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Infinite order decoupling of random chaoses in Banach space

arXiv:math/9211212v1 [math.FA] 18 Nov 19921Infinite order decoupling of random chaoses in Banach spaceJerzy Szulga 1We prove a number of decoupling inequalities for nonhomogeneous random polynomialswith coefficients in Banach space. Degrees of homogeneous components enter into com-parison as exponents of multipliers of terms of certain Poincar´e-type polynomials.

It turnsout that the fulfillment of most of types of decoupling inequalities may depend on thegeometry of Banach space.KEY WORDS: decoupling principle, symmetric tensor products, random polynomials,multiple random series, multiple stochastic integrals, random multilinear forms, randomchaos, Gaussian chaos, Rademacher chaos, stable chaos, multiple Wiener integral, multiplestable integral, Mazur-Orlicz polarization formula, symmetrization, Banach space, Banachlattice, Krivine’s type, rearrengement invariant space, convexity, Orlicz space, Rademachersequence, Gaussian law, Walsh polynomials, empirical measure.Contents1INTRODUCTION22RANDOM TENSOR PRODUCTS42.1Notation . .

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. .42.2Domination of random polynomials .

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.62.3Lower and upper decoupling inequalities . .

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.93SLICING AND DECOUPLING103.1Slicing . .

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.133.4Reduction to Rademacher chaoses . .

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.173.6.1Rademacher versus Gaussian chaoses. .

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. .194STABLE CHAOSES214.1Auxiliary definitions and inequalities.

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. .225CONCLUDING REMARKS255.1Multiple stochastic integrals .

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.26AMS (1980) Classification Primary: 60B11, 60H07, 15A69 Secondary: 46M05 , 46E30, 60E15, 62H05, 62G30,42A55. 15A52, 26D151Dept.

of Mathematics, Auburn University, Auburn, AL 36849

INFINITE ORDER DECOUPLING21INTRODUCTIONThe concept of decoupling stems from the martingale theory (cf. the survey [Bur86]).

First de-coupling inequalities for multiple random forms were proved in [MT86, MT87, Kwa87], and manyvariants have been published since the time when the above papers were published (to name but asample, cf. [DA87, Hit88, Zin86, NP87, dlPn90, KS89, Szu91a, RW86, RST91, Szu92], and furtherreferences in there).

So far, all known results have involved a two-sided estimate of Lp-(or Orlicz)norms of suitable k-homogeneous multilinear forms (or multiple integrals), where k is arbitrary butfixed. Decoupling constants are degree-dependent and escape to infinity.

If the degree increases,the strength of the decoupling principle seems to decline.In this paper we will show how to overcome this deficiency (one cannot expect that decouplingconstants remain bounded). Our approach is based on a suitable normalization of polynomialsQ(X; t) = Q0 + tQ1(X) + .

. .

+ tnQn(X),where X = [Xij] is a matrix of random variables, with independent rows, and Qk is a Banachspace-valued homogeneous polynomial of degree k (a k-linear form). Under suitable integrabilityand symmetry assumptions the presented decoupling principle compares norms (e.g., Orlicz norms)of Q(X, t) and of the polynomial Q(X),∥Q(X)∥∼∥Q(X)∥where the matrix X is a “decoupled” version of X, i.e., the columns of X are replaced by theirindependent copies.

For example, on the real line, for Rademacher or standard Gaussian randomvariables, we check directly thatE|Xk≥0Qk|2 = E|Xk≥0Qk|2(1.1)whereQk =1√k!Xi1,...,ik,ij̸=ij′fk(i1, . .

. , ik)X1i1 · · · Xkik,if fk is symmetric, orXi1<...

. .

, ik)X1i1 · · · Xkik,if fk is tetrahedraland Qk follows the same pattern, respectively, but without the multiplier 1/√k! in the symmetriccase.

Also, L2-norm can be replaced by Lp-norm, p > 1, at cost of multiplying each k-homogeneouspolynomial by a constant ckp.Degrees of specific components will enter into formulas as exponents of certain multipliers. Wereplace, so to speak, external constants by internal constants or, more precisely, by sequences ofconstants.

Asymptotic behavior of such sequences is of interest, and the exponential growth ismost desirable. If columns of X are identical, we call Q(X) a “ random chaos”, and when theyare independent (desirably – identically distributed), a “decoupled random chaos”.

A tetrahedraldecoupled chaos can be written as a sort of lacunary chaos, by a monotone (non-unique, in general)change of ordering on all tetrahedra, from the coordinatewise ordering to a linear ordering. Ananalogous procedure for symmetric chaoses is possible “locally”, i.e., for a fixed and finite order,and when only a finite number of random variables is involved.

INFINITE ORDER DECOUPLING3Thus, decoupling inequalities can be viewed as embedding-projection theorems. In the infiniteorder decoupling we require that projections are contractions.

We will observe new phenomena,absent in the homogeneous (or finite order) case. First of all, in the infinite order decoupling, twotypes of inequalities (“the lower decoupling” – domination by the chaos, and “the upper decoupling”– domination of the chaos) determine two distinct problems.

Already homogeneous tetrahedral andsymmetric chaoses behave differently (cf. Bourgain’s example in [MT87]).A robust lower decoupling inequality is satisfied, i.e., the inequality is fulfilled in any Banachspace and for an arbitrary symmetric integrable polynomial chaos.

At the same time, the fulfillmentof a robust upper decoupling inequality is still uncertain. In order to study the upper decouplingprinciple, we introduce a class of Banach spaces that are characterized by a certain inequalityinvolving linear forms in independent random variables with vector coefficients (in some aspects,the property is similar to classical properties of Banach spaces, like Rademacher type and cotype,smoothness or convexity of norm, etc.The main feature of the new class of spaces is that random polynomials admit a “horizontalslicing”, reducing the study to that of sums of independent random variables.

In the introducedclass of Banach spaces a sign-randomized upper decoupling inequality holds. The class of spacesallowing slicing of random polynomials is, unfortunately, geometry-dependent, and very fragile.It is sensitive to an equivalent renorming (just an addition of a two-dimensional normed spacewith ℓ∞- or ℓ1- norm terminates the property), in contrast to the homogeneous case.

Therefore,a positive result will always require the existence of a suitable equivalent norm (cf. “smooth” vs.“smoothable”, or “convex” vs. “convexifiable”).

The family of Banach spaces, satisfying the slicingrequirements, contains Banach lattices of finite cotype and sufficiently convex norm. This classturns out to be suitable even for the tetrahedral lower decoupling.In Section 2 we introduce the nonhomogeneous tensor product notation, define the domination,and derive some basic relations.

We refer to [PA91] and the literature included there for a treatmentof Gaussian symmetric tensors.The comparison will be given in terms of the aforementionedPoincar´e -type polynomials or, equivalently, in terms of a semigroup of contractions, associatedwith a random polynomial.The new results are gathered in Sections 3 and 4. The employed techniques in the nonho-mogeneous case are different from techniques related to the homogeneous case.

First of all, theapplicability of conditional expectations is limited. Secondly, the type of domination forbids use ofexternal constants.

We will use a “slicing technique”, which reduces the study to a case of certainsums of random variables.In Section 5 we will indicate some directions in a further study of decoupling inequalities, andshow that many assertions can be directly obtained from results of this paper. For example, onecan formulate decoupling inequalities in the language of infinite order stochastic multiple integrals.We will also collect some observations that do not fit into the main line of the paper, although maybe of some interest.Let us point out that a widely understood convexity is the principal feature implicit in mostof applied techniques.

