).
2 Preliminaries
2.1 Some General Notation
We letN = {1,2, . . .}andN0 = {0,1,2, . . .}. For a vector space V and B ⊆ V, we set −B := {x ∈ V: −x ∈ B}.Given a non-empty set, we let 2 denote the system of all subsets ofand use AB :=(A\B)∪(B\A)for A,B ∈ 2. A system of pair-wise disjoint subsets(Ai)i∈I ofis apartitionof, where I is an arbitrary index set andAi = ∅is allowed, if
i∈I Ai =. If(M,d)is a metric space we defined: M ×2M → [0,∞]by settingd(x,A):=inf{d(x,y): y∈ A}for all (x,A)∈ M×2M. IfVis a Banach space and(M,d)a metric space, thenC(M;V)is the space of continuous maps fromM toV, andCb(M;V)the subspace of bounded continuous maps fromM toV.
Banach Space- Valued random variables: For a Banach space X, we let B(X)denote the Borelσ-algebra generated by the norm-open sets. Forx ∈ X and ε > 0, we set Bx,ε := {y ∈ X: x−yX < ε}. For B ∈ B(X), we letB¯ denote the norm-closure of B, we let Bo denote the interior and∂B := ¯B\Bo. Given a probability space (,F,P) and a measurable space (S, ), an F/-measurable mappingξ: → S is called anS-valued random variable. For a random variable ξ :→ S, the law ofξis denoted byL(ξ)(A):=P(ξ ∈ A)for A∈.
Lebesgue spaces: ForXa separable Banach space and(S, )a measurable space, we defineL0((S, );X)to be the space of/B(X)-measurable mappings fromSto X. If(S, )is equipped with aσ-finite measureμandp∈(0,∞), then we define Lp((S, , μ);X):=
ξ∈L0((S, , μ);X): ξLpp((S,,μ);X):=
S
ξXpdμ <∞
. If there is no risk of confusion, we write for exampleLp(μ;X)orLp(;X)as short- hand notation forLp((S, , μ);X), and we setLp((S, , μ)):=Lp((S, , μ);R).
Probability measures on Banach spaces:
(1) Given an index setI = ∅, a family(μi)i∈I ⊆Pp(X)(Pp(X)was introduced in Sect.1) isuniformlyLp-integrableif
Klim→∞sup
i∈I
{xX≥K} xpXdμi(x)=0.
Accordingly, a family of X-valued random variables (ξi)i∈I is uniformly Lp- integrableif(L(ξi))i∈I is uniformlyLp-integrable.
(2) Forμ ∈ P(X)andμn ∈ P(X),n ∈ N, we writeμn →w∗ μasn → ∞ifμn
converges weakly toμ, i.e., if limn→∞
X f(x)dμn(x)=
X f(x)dμ(x)for all f ∈Cb(X;R). Moreover, for a sequence ofX-valued random variables(ξn)n∈N
and an X-valued random variable ξ (possibly defined on different probability spaces) we writeξn→w∗ ξ asn→ ∞provided thatL(ξn)→w∗L(ξ)asn→ ∞.
We shall frequently use the following well-known result, which relatesLp-uniform integrability and convergence of moments:
Lemma 2.1 Let p ∈(0,∞), let X be a separable Banach space, and letμ,(μn)n∈N
be a sequence inPp(X)such thatμn→w∗ μ. Then, the following are equivalent:
(i)
Xxpdμn →
Xxpdμ. (ii) (μn)n∈Nis uniformlyLp-integrable.
Proof Apply, e.g., [19, Lemma 4.11 (in (5) lim sup can be replaced by sup)] to the random variablesξ, ξ1, ξ2, . . .whereξ = ζXpandξn= ζnXp, and whereL(ζ )=μ
andL(ζn)=μn.
Stochastic basis: We use the notion of astochastic basis(,F,P,F), which is a probability space(,F,P)equipped with a filtrationF =(Fn)n∈N0,F0 ⊆F1⊆
· · · ⊆F, and where we setF∞:=σ
n∈N0Fn . For measurable spaces(,F)and (S,S), andξ =(ξn)n∈Na sequence ofS-valued random variables on(,F), we let Fξ =(Fnξ)n∈N0 denote the natural filtration generated byξ, i.e.,F0ξ := {∅, }and Fnξ :=σ (ξ1, . . . , ξn)forn∈N, andF∞ξ :=σ(ξn :n∈N).
2.2 Stochastic Kernels
We provide some details for regular versions of conditional probabilities we shall use later.