This includes the setting of Banach spaces and existence of moments (in-tegrability) of involved random variables. One can find a number of decoupling principles in the

INFINITE ORDER DECOUPLING4literature, where the convexity is of no concern, and the focus is on positivity, not on symmetry.However, most of the known results have been obtained so far at the cost of the limitation to thereal line (see [KS89] for a discussion on the latter topic). In some special cases, e.g., for Gaussianhomogeneous polynomials, a decoupling principle applies to probabilities P(· /∈K), where K isa convex symmetric set in a Banach space [Kwa87]; the aforementioned paper [DA87] deals withp-stable random variables and spaces Lr, 0 < r < p, etc.2RANDOM TENSOR PRODUCTS2.1NotationThroughout the paper ε = (εi) denotes a Rademacher sequence, that is, εi are independent randomvariables taking values ±1 with probability 1/2.Walsh functions are products of Rademacherrandom variables.

We will denote by X a sequence, and by X = [X1, . .

. , Xn] a matrix, of realrandom variables.

(E, ∥·∥E) denotes a real Banach space. By (L, ∥·∥L) we denote a rearrangementinvariant Banach space of integrable random variables, L ⊂L1(P), defined on a probability space(Ω, F, P), rich enough to carry independent sequences, and with a separable σ-field.

In fact, wewill use only specific properties of L, ensured by the above restrictions. (L)Conditional expectations are contractions acting in LL(E) denotes the Banach space of E-valued random variables (i.e., strongly measurable mappingsfrom Ωinto E) whose norms belong to L a.s., and let ∥θ∥L(E) = ∥∥θ∥E ∥L.

Whenever it causesno ambiguity, we omit the subscript.Let N = { 1, 2, . .

. } be the set of natural numbers, and N = N ∪{ 0 }.

For m, n ∈N, put[m, n] = { m, m + 1, . .

. , n }.

Throughout the paper, the bold-face Greek characters α, β, . .

., etc.,will denote subsets of N, identified with { 0, 1 }-valued sequences:N ⊃α ←→α = (α1, α2, . .

.) ∈{ 0, 1 }N .Denote |α| = #α = Pi αi, and α′ = (1 −α1, 1 −α2, .

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). The following convention will be veryhelpful.

Let E be a nonvoid set. Suppose that 0, 1 ∈E.

Define two operations { 0, 1 } × E →E:0x = 0,1x = xandx0 = 1,x1 = x (by convention, 00 = 1).If ∗is any operation in an abstract set Z, then we will use the same for functions taking values inZ. In particular, if ∗: E × Y →Z, then, by writing ∗: EN × Y N →ZN , we understand theaction of ∗coordinatewise.

For example, (X ∗y)i = xi ∗yi, i ∈N, X = (xi), y = (yi). Also, forX ∈EN and α ∈{ 0, 1 }N , αX = (αixi) and Xα = (xαii ).

In section 3.4, the term SX, where Sand X are (N ×n) matrices, according to our convention, will denote a new (N ×n) matrix, whoseentries are products of entries of S and X.Identifying α and a constant sequence (α, α, . .

. ), we have then αX = (αx1, αx2, .

. .

).Forα ⊂N, we identify Nα and the subset { αi : i ∈N } of NN (the empty set is identified with{ (0, 0, 0, . .

.) }.

We will consider functions f = (fα : α ∈{ 0, 1 }N ), where fα : Nα →E (f∅is

INFINITE ORDER DECOUPLING5an element of E, and, if necessary, f may be identified with suitable functions f : 2N × NN →E, f(α, i) = fα(αi)). For definitness, we request that all functions f = (fα) under considerationhave a finite support (i.e.

f(α, i) = 0 for all but finitely many α and i.In this paper we will see an abundance of summation, averaging, and integration on severallevels, in order to diminish the notational burden, we introduce a variety of summing brackets (ofcourse, we might replace all following brackets by just one but, by doing so, we would cause aserious visual dissonance). DefineDfαE=DfαEα =Xαifα(αi)andDDfEE=XαDfαEα.and, for x = (xi) ∈EN ,(( x )) =XixiAll functions f = (fα), appearing in the sequel are assumed to vanish on diagonals, i.e., fα(αi) = 0if at least two nonzero-arguments (αi)k are equal.

Define the symmetrizator bf, which unifies valuesof functions fα on distinct tetrahedra, by the formulacfα =1|α|!Xσσfα,where the sum is taken over all permutations σ of α, and σfα = fα ◦σ. If bf = f, then thefunctions is called symmetric.

Call a function f tetrahedral, if it may take nonzero values only onthe main tetrahedron: α = { 1, . .

. , |α| }, i1 < .

. .

< i|α| .The random matrix X = (Xij : i ∈[1, n], j ∈N) ∈R[1,n]×N, considered before, can bewritten as X = [X1, X2, . .

.] ∈(RN )[1,n], where Xi = (Xij : j ∈N).

Define a tensor productX⊗= (X⊗α) on R[1,n]×N by the formulaX⊗α(αi) = Xα11 i1 · · · Xαnn in,and the symmetric tensor product, by Xb⊗= dX⊗. By convention, a single sequence X can be viewedas a matrix (the sequence) [X, .

. .

, X]. Whence the tensor products X⊗and its symmetrizationsare well defined.

One can consider other type of symmetrizators U and the induced symmetrictensor products U⊗(see Section 5).The Mazur-Orlicz polarization formula can be written as follows (it is fulfilled in any commu-tative algebra).Xb⊗α =1k!Xβ⊂α(−1)k−|β|(( βX ))⊗α,=2−n 1k!Xβ⊂[1,n](−1)k−|βα|(( 2βαX ))⊗α,(2.1)where |α| = k. Endowing 2[1,n] with the uniform probability, the functionsri(β) = (−1)βi,β ∈2[1,n],i = 1, . .

. , n

INFINITE ORDER DECOUPLING6are representations of the first n Rademacher random variables. We define Walsh functions asproducts of Rademacher functions:wα(β) df=Yi∈αri(β) = (−1)⊗αβ,β ∈2[1,n].

(2.2)Notice the presence of Walsh functions in the Mazur-Orlicz polarization formula.By S0k = S0k(X) denote the σ-field generated by the familyngα(Xα) : α ⊂[1, n], |α| = k, gα = ˆgα, g : Rk →Ro,and let Sk = Sk(X) be the σ-field generated by S01 ∪. .

. ∪S0k, and S be spanned by Sk Sk.Proposition 2.1 Let [X, X] have independent rows and i.i.d columns.Let gα be a functionvanishing on diagonals and equal 1 offdiagonals.

Then the following equalities hold.gα(( βαX ))⊗α = E[ gα(( αX ))⊗α | Xβ ];(2.3)gα(X + X′)⊗α = E[ gα(2X)⊗α | X + X′ ],(2.4)where X′ is an independent copy of X;gαX⊗α = E[gα (X + X′)⊗α | X ](2.5)provided EX′ = 0, and X and X′ are independent;DDfXb⊗EE=DDfE[X⊗|S(X)]EE= E[DDfX⊗EE|S(X)](2.6)Proof.Conditions (2.3), (2.4), and (2.5) follow immediately. In order to see the fulfillment ofthe remaining condition, it suffices to implement the following simple rule.

For two σ-fields F1, F2,and random variables Zm, m = 1, . .

. , M, if E[ Zm | F1 ] = E[ Z1 | F1 ], and Pm Zm is F1-measurable,then E[ Zi | σ(F1∪F2] = Pm Zi/M.

This rule allows us to reduce each situation to the homogeneouscase, and then the proof is direct.2.2Domination of random polynomialsThe decoupling principle for nonhomogeneous polynomials will be defined in terms of a more generalconcept of domination, applied to certain Poincar´e-type polynomials (e.g., cf. various variants ofhypercontraction [Gro73, KS88, KS91] or Malliavin’s calculus, cf.

[Sug88]).First, we decide a setting of domination.Let (E, ∥· ∥) be a Banach space, L ⊂L0(P)be an algebra of real random variables, endowed with a positive functional ϕ, and L(E) = ξ ∈L0(E) : ∥ξ∥∈L a.s. Define the functional Φ(ξ) = ϕ( ∥ξ∥).Usual examples consist of Lp-norms or quasi-norms, 0 ≤p ≤∞, Orlicz (or more generalrearrangement invariant) norms, distribution tails φ(θ; t) = P(|θ| > t), etc. (cf.

[KW92, Chapters3, 5] for more examples).Consider E-valued random polynomialsDDfX⊗EE, where f = (fα : α ⊂N,fα : Nα →E,fα ≡0 for all but finitely many finite sets α, and f belong to a certain category F of functions,

INFINITE ORDER DECOUPLING7realized on the class of Banach spaces. The role of a constant is to be played by a real valuedfunction c = (cα) : 2N × N →R ∈U, where U is realized on R. The system S = (E, L, ϕ, F)becomes the setting of domination.Definition.Say that X is dominated by X′ in the setting S with a constant c (X ⪯S,c X′,in short), if, for every n ≥0,Φ(DDfX⊗EE) ≤Φ(DDfcX′⊗EE),f ∈F.If the constant is of the form cα = c|α|, where c is a positive number we will say that the X isexponentially dominated by X′.It is easy to see that linear forms in zero mean integrable random variables are comparablewith their symmetrized counterparts.

A similar property is enjoyed by random chaoses. Let Φ bea positive convex functional defined on L1(E) turning conditional expectations into contractions,i.e.,Φ(E[X|F]) ≤Φ(X)(2.7)(for example, Φ(X) = ∥X∥L(E), where L is a rearrangement invariant space of real random vari-ables, or Φ(X) = Eϕ(∥X∥E), where ϕ is an increasing convex function).Proposition 2.2 Let X and X′ be independent identically distributed matrices with independentrows and interchangeable columns (this includes the case of matrices with identical columns).

letf = (fα) be an E-valued function vanishing on diagonals.1. ThenΦ(DDf(X −EX)⊗EE) ≤Φ(DDf(X −X′)⊗EE).(2.8)2.

If f is symmetric, and X has independent columns, then there exists a Walsh system wα,independent of X and X′, such thatΦ(DDf(X −X′)⊗EE) ≤Φ(DDfw(2X)⊗EE). (2.9)where (fw)α(i) = fα(i)wα.3.

If f is symmetric, and EX = 0, thenΦ(DDf(X −X′)⊗EE) ≤Φ(DDfw(4X)⊗EE). (2.10)Proof.Inequality (2.8) follows by convexity and contractivity of conditional expectations.

INFINITE ORDER DECOUPLING8Inequality (2.9) follows from the estimatesΦ(DDfX −X′⊗EE)= Φ(Xα⊂[1,n]Dfα(X −X′)⊗α E)= Φ(Xα⊂[1,n]DfαXβ≤αX⊗β ⊗(−X′)⊗(α−β) E)= Φ(Xα⊂[1,n]Dfα2|α|−nXβ⊂[1,n]X⊗βα ⊗(−X′)⊗(αβ′) E)≤12nXβ⊂[1,n]Φ(Xα⊂[1,n]Dfα2|α|X⊗βα ⊗(X′)⊗(αβ′)(−1)⊗(αβ′) E)=12nXβ⊂[1,n]Φ(Xα⊂[1,n]wα(β)Dfα(2X)⊗α E)= Φ(Xα⊂[1,n]Dfαwα(2X)⊗α E).The proof of (2.10) is similar.However, the assumption EX = 0 is essential in the followingargument. We haveΦ(DDf(X −X′)⊗EE) = Φ(Xα⊂[1,n]Dfα(X −X′)⊗α E)= Φ(Xα⊂[1,n]DfαXβ≤αX⊗β ⊗(−X′)⊗(α−β) E)= Φ(Xα⊂[1,n]Dfα2|α|−nXβ⊂[1,n]X⊗βα ⊗(−X′)⊗(αβ′) E)≤12nXβ⊂[1,n]Φ(Xα⊂[1,n]Dfα2|α|X⊗βα ⊗(X′)⊗(αβ′)(−1)⊗(αβ′) E)≤12nXβ⊂[1,n]Φ(Xα⊂[1,n]Dfα2|α|(X + X′)⊗βα ⊗(X + X′)⊗(αβ′)(−1)⊗(αβ′) E)= 12nXβ⊂[1,n]Φ(Xα⊂[1,n]wα(β)2|α|DfαE[(2X)⊗α|X + X′]E)≤Φ(Xα⊂[1,n]Dfαwα(4X)⊗α E).The proof has been completed.The following contraction principle is well known in the one dimensional case (cf.

[Kah68] forthe real case, and [HJ74], for the vector case).Theorem 2.3 Let ϕ : R+ →R+ be a convex increasing function. Let X be a matrix of realsymmetric random variables with independent rows and either independent or identical columns.Then for every E-valued function f ∈FS (or FT ), and bounded real function g = (gα) withc = ∥g∥∞, of the form gk(i) = gk1(i1) · · · gkk(ik), we haveEϕ∥DDfgX⊗EE∥≤Eϕ∥DDf(cX)⊗EE∥

INFINITE ORDER DECOUPLING9Proof.In the case of a homogeneous chaos, i.e., when columns of the matrix X are identical,the result appeared in [Kwa87], while for nonhomogeneous chaoses, in [KS88].The case withindependent columns follows by a spreading argument. That is, there exists an enumeration offunctions f such that the polynomialDDfX⊗EEcan be written as a polynomialDDf ′X⊗EE.

Hence,we arrive in the previous situation.Say that tails of two random variable X and Y are comparable, if, for some constants K > 0and t0 ≥0P(|X| > t) ≤KP(|Y | > Kt)andP(|Y | > t) ≤KP(|X| > Kt),t ≥t0.Notice that, at cost of increasing the constant K, one may assume that the above estimates holdfor every t > 0. Thus, there exist probability spaces and copies X′, X′′ and Y ′, Y ′′ of X and Y ,respectively, such that |X′| ≤K′|Y ′| and |Y ′′| ≤K′|X′′| a.s.

In particular, the upper decouplinginequality is satisfied simultaneously for chaoses spanned by X and Y , provided components ofboth sequences are independent and have comparable tails.Corollary 2.4 Let X1 and X2 be matrices of real symmetric random variables with independentrows and independent or identical columns. Suppose that corresponding entries of both matriceshave comparable tails.