Definition 2.2 LetXbe a separable Banach space and(S, )a measurable space. A mappingκ: S×B(X) → [0,1]is a/B(X)-measurable kernel if and only if the following two conditions hold:
(i) For allω∈S, it holds thatκ[ω,·] ∈P(X).
(ii) For allB ∈B(X), the mapω→κ[ω,B]is/B(R)-measurable.
Remark 2.3 Let the space(S, )be equipped with a probability measurePand let ⊆B(X)be a countableπ-system that generatesB(X). For two kernelsκ, κ:S → P(X), the following assertions are equivalent:
(i) κ[ω,B] =κ[ω,B]forP-almost allω∈S, for allB∈. (ii) κ[ω,·] =κ[ω,·]forP-almost allω∈S.
We need the existence of kernels describing conditional probabilities:
Theorem 2.4 [19, Theorem 6.3]Let X be a separable Banach space, (,F,P)a probability space,G⊆Fa sub-σ-algebra, and letξ :→X be a random variable.
Then, there is aG/B(P(X))-measurable kernelκ: →P(X)satisfying κ[·,B] =P(ξ∈ B|G) a.s.
for all B∈B(X). Ifκ:→P(X)is another kernel with this property, thenκ=κ a.s.
We refer toκas aregular conditional probability kernelforL(ξ|G).
2.3 Decoupling
We briefly recall the concept of decoupled tangent sequences as introduced by Kwapie´n and Woyczy´nski [24]. For more details, we refer to [10,25] and the references therein.
Definition 2.5 Let X be a separable Banach space, let (,F,P, (Fn)n∈N0) be a stochastic basis, and let(dn)n∈Nbe an(Fn)n∈N-adapted sequence ofX-valued ran- dom variables on(,F,P). A sequence ofX-valued and(Fn)n∈N-adapted random variables(en)n∈Non(,F,P)is called an(Fn)n∈N0-decoupled tangent sequenceof (dn)n∈Nprovided there exists aσ-algebraH⊆F satisfyingσ ((dn)n∈N)⊆Hsuch that the following two conditions are satisfied:
(i) Tangency: For alln∈Nand allB∈B(X), one has
P(dn∈ B|Fn−1)=P(en∈ B|Fn−1)=P(en∈ B|H) a.s.
(ii) Conditional independence: For allN ∈NandB1, . . . ,BN ∈B(X)one has P(e1∈B1, . . . ,eN ∈BN|H)=P(e1∈ B1|H) . . .P(eN∈ BN|H) a.s.
A construction of a decoupled tangent sequence is presented in [25, Section 4.3].
Example 2.6 Let(,F,P, (Fn)n∈N0)be a stochastic basis,(ϕn)n∈Nand(ϕn)n∈Ntwo independent and identically distributed sequences of independent,R-valued random variables such that ϕn andϕn are Fn-measurable and independent of Fn−1for all n ∈N, and let(vn)n∈N0 be an(Fn)n∈N0-adapted sequence of X-valued random vari- ables independent of(ϕn)n∈N. Then,(ϕnvn−1)n∈Nis an(Fn)n∈N0-decoupled tangent sequence of(ϕnvn−1)n∈N, where one may take
H:=σ((ϕn)n∈N, (vn)n∈N0).
Similarly, (ϕn)n∈N and(ϕn)n∈N could be X-valued random variables and(vn)n∈N0
R-valued.
3 A Factorization for Regular Conditional Probabilities
Theorem3.1 extends [19, Lemma 3.22] and can be viewed as a strong version of Montgomery-Smith’s distributional result [29, Theorem 2.1]. Theorem3.1is used to prove Theorem4.3, where it yields a refined argument for the existence of a decoupled tangent sequence. In this sense, it also contributes to [24] (cf. [10, Proposition 6.1.5]).
Theorem 3.1 Let (,F,P) be a probability space, G ⊆ F be a σ-algebra, let d ∈ L0(F;R)satisfy d() ⊆ [0,1), and letκ: ×B([0,1)) → [0,1]be a reg- ular conditional probability kernel forL(d|G). Let(,¯ F,¯ P)¯ :=(×(0,1],F⊗ B((0,1]),P⊗λ), whereλis the Lebesgue measure onB((0,1]). Set[0,0):= ∅and
define H: ¯→ [0,1], d0:× [0,1] → [0,1]by
H(ω,s):=κ[ω,[0,d(ω))] +sκ[ω,{d(ω)}], (8) d0(ω,h):=inf{x∈ [0,1]:κ[ω,[0,x]] ≥h}. (9) Then,
(i) H isF/B([0,¯ 1])-measurable, independent ofG⊗{∅, (0,1]}, and uniformly[0,1]
distributed,
(ii) d0isG⊗B([0,1])/B([0,1])-measurable, and
(iii) there is anN ∈ F withP(N) = 0such that d0(ω,H(ω,s)) = d(ω) for all (ω,s)∈(\N)×(0,1].