Then, for any symmetric or tetrahedral function f, polynomials in X1 andX2 are comparable, i.e., for any increasing function φ : R+ →R+,Eϕ∥DDfX⊗iEE∥≤Eϕ∥DDf(cXj)⊗EE∥,i, j ∈{1, 2}, for some constant c, depending on the tail domination constant K.2.3Lower and upper decoupling inequalitiesDefinition. Let X be a sequence of real independent random variables and X be a matrix whosecolumns are independent copies of X.

Let F be a class of functions fα = (fα), fα : Nα →E.Denote by UD = UD(E; Φ; F) (respectively, LD = LD(E; Φ; F)) the class of sequences X = (Xi)of independent random variables (more exactly, the class of product probability measures) suchthat that the upper decoupling inequality (respectively, the lower decoupling inequality) holds on F,i.e., there exists a constant c such that, for every n ∈N and f ∈F, one hasEΦ(DDfX⊗EE) ≤EΦ(DDf(cX⊗EE)(respectively,EΦ(DDf(X⊗EE) ≤EΦ(DDf(cX)⊗EE) ).If the considered sequences have components with the same probability distribution µ, we willsay that µ (or a random variable with the distribution µ) satisfies the upper (respectively, lower)decoupling inequality.The most important classes are FS, the class of symmetric functions, and FT , the class oftetrahedral functions (recall that we always assume that functions vanish on diagonals). One canconsider also other classes (cf.Section 5).More precisely, the decoupling introduced above isunderstood in the sense of the exponential domination.

Note that in most cases of interest that isa desired property. By the same token one can discuss the decoupling in a weaker sense (with a

INFINITE ORDER DECOUPLING10“constant” cα being not necessarily of the exponential type), but then both sides of decoupling,the lower and upper inequality, should be treated separately.Proposition 2.2 indicated that in case of insufficient symmetry it is necessary to randomizesigns of consecutive homogeneous components of a random chaos.In the proposition, such arandomization does not affect internal components of homogeneous polynomials. However, as willbe shown, frequently one needs random signs within each and every homogeneous term, and theintricacy of such a randomization may vary.

for example, one may use Walsh multipliers inducedeither by one sequence ε, or by a matrix [ε1, ε2, . .

.] with independent Rademacher columns, or,instead of Walsh functions, one may require plain Rademacher family, indexed by the multi-indexαi.

The latter sign-randomization is the only known way, so far, of extending Theorem 2.3 tofunctional multipliers g whose arguments are not separated (cf. [KS86]).3SLICING AND DECOUPLING3.1SlicingLet C be a class of finite random real sequences and N be an integer.

Say that an n × N randomreal matrix X is C-sliceable, if its rows are independent and belong to C. For α ⊂[1, N], denoteα∗= max α = max { i ∈[1, N] : αi = 1 } (max ∅df= 0).Lemma 3.1 Let E be a measurable vector space and Φ : E →R+ be a measurable function. LetC and eC be classes of finite random sequences such thatEΦ x +mXi=1ξixi!≤EΦ x +mXi=1eξixi!

(3.1)for every integer m, { xi } ⊂E, (ξi) ∈C, and (eξi) ∈eC. Let X and eX be C- and eC-sliceable n × Nrandom matrices, respectively, and f = (fα) : α ⊂[1, N]) be an E-valued symmetric function.ThenEΦ DDfX⊗EE ≤EΦ DDf eX⊗EE (3.2)Proof.The statement will be proved by induction with respect to n. Without loss of generality,we may assume that X and eX are independent and defined on a product space, and functionsfα(αi) vanish unless i1 < i2 < i3 .

. ..For n = 1, (3.2) coincides with (3.1).

Suppose that (3.2) holds for every C-sliceable matrix Xand every eC-sliceable matrix eX. We note the decomposition:fα(αi) = fα(αi)1I{ iα∗≤n−1 } + fα(αi)1I{ iα∗=n }Hence, denotingf (n)α (αi) = fα(αi)1I{ iα∗≤n },andefα\{α∗}(αi) = ef (n−1)α\{α∗}1I{ iα∗=n }

INFINITE ORDER DECOUPLING11we haveDDXα EE=Xα⊂[1,N]DfαX⊗α E=Xα⊂[1,N]Df (n−1)αXα E+Xα⊂[1,N]D ef (n−1)α\{α∗}X⊗α\{α∗} EXα∗,n.Now, we use the Fubini’s theorem, combined, first, with the inductive assumption, and then, withcondition (3.1). This completes the proof.Remarks 1 Several special cases and variations of the above lemma will be of particular interest.1.

Let X = [X1, . .

. , Xn], where Xk is an n × k random matrix, and eX have the same structure.Assume that both matrices are C- and eC-sliceable, respectively, and let N = 1 + .

. .

+ n. LetE and Φ be as in Lemma 3.1. (a) If condition (3.1) is fulfilled then, for every symmetric function f = (fk : 0 ≤k ≤n), wehaveEΦ nXk=0DfkX⊗kkE!≤EΦ nXk=0Dfk eX⊗kkE!

(3.3)(b) Assume, additionally, that columns of X and eX are independent, and the classes C andeC are closed under independent extensions (i.e., if ξ, ξ′ ∈C, and ξ is independent of ξ′,then (ξ, ξ′) ∈C). Then the following inequality is sufficient for (3.3).EΦ(x + ξy) ≤EΦ(x + eξy)(3.4)2.

Let 0 < q < p < ∞and C, eC, X, eX be as in the lemma or as in the special case describedabove (in Remark 1.1). Assume that∥x +mXi=1ξixi∥p ≤∥x +mXi=1eξixi∥q.

(3.5)Then∥DDfX⊗EE∥p ≤∥DDf eX⊗EE∥q,(3.6)and, in the special case (Remark 1.1),∥nXk=0DfkX⊗kkE∥p ≤∥nXk=0Dfk eX⊗kkE∥q. (3.7)Consider the assumption in Remark 1.2.

Then (3.7) is fulfilled provided∥x + ξy∥p ≤∥x + eξy∥q(3.8)holds. If there exists a constant c such that eξ = cξ, relations (3.5)–(3.8) are called hyper-contraction inequalities, and ξ is called a hypercontractive random variable.

Gaussian andRademacher random variables are hypercontractive with constants c = cp,q = ((p −1)/(q −1)1/2, 1 < q < p < ∞(cf. [Bor84, Gro73, KS88, KS91]).

A symmetric α-stable random vari-able is hypercontractive in any normed space with exponents q, p ∈(hα, α), where hα = 0,for α ≤1, and hα < 1, for every α < 2 [Szu90].

INFINITE ORDER DECOUPLING123.2Tail estimatesIn [KW92] the following relation between two E-valued random vectors X and Y is called the Φdomination of X by Y :EΦ(x + X) ≤EΦ(x + Y ),x ∈E.In case when Φ(·) = ∥· ∥p, for some p > 0, we will use the phrase “(E + Lp(E))-domination (todistinguish the notion from the comparison of moments).The fulfillment of Φ-domination yields immediately the same relation for sums of finite copies ofX and Y . In [Szu92] we used that fact to prove that the Φ-domination of two type of random chaosesgenerated by hypercontractive random variables implies the tail domination.

We will rephrase thatresult in a more general manner, pointing out the assumptions needed for the fulfillment of the taildecoupling.Theorem 3.2 Let a class X of random vectors X be (E + Lp(E))-dominated by a class Y ofrandom vectors Y in c0 (or, equivalently, in every separable Banach space). Let Y satisfy the, socalled, Marcinkiewicz-Paley-Zygmund (MPZ) condition, i.e.m = supY ∈Y∥Y ∥p∥Y ∥q< ∞for some (equivalently, all) q < p. Then, for some constants c, C > 0, for every Y ∈Y, there existst0 = t0(L(∥Y ∥), such thatP(∥X∥> ct) ≤CP(∥Y ∥> t),t ≥t0.If, additionally, Y is bounded in L0(E), then the class X is tail-dominated by the class Y (i.e., thenumber t0 above does not depend on a particular choice of Y ∈Y.We omit the proof, since its steps are exactly the same as steps in the proof of Theorem 5.3 in[Szu92].

Also, as in [Szu92], we obtain immediately the following corollaries.Corollary 3.3 Let assumptions of Theorem 3.2 be fulfilled, including the boundedness of Y inL0(E).1. Let ϕ : R+ →R+ be an increasing function of moderate growth, and φ(0) = 0.

Then, forsome C′ > 0,Eϕ(∥X∥) ≤C′Eϕ(∥Y ∥),X ∈X,Y ∈Y,If the growth is not moderate, then we still preserve the implicationEϕ(∥Y ∥) < ∞⇒Eϕ(∥cY ∥) < ∞,X ∈X,Y ∈Y,for some universal constant c > 0.2. The L0-boundedness of Y implies the L0-boundedness of X.

If Y is tight, so is X.3. The domination in the sense of tightness also holds in any separable Fr´echet (i.e., metriz-able complete locally convex) space, with the topology generated by a countable family ofseminorms (cf.

[Rud73]), provided the (E + Lp(E))-domination is fulfilled and the uniformMarcinkiewicz-Paley-Zygmund condition is fulfilled for all seminorms.

INFINITE ORDER DECOUPLING13It is clear how this pattern applies to decoupling inequalities. If a (lower or upper) decouplinginequality holds in every Banach space, and a random chaos is induced by hypercontractive randomvariables, then the same type of decoupling holds by means described in the above corollary.3.3Lower decouplingWe assume in this subsection that L is an Orlicz space Lϕ such that ϕ satisfies a strong convexityconditionfor some a < 1, ϕa is convex(3.9)Note that (3.9) means that, for some p > 1, limt→∞ϕ(t)/tp = ∞.

In particular, for a moderatelyincreasing ϕ (i.e., for separable Lϕ), L is uniformly convex. We begin with an auxiliary result.Lemma 3.4 Let L and ϕ satisfy (3.9).Let θ = (θi) be a sequence of integrable independentidentically distributed random variables.

Put Sn = θ1 + . .

. + θn and ξ = supn |Sn|/n, and let(ε, ε1, ε2, .

. .) be a Rademacher sequence independent of (θi).

Then, there exists a constant cϕ, suchthat(i) For every x, y ∈EEϕ(∥x + εξy∥) ≤Eϕ(∥x + cϕεθy∥). (ii) For every n ∈N, x, x1, .

. .

, xn ∈E,Eϕ(∥x +nXi=0εiSii xi∥) ≤Eϕ(∥x + cϕθnXi=0εixi∥).Proof.Assertion (ii) follows immediately from (i), the Fubini’s theorem, and the contractionprinciple for a Rademacher sequence.We will prove (i). The function [0, ∞) ∋t 7→ψ(t) = Eϕ(∥x + εty∥) −ϕ(∥x∥) is convex andincreasing.

HenceEψ(ξ) ≤CEψ(θ),since the sequence M1 = Sn/n, M2 = Sn−1/(n −1), . .

. , Mn−1 = S2/2, Mn = S1 = θ1 forms amartingale with respect to the natural filtration.It is an elementary exercise to prove that the following transformations inherit property (3.9):the shift φ −a, the composition φ ◦ψ with another convex function, averagesR φω(·)µ(dω) withrespect to probability measures µ and a (measurable) family {φω} of functions with property (3.9).Hence the function [0, ∞) ∋t 7→Eφ(∥x + εty∥) −φ(∥x∥) has property (3.9), where x, y ∈E and εis a Rademacher random variable.

Therefore, by Doob’s inequality,P(ξ > t) = P(ψ(ξ) > ψ(t)) ≤E[ψa(θ); ξ > t)ψa(t). (3.10)Then, we infer from (3.10) and H¨older’s inequality thatEφ(∥x + εyξ∥) −φ(1)= Eψ(ξ) =Z ∞0P(ξ > t) dψ(t)≤Z ∞0E[ψa(θ); ξ > t)ψa(t)dψ(t)=(1 −a)−1E[ψa(θ)ψ1−a(ξ)]≤(1 −a)−1(E[ψ(θ)])a(E[ψ(ξ)])1−a.

INFINITE ORDER DECOUPLING14Define a0 = inf { a ∈(0, 1) : φa is convex } . Then, letting a →a0, and using the convexity, weobtainEψ(ξ) ≤(1 −a0)−1/a0Eψ(θ) ≤Eψ((1 −a0)−1/a0θ).The lemma has been proved.Remark 2 The constant c = cϕ depends on the exponent a, appearing in (3.9), or more precisely,on a0.

Also, c = ∞, if a0 = 1, in general. The convexity assumption concerning ϕ is necessaryfor (i), if we do not restrict the class of distributions of θ.

Consider, for example, L = L1. Then(i) implies that E supi |θi|/i < ∞, if E|θ| < ∞.A symmetric random variable θ with the tailP(|θ| > t) = (t log2 t)−1, t ≥e, produces a quick counterexample.We will let the generality of the proof of the following theorem slightly exceed our currentneeds.

The reason will be explained in Section 5. Recall (see (1.1)) that in the real case the lowerdecoupling for Gaussian or Rademacher chaoses holds with a constant ck = 1/√k!.Theorem 3.5 Let the matrix [X, X] have i.i.d.

columns. Let ϕ satisfy (3.9).

Then the sign-randomized weak lower decoupling inequality holds, i.e., for Walsh function w = (wk), independentof [X, X], we have∥nXk=0wkDfkX⊗k E∥≤∥nXk=0wk(2ck)kk!DfkX⊗k E∥,(3.11)for every symmetric function f = (fk) vanishing on diagonals, where c = cϕ depends only on thefunction ϕ. If the underlying random variables are symmetric, then the Walsh functions can beomitted.Proof.We begin with the Mazur-Orlicz polarization formula.∥Xα⊂[1,n]DfαX⊗α E∥=∥Xα⊂[1,n]Dfα1|α|!Xβ≤α(−1)|α−β|(( βX ))⊗α E∥=∥2−nXβ≤[1,n]DXα⊂[1,n]1|α|!

(−1)|α−βα|fα(( 2βαX ))⊗α E∥≤2−nXβ≤[1,n]∥Xα⊂[1,n]1|α|! (−1)|α−βα|Dfα(( 2βαX ))⊗α E∥For a fixed β, we have (cf.

(2.3))(( βαX ))⊗α = E[ (( αX ))⊗α | Xβ ].Recall that wα(β) = (−1)αβ are Walsh functions. Since the mapping β 7→β′ = 1 −β is measurepreserving, hence, by the contractivity of conditional expectations and Fubini’s theorem, we have∥Xα⊂[1,n]DfαX(⊗α) E∥≤∥Xα⊂[1,n]wα2|α||α|!Dfα(( αX ))⊗α E∥.At this moment we give up the generality and notice that fα (|α| = k) vanish unless α = [1, k].