Before we prove this theorem, let us comment on Item (i). There are two extreme cases. The first one isG := {∅, }. In this case, we get that κ[ω;(−∞,x]]is the distribution function of the law ofd and here it is known that the distribution ofHis the uniform distribution on[0,1]. The other extreme case isG=Fand here we can takeκ[ω,B] := 1{d(ω)∈B} which implies thatH(ω,s) =s. Our result interpolates between these two extreme cases.
Proof of Theorem3.1 (i) For all n ∈ N and ∈ {1, . . . ,2n}, let An, := [(− 1)2−n, 2−n). DefineHn: ¯→ [0,1]by
Hn(ω,s):=
2n
=1
1{d∈An,}(ω)
κ[ω,[0, (−1)2−n)] +sκ[ω,An,] ,
so that for all(ω,s)∈ ¯it holds that
|Hn(ω,s)−H(ω,s)| ≤
2n
=1
1{d∈An,}(ω)(1+s)κ[ω,An,\{d(ω)}] →0 as n→ ∞.
TheHnareF/B([0,¯ 1])-measurable, soHis as point-wise limit (the measurability ofHcan be seen directly as well). Letn ∈N,G∈GandB ∈B([0,1]). Because b1
01{a+sb∈B}ds=λ(B∩ [a,a+b])fora,b∈ [0,1]witha+b≤1 (whereλ denotes the Lebesgue measure), we get
P((¯ G×(0,1])∩ {Hn∈B})=
2n
=1
Gλ B∩
κ[·,[0, (−1)2−n)], κ[·,[0, 2−n)] dP
=P(G)·λ(B).
This proves that Hn is uniformly [0,1] distributed and independent of G ⊗ {∅, (0,1]}for alln ∈ N. This completes the proof of (i), as H is the point-wise limit of(Hn)n∈N (twoR-valued random variablesξ1, ξ2are independent if and only if for all f,g∈Cb(R)it holds thatE[f(ξ1)g(ξ2)] =E[f(ξ1)]E[f(ξ2)]).
(ii) For allx∈ [0,1], note that
{d0≤x} = {(ω,h)∈× [0,1]:κ[ω,[0,x]] −h ≥0} ∈G⊗B([0,1]). (10) (iii) It follows from (10) and the definition ofHthat we have, for allx∈ [0,1], that
{(ω,s)∈ ¯:d0(ω,H(ω,s))≤x}
= {(ω,s)∈ ¯:κ[ω,[0,x]] ≥κ[ω,[0,d(ω))] +sκ[ω,{d(ω)}]}
can be written asBx×(0,1]for some uniqueBx ∈Fand that we have that Bx×(0,1] ⊇ {(ω,s)∈ ¯:d(ω)≤x} =:Cx×(0,1].
On the other hand from the fact that the image measure of the map(ω,s) → (ω,H(ω,s))as a map frominto× [0,1]equalsP⊗λ, we obtain, for all x∈ [0,1], that
P(Bx)=P(Bx×(0,1])=E 1
0
1{d0(ω,h)≤x}dhdP(ω)
=E 1
0
1{κ[ω,[0,x]]≥h}dhdP(ω)=Eκ[·,[0,x]] =P(Cx).
It follows thatP(Bx\Cx) =0 for all x ∈ [0,1]. LetN := ∪q∈Q∩[0,1)(Bq\Cq) so that P(N) = 0. Then, observing that Bx = ∩q∈Q∩[x,1)Bq (this follows from Bx ×(0,1] = {d0(·,H(·,·)) ≤ x} = ∩q∈Q∩[x,1){d0(·,H(·,·)) ≤ q} = (∩q∈Q∩[x,1)Bq)×(0,1] and the uniqueness of the sets Br, r ∈ [0,1]) and Cx = ∩q∈Q∩[x,1)Cqfor allx ∈ [0,1), we have for all(ω,s)∈(\N)×(0,1]
thatd0(ω,H(ω,s))=d(ω).
Corollary 3.2 Let(,F,P)be a probability space,G⊆Faσ-algebra, X a separable Banach space, d ∈L0(F;X). Let(,¯ F,¯ P)¯ :=(×(0,1],F⊗B((0,1]),P⊗λ), whereλis the Lebesgue measure on B((0,1]). Then, there exist random variables H: ¯→ [0,1], d0: × [0,1] →X such that
(i) H is uniformly[0,1]distributed and independent ofG⊗ {∅,[0,1]}, (ii) d0isG⊗B([0,1])/B(X)-measurable, and
(iii) there is anN ∈ F withP(N) = 0such that d(ω) = d0(ω,H(ω,s)) for all (ω,s)∈(\N)×(0,1].