INFINITE ORDER DECOUPLING15Now it suffices to apply the Slicing Lemma 3.1. Let g = (gk) be a symmetric function, gk :N k →E, (3.9) be fulfilled and w = (wk) be a Walsh sequence independent of X.

Then, for aconstant c = cϕ,∥nXk=0wkDgkX1 + . .

. + Xkk⊗k E∥≤∥nXk=0wkDgk(cX)⊗k E∥.

(3.12)The proof is completed.Corollary 3.6 Let assumptions of Theorem 3.5 be fulfilled, where ϕ(t) = tp, p > 1. Denote byQ(f) the coupled, and by Q(f), the decoupled chaos, as appear, respectively, in the right andleft hand side of inequality (3.11).

Assume that components Xi of X are hypercontractive, withhypercontractivity constants uniformly bounded away from 0. Then the following conditions arefulfilled.1.

There is a constant C, depending only on the hypercontractivity constants, and a sequentialconstant c = (ck), depending only on p, such that, for any non-decreasing moderately growingfunction ϕ : R+ →R+, every f ∈FS,Eϕ(∥Q(f)∥) ≤CEϕ(∥Q(cf)∥)(if ϕ does not grow moderately, the finitness of the Orlicz modular is preserved).2. The stochastic boundedness of a family of coupled polynomial chaoses { Q(fa) : a ∈A } im-plies the same fornQ(dfa) : a ∈Ao, where d = c−1, and c appears in the preceding state-ment.

By the same token, the tightness of the first family yields the tightness of the secondfamily.Proof.It suffices to interpret appropriately Corollary 2.4.3.4Reduction to Rademacher chaosesWe will focus on a search of reasonably wide classes of Banach spaces, which support the expo-nential upper decoupling. Recall that X ∈UD = UD(E; Φ; F) (X satisfies the upper decouplinginequality), ifEΦ(DDfX⊗EE) ≤EΦ(DDf(cX⊗EE)for every function from class F.The most important are classes FS, of symmetric functions,and FT , of tetrahedral functions.

Denote by µ = L(X) the distribution of a sequence X. Letϕ : R+ →R+ be a measurable function. Denote by RU = RU(µ, ϕ) (respectively, RL = RL(µ, ϕ)the class of Banach spaces such that, for some constant c > 0, the inequalityEϕ(∥x + XXiεixi∥) ≤Eϕ(∥x + cXXixi∥),(3.13)(respectively,Eϕ(∥x +XXixi∥) ≤Eϕ(∥x + cXXiεixi∥))(3.14)

INFINITE ORDER DECOUPLING16is fulfilled, for every x ∈E, and for all finite families { xi } ⊂E.The following result shows the importance of the introduced classes. Its proof is a direct conse-quence of the Slicing Lemma 3.1, and, for tetrahedral functions, of equality (2.6) from Proposition2.1.Proposition 3.7 Let X be a sequence of independent symmetric random variables, and X be amatrix whose columns are independent copies of X, f = (fα) be a symmetric function with valuesin E, ϕ : E →R+ be a measurable function.

Denote by ε = (εi) be a Rademacher sequenceindependent of X, and by S, a Rademacher matrix, independent of [X, X].If E ∈RU(µ, ϕ)(respectively, E ∈RL(µ, ϕ)), thenEϕDDf(XS)⊗EE ≤EϕDDf(cX)⊗EE = EϕDDf(cSX)⊗EE (respectively, the converse implication is valid, with c replaced by c−1).If, additionally, ϕ isconvex, the latter inequality is fulfilled also for tetrahedral functions (respectively, the fulfillmentof the latter inequality for tetrahedral functions implies the same, for symmetric functions).3.5Limitations of the reductionInequalities (3.14) and (3.13) may fail in some Banach spaces, and for some random sequences.First, we note the following immediate consequence of Proposition 3.7.Lemma 3.8 Let X, X, and ϕ be as in Proposition 3.7. Let E ∈RU(µ, ϕ) (respectively, E ∈RL(µ, ϕ).

ThenEϕ(∥x +mXj=1nXi=1Xjεijxij∥) ≤Eϕ(∥x + cmXj=1nXi=1Xijxij∥)(3.15)(respectively,Eϕ(∥x +mXj=1nXi=1Xijxij∥) ≤Eϕ(∥x + cmXj=1nXi=1Xjεijxij∥))(3.16)for every m, n ∈N, and every matrix [xij] of vectors of E.Proposition 3.9 Let X, X be as in Proposition 3.7, ϕ(t) = t2, and G = (G, G1, G2, . .

. ), andY = (Y, Y1, Y2, .

. .

), respectively, be a sequence of i.i.d. standard normal, and a sequence of i.i.d.exponential random variables with parameter 2, respectively, such that G and Y are independentof X.1.

Let X be square integrable. Assume that E ∈RU(µ, ϕ).

Then the following conditions arefulfilled. (i)E∥x + XGy∥2 ≤E∥x + cGy∥2,x, y ∈E;(3.17)(ii)E∥x + εY y∥2 ≤E∥x + c′Gy∥2,x, y ∈E,(3.18)where c′ may be a new constant;

INFINITE ORDER DECOUPLING17(iii)E∥x +XiεiYixi∥2 ≤E∥x + c′ XiGixi∥2,x, y ∈E(3.19)(iv) E does not contain isomorphic copies of ℓ∞n , uniform in n, neither it contains two-dimensional subspaces isometric to ℓ∞2 or ℓ12.2. Assume that E ∈RL(µ, ϕ), where ϕ be a nondegenerate nondecreasing function such thatlim supt→∞ϕ(t)e−at2 = 0 for some a > 0.

Then X is square integrable.Proof.1. Inequality (3.17) follows by Lemma 3.8 and the (real) Central Limit Theorem.

Thepassage to the limit can be justified by a routine uniform integrability argument (cf., e.g., [Bil68,Theorems 5.3 and 5.4]). By the same token, one may assume that X in (3.17) has the normaldistribution.

An exponential random variable Y with intensity λ = 2 has the tail comparable tothe tail of the product of two independent Gaussian random variable (cf. [Yos80, pp.

243-244]),which proves (3.18), in view of Corollary 2.4. Estimate (3.19) follows by the Fubini’s theorem anditeration.Suppose E = ℓ∞2(i.e., E is just R2 with sup-norm).Take orthogonal y, x in (3.18) with∥x∥= 1, ∥y∥= 1/u < 1, and subtract 1 from both sides of (3.17).

Then the right hand side is oforder exp−u2/2, while the left hand side is of order exp {−2u}, for u →∞, which produces acontradiction.Inequality (3.19) yields the domination of sums of symmetrized independent exponential randomvariables by sums of independent Gaussian random variables. Clearly, this is impossible in ℓ∞n (itsuffices to take orthogonal xi’s, and apply the classical estimates for suprema of independent randomvariable, cf., e.g., [VCT87, Lemma V.3.2]).Finally, inequality (3.18) does not hold in ℓ12, since by the Ferguson-Hertz embedding theoremevery two-dimensional normed space can be isometrically embedded into L1 (cf.

[Fer62, Her63],see also [KS91]). One can construct a direct counterexample, too.2.

By choosing xi = tnx/√n, i = 1, . .

. , n, where ∥x∥= 1, in the defining inequality of the classRL, we infer that, for some constant c′Eϕ(tn| Pni=1 Xi|√n) ≤c′Eϕ(tn| Pni=1 εi|√n),(3.20)for every real sequence tn →0.

Because of the regular variability of ϕ at ∞, the right hand sideconverges to 0, hence the sequence (Pni=1 Xi/√n) is bounded in L0 (i.e., tight), by Chebyshev’sinequality. This is possible only if X ∈L2.3.6Reduction in some spaces3.6.1Rademacher versus Gaussian chaosesSo far, we have established a class of Banach spaces, where the exponential upper decouplinginequality is fulfilled.Now, we will show that class is reasonably wide.In general, the upperdecoupling may depend also on distributions of involved random variables (whether it does, is an

INFINITE ORDER DECOUPLING18open question at this time). Before we proceed further, in order to avoid unnecessary redundancy,we will determine some dependence (far from being complete) between decoupling inequalities forrandom chaoses spanned by random variables with different distributions.Proposition 3.10(i) The class UDS = UD(E; ∥· ∥p, FS) is closed under products and sums ofi.i.d.

sequences, i.e., if X and X′ are equidistributed independent sequences, and X, X′ ∈UDS with a constant c, then XX′ = (XiX′i) ∈UD and X + X′ ∈UDS with the constant c.(ii) In addition to the above properties, the class UDT is also closed under linear combinationsof independent sequences, i.e., if X and X′ are independent sequences, and X, X′ ∈UDTwith constants c, c′, then, for every numerical sequences a and b, aX + bX′ ∈UDT with theconstant c.(iii) Denote ψ(t) = ψx,y(t) = EΦ(x + εty). If X(m) ∈UD with the same constant c, the distri-butions of X(m) converge weakly to the distribution of X, and the familynψ(X(m))oisuniformly integrable, then X ∈UD with a constant which is less or equal c.(iv) If Φ is convex, then UDS ⊂UDTProof.The closeness under the product is easy to see and follows immediately by Fubini’stheorem.In order to prove the additivity in (i), let us consider a (2n × 2n)-matrix"XYX′Y ′#where Y and Y ′ are independent copies of X, and the sequence (X, Y ), where Y is an independentcopy of X.

Then it suffices to change the enumeration of arguments of functions fk(·), putting, inparticular fk = 0 for k ∈[n + 1, 2n].For additivity in (ii), we rather use the following (2n2 × n)-matrixX1∗∗. .

.∗X′1∗∗. .

.∗∗X2. .

.∗∗X′2. .

.∗∗∗X3. .

.∗∗∗X′3. .

.∗. .

.. . .. .

.. . .. .

.∗∗∗. .

.Xn∗∗∗. .

.X′n,where the symbols ∗indicate the presence of mutually independent copies of corresponding portionsof columns.Note that the lack of symmetry assumption in (ii) enables us to use arbitrary sequential multi-pliers a and b, while both sequences must be constant under the symmetry assumption.Assertion (iii) follows from basic properties of weak convergence (cf. [Bil68, Theorems 5.3 and5.4]).

INFINITE ORDER DECOUPLING19Assertion (iv) follows from (2.6).Corollary 3.11 If the upper decoupling inequality for symmetric (or triangular functions) is satis-fied for some zero-mean probability law with finite variance, e.g., by a Rademacher random variable,then it is satisfied by the Gaussian law.Proof.The assertion follows from Proposition 3.10, (i) or (ii), the Central Limit Theorem, andProposition 3.10(iii).Corollary 3.12 Let the assumptions of Theorem 3.11 be fulfilled. Then the upper decouplinginequality of the same type (i.e., either for symmetric or triangular functions) is satisfied by allsymmetrized Gamma(m)-distributions, m = 1 (exponential law), 2, 3, .

. .Proof.Indeed, the product of two independent Gaussian random variables is comparable to arandom variable with exponential distribution (cf.

e.g. [Yos80, pp.243-244]).

Hence the assertionfollows by Proposition 3.10.3.6.2Banach latticesLet E be a Banach lattice. Then, for every continuous positive homogeneous function ψ : Rn →R,the expression ψ(x1, .

. .

, xn) ∈E, x1, . .

. , xn ∈E, is well defined, in particular,(nXi=1|xi|p)1/p;(E|nXi=1xiθi|p)1/p,where θi ∈Lp are real random variables, and 0 < p ≤∞([Kri74], also see [LT79]).

Any inequality orequality that is valid in the real case, carries over to Banach lattices (when E is a space of functions,these constructions, in general, can be viewed pointwise, both intuitively and rigorously).Recall the Krivine’s notion of type ≥p and ≤p (p-convex and p-concave in [LT79]). A Banachlattice is said to be of type ≥p (respectively, of type ≤p), 1 ≤p ≤∞if∥(nXi=1|xi|p)1/p∥≤C(nXi=1∥xi∥p)1/p(respectively, the inverse inequality holds).

These properties refer to a degree of convexity of theunit sphere, compared to the unit sphere in Lp. For example, Lr is of type ≤p, when r ≤p, andof type ≥p, when r ≥p, 0 < r ≤∞.Theorem 3.13 Every Banach lattice E of type ≤q < ∞and type p > 1 admits an equivalentnorm for which both upper and lower (and both symmetric and tetrahedral) decoupling inequalitieshold for the Rademacher (hence Gaussian) law, by means of comparison in Lr, 1 < r < ∞.

Moreprecisely, for such a norm, there exists a constant c such that∥Xk≥0DQk(f/c)∥≤∥Xk≥0DQk(f)∥≤∥Xk≥0DQk(cf)∥,

INFINITE ORDER DECOUPLING20whereQk(fk) =1√k!DfkX⊗k Eif fk is symmetricDfkX⊗k Eif fk is tetrahedral,and Qk(fk) =DfkX⊗k Ein both cases.Proof.Essentially, we reduce the problem to the situation on the real line (cf. (1.1)).

By Figieland Johnson theorem ([FJ74], see also [LT79, Theorem 1.d.8]), a Banach lattice E, which is of type≥p and ≤q, 1 < p ≤q < ∞, admits an equivalent norm, making both constants C, appearing inthe definition, equal to 1. So, assume that is the case.

Then, we have∥(E|nXi=1xiθi|p)1/p∥≤(E∥nXi=1xiθi∥p)1/pand(E∥nXi=1xiθi∥q)1/q ≤∥(E|nXi=1xiθi|q)1/q∥for any collection of suitably integrable random variables (θi). We will apply both inequalitiesto Rademacher chaoses and use the hypercontractivity of Rademacher chaos.Denote cr,q =max(1, ((r −1)/(q −1))1/2).The proof is similar in the symmetric and tetrahedral case, andalso for the upper and lower decoupling.

We will give details only in one case, say, for tetrahedralf and the upper inequality. We haveEDDfε⊗EE r1/r≤E|DDf(cr,qε)⊗EE|q1/q≤(E|DDf(cr,qcq,2ε)⊗EE|2)1/2 =DD|f|2(cr,qcq,2)⊗E)1/2≤(E|DDf(cr,qcq,2S)⊗EE|2)1/2 ≤(E|DDf(cr,qcq,2c2,pS)⊗EE|p)1/p≤EDDf(cr,qcq,2c2,pS)⊗EE p1/p≤EDDf(cr,qcq,2c2,pcp,rS)⊗EE r1/r.Other cases follow by an almost verbatim argument.The class of Banach lattices, appearing in the theorem, can be enlarged to uniformly convexspaces with a local unconditional structure (LUST) (i.e., such that E can be embedded into thedual of a Banach lattice, cf.