Proof This is an immediate consequence of Theorem3.1and the fact that that X is Borel-isomorphic to[0,1), see, e.g., [11, Theorem 13.1.1].
4 A Reduction of General Decoupling to Haar-Type Series
Before we turn to our main Theorem4.3, we discuss some properties of the extension ofP toPp-ext(see Definition1.3). For this, we need
Lemma 4.1 Assume a metric space(M,d)and a continuous map∗ : M×M → M with(x∗y)∗z=x∗(y∗z)for x,y,z∈ M. Let∅ =P ⊆M and
P∗:=cld({x1∗ · · · ∗xL :x1, . . . ,xL ∈P,L ∈N})
where the closure on the right side is taken with respect to d. Then, one has(P∗)∗=P∗ andP∗is the smallest d-closed setQwithQ⊇Pandμ∗ν∈Qfor allμ, ν∈Q.
Proof The equality (P∗)∗ = P∗ follows from the continuity of∗ and a standard diagonalization procedure. This also implies thatμ∗ν∈P∗for allμ, ν∈P∗. Now let us assume a setQas in the assertion. Then,x1∗· · ·∗xL ∈Qfor allx1, . . . ,xL ∈P.
AsQis closed we deduceP∗⊆Q.
Lemma4.2reveals some basic properties ofPp-ext. To this end, forp∈(0,∞)we introduce onPp(X)⊆P(X)the metric
dp(μ, ν):=d0(μ, ν)+
X
xpdμ(x)−
X
xpdν(x)
(11) whered0is a fixed metric onP(X)that metricizes thew∗-convergence, see for example [30, Theorem II.6.2].
Lemma 4.2 Let X be a separable Banach space, p∈(0,∞), and letP ⊆Pp(X)be non-empty. Then,
(i) (Pp-ext)p-ext=Pp-extand
(ii) Pp-extis the smallest dp-closed setQwithQ⊇Pandμ∗ν∈Qfor allμ, ν∈Q.
Proof We will verify that the convolution is continuous with respect todp, the assertion then follows from Lemma4.1. To verify this, we letμ, ν, μn, νn ∈ Pp(X),n ∈ N, such that limn→∞dp(μ, μn)=limn→∞dp(ν, νn)=0. It is know thatμn∗νn →w∗
μ∗νas well (one can use [19, Theorem 4.30]). Because for K > 0 we have, with h(x,y):=max{xX,yX},
{x+yX≥K}x+ypXdμn(x)dνn(y)≤2p
{h(x,y)≥K/2}hp(x,y)dμn(x)dνn(y)
≤2p
{xX≥K/2}xpXdμn(x)+2p
{yX≥K/2}yXpdνn(y),
cf. [1, p. 217], by Lemma2.1we get thatμn∗νnis uniformlyLp-integrable and thus, again by Lemma2.1, we obtain the convergence of the p-th moments.
Now we formulate the main result of this section. See Definition1.2for the definition ofAp(,F;X,Pp-ext).
Theorem 4.3 Let X be a separable Banach space and letλ∈C(X×X;R),λ∈, for an arbitrary non-empty index set. Suppose that there exist a p ∈ (0,∞)and constants Cλ∈(0,∞),λ∈, such that
|λ(x,y)| ≤Cλ(1+ xXp+ ypX)
for all(x,y)∈ X×X , and letP ⊆Pp(X)withδ0∈P. Then, the following assertions are equivalent:
(i) For every stochastic basis(,F,P,F)with F = (Fn)n∈N0 and every finitely supported2(dn)n∈N∈Ap(,F;X,Pp-ext)it holds that
λ∈supEλ
∞
n=1
dn, ∞ n=1
en
≤0, (12)
provided that(en)n∈Nis anF-decoupled tangent sequence of(dn)n∈N.
(ii) For every probability space(,F,P), every finitely supported sequence of inde- pendent random variablesϕ=(ϕn)n∈NinLp(P;X)satisfyingL(ϕn)∈Pfor all n∈N, and every An∈Fnϕ, n∈N0, it holds that
λ∈supEλ ∞
n=1
ϕn1An−1,∞
n=1
ϕn1An−1
≤0, (13)
where(ϕn)n∈Nis an independent copy of(ϕn)n∈N.
Proof Proof of (i)⇒(ii). In (ii), we have(1An−1ϕn)n∈N ∈Ap(,Fϕ,ϕ;X,P)with Fϕ,ϕ=(Fnϕ,ϕ)n∈N0 whereF0ϕ,ϕ := {∅, }andFnϕ,ϕ :=σ (ϕ1, ϕ1, . . . , ϕn, ϕn)for n∈N. Therefore, the implication (ii)⇒(i) follows by Example2.6.
The implication (ii)⇒(i) will be proved in Appendix A. Theorem4.3allows us to prove Theorem1.4from Sect.1:
Proof of Theorem1.4 The statement for generalfollows from the case= {λ0}so that we may assume this case and let := λ0 and := . By the lower and upper semi-continuity, we can find continuous , : [0,∞) → [0,∞), ∈ N, such that (ξ) ↑ (ξ)andC(1+ |ξ|p)≥ (ξ)↓ (ξ)for allξ ∈ [0,∞). Next, we set(x,y):= (SxY)− (T yZ),∈N. Then, the monotone convergence theorem implies that for allξ, η∈Lp(X)the conditions sup∈NE(ξ, η)≤0 and E
(SξY)− (TηZ)
≤0 are equivalent.
Let us list some common choices ofPin the setting of decoupling inequalities. To do so, we exploit the following lemma:
2 Recall that this means that there is anN∈Nwithdn≡0 forn>N.
Lemma 4.4 Let C,p∈(0,∞), let X be a separable Banach space, let(,F,P)be a probability space, and let∈C(X;R)be such that
|(x)| ≤C(1+ xpX) (14)
for all x ∈ X . Assumeξ, ξn ∈ Lp(P;X), n∈ N, such thatξn →w∗ ξ as n → ∞and that(ξn)n∈Nis uniformlyLp-integrable. Then,
nlim→∞E(ξn)=E(ξ). (15) Proof It follows from the uniformLp-integrability of(ξn)n∈Nand estimate (14) that ((ξn))n∈Nis uniformlyL1-integrable. Moreover, note that(ξn)→w∗ (ξ)asn →
∞, so that we may apply Lemma2.1for p=1.
Note that ifξn → ξ inLp(P;X), ξn, ξ ∈ Lp(P;X), then the assumptions on (ξn)n∈Nandξ in Lemma4.4are satisfied (see [19, Lemma 4.7]).
Example 4.5 (Adapted processes) If p ∈ (0,∞)andP =Pp(X), thenPp-ext = P by Lemma4.4 and the space Ap(,F;X,P) consists of all (Fn)n∈N-adapted processes(dn)n∈NinLp(P;X).
Example 4.6 (Lp-martingales) If p∈ [1,∞)andPconsists of all mean zero mea- sures inPp(X), thenPp-ext =P by Lemma4.4(one can test with(x):= x,a, where a ∈ X and X is the norm-dual) and Ap(,F;X,P) consists of all Lp- integrableF-martingale difference sequences.
Example 4.7 (Conditionally symmetric adapted processes) Suppose p ∈ (0,∞)andPconsists of all symmetric measures inPp(X). As a measureμ∈P(X) is symmetric if and only if for all f ∈ Cb(X;R)it holds that
X f(x)dμ(x) =
X f(−x)dμ(x), it follows thatPp-ext =P. Moreover, the setAp(,F;X,P)con- sists of all sequences of X-valued (Fn)n∈N-adapted sequences of random variables (dn)n∈N such that dn ∈ Lp(P;X)anddn isFn−1-conditionally symmetric for all n ∈N, i.e., for alln∈Nand allB ∈B(X)it holds thatP(dn ∈ B|Fn−1)=P(dn ∈
−B|Fn−1)a.s.
Example 4.8 (One- dimensional laws) If p∈(0,∞),∅ =P0⊆Pp(R), and P =P(P0,X):=
μ∈Pp(X): ∃μ0∈P0,x∈ X: μ(·)=μ0
{r ∈R:r x ∈ ·} ,
then an X-valued random variableϕ satisfiesL(ϕ) ∈ P if and only if there exists anx ∈ X and aR-valued random variableϕ0such thatϕ = xϕ0andL(ϕ0)∈ P0. Moreover,Ap(,F;X,P)contains all sequences of the form(ϕnvn−1)n∈N where (ϕn)n∈Nis an(Fn)n∈N-adapted sequence ofR-valued random variables such thatϕn
is independent ofFn−1andL(ϕn)∈ P0, andvn−1 ∈ Lp(Fn−1;X)for alln ∈ N.
Finally, it holds thatP((P0)p-ext,X)⊆Pp-ext.