, e.g., [GL74]). In fact, the context of Banach lattices, as appear in thetheorem, makes the problem of decoupling rather trivial.

Any tetrahedral (respectively, symmetricand of finite order) Rademacher or Gaussian chaos is exponentially equivalent, in the sense of theintroduced domination in any Lr, 1 < r < ∞, to an infinite (respectively, finite) Rademacher orGaussian sumXk=0Xi∈Nkfk(i)Xi,wherenXi : i ⊂NN , i finiteois a family of independent Rademacher or Gaussian variables. Inparticular, for the aforementioned class of Banach spaces, after a renorming, in a trivial manner aninfinite order contraction principle holds for Rademacher or Gaussian chaosesE∥DDfgX⊗EE∥r ≤E∥DDf(cX)⊗EE∥r,

INFINITE ORDER DECOUPLING21where c is a suitable constant, and g = (gk), gk : N k →[−1, 1] is an arbitrary measurable function.Such the contraction principle fails in general, e.g., if E = c0, and even for a single k-homogeneouscomponent, k ≥2 [KS86].Remark 3 A related procedure can be applied for random variables with sufficiently high mo-ments.That is, if E is as in Proposition 3.13, and θ ∈Lr, r ≥2, is a symmetric randomvariable such that r > q0 = inf { q : E is of type ≤q }, then θ is hypercontractive with constantscr,q(θ) = ∥θ∥r/∥θ∥qcr,q [KS88], which would replace constants cr,q in the proof. By a similar ar-gument to the one used in the proof of Proposition 3.9, one can show that Lq /∈RU(L(X), Lr), ifq ≥q0 > r0df= sup { r : θ ∈Lr } (Lq in the latter formula can be replaced by any Banach spacescontaining isomorphic copies of finite dimensional spaces ℓqn, uniform in n).

This may suggest thatthe upper decoupling inequality fails in such spaces yet the problem remains open.We do not know whether the upper decoupling inequalities for Gaussian and Rademacherchaoses are equivalent. For tetrahedral functions, even in the homogeneous case, the lower de-coupling inequality may fail (cf.

Bourgain’s example included in [MT87], or [KW92, Section 6.9]).4STABLE CHAOSES4.1Auxiliary definitions and inequalitiesIn this section X denotes a symmetric standard α-stable (SαS, in short) random variable, i.e.,E exp {itX} = exp {−|t|α}, and Y denotes a symmetric α-Pareto random variable, i.e., P(|Y | >t) = t−α, t ≥1 (SαP, in short) . It is known that tails of SαS and SαP random variables arecomparable, i.e.

P(|X| > t) ≤KP(|Y | > Kt) and P(|Y | > t) ≤KP(|X| > Kt), t ≥t0 (we mayassume that the above estimate are valid for all t ≥0). Hence, for tetrahedral or symmetric infiniteorder polynomial Q(f, ·) we have∥Q(f, Y/c)∥p ≤∥Q(f, X)∥p ≤∥Q(f, cY )∥p.

(4.1)This remark will enable us to switch freely (in the sense of the exponential domination) betweenstable and Pareto chaoses, and benefit from algebraic properties of stable random variables, oranalytic properties of Pareto random variables. We will need also the following estimate (cf.

[KS88,Szu90, Szu91b] for similar inequalities.Lemma 4.1 Let 0 < s < α < 2. There exists a constant a = a(α, s) such that, for every sequenceof i.i.d.

SαS (or SαP) random variables, the inequality(∥x∥α + aXi∥xi∥α)1/α ≤∥x +XiXixi∥sis fulfilled, for all x, x1, x2, . .

. ∈E.Proof.We will apply a hypercontractive iteration for Pareto random variables, and then usethe fact that SαP law belongs to the normal domain of attraction of the SαS law (cf.

the afore-mentioned papers for details). It suffices to verify the inequality(1 + atα)s/α ≤E∥x + Y ty∥s,(4.2)

INFINITE ORDER DECOUPLING22where ∥x∥= ∥y∥= 1, 0 < t ≤1. The inequality follows by combining the estimateE∥x + Y ty∥s −1 ≥tαinf∥x∥=∥y∥=1 E[∥x + Y y∥s −1; |Y | ≥2] ≥tαE(|Y | −1|s −1)+with the inequality (1 + tα)s/α −1 ≤s/αtα, which holds for all t ≥0.

Puta = αE(| |Y | −1 |s −1)+/s.This completes the proof.We will see that the fulfillment of decoupling inequalities may depend on the convexity andsmoothness of the norm. A norm of a Banach space E is called p-smooth (cf.

[Ass75, LT79]),1 < p ≤2, if(E∥x + εty∥2)1/2 ≤(1 + Ctp)1/pwhere ∥x∥= ∥y∥= 1, t > 0 (it suffices to consider only t ≤1), and ε is a Rademacher randomvariable. By hypercontractivity, the L2-norm on the left hand side can be replaced by any Ls-norm,1 < s < ∞.

A Banach E is called p-smoothable, if it admits an equivalent p-smooth norm. It will beconvenient to extend trivially the notion of smoothness to the case p = 1 (every norm is 1-smooth).For a Banach lattice E, letk0 = inf { q : Eis of Krivine’s type ≤q } ≤∞, k0 = sup { p : Eis of Krivine’s type ≥p } ≥1.Clearly, k0 ≤k0.

Say that a Banach space is of infinite cotype, if it contains subspaces isomorphicto ℓ∞n uniform in n. Otherwise, E is said to be a space of finite cotype. A Banach lattice is of finitecotype if and only if it is of Krivine’s type ≤q, for some q < ∞[LT79].4.2Symmetric decouplingLet us extract a suitable fragment from Theorem 3.5.Theorem 4.2 (Lower Symmetric Decoupling) Let 1 < p < α < 2.The lower decouplinginequality in Lp for symmetric SαS and SαP chaoses is fulfilled with constants ck = dk, for somed > 0.The obtained constant is the best we know, even in the real case.

However, for a single homo-geneous chaos, the estimate can be significantly improved (cf., e.g., [DA80]). Like before, in theRademacher or Gaussian case, we can prove the upper decoupling inequality only in some Banachspaces.

Surprisingly, the upper decoupling inequality for nonintegrable stable chaoses is a trivialconsequence of the slicing techniques, and holds in an arbitrary Banach space. Recall that any SαS(or SαP) random variable is hypercontractive in any normed space with exponents q, p ∈(hα, α),and hα = 0, for α ≤1.Theorem 4.3 (Upper Symmetric Decoupling) Let E be a Banach space.

Consider SαS (orSαP) chaoses in symmetric functions and the norm Ls, hα < s < α.

INFINITE ORDER DECOUPLING23(i) For α ≤1, the exponential upper decoupling inequality for symmetric SαS (or SαP) chaosesholds in every Banach space. (ii) Let E be p-smoothable, 1 ≤p ≤2, and 0 < α ≤p.

Then E ∈RU(µ, s). In particular, if anupper symmetric decoupling inequality holds for Rademacher chaoses, with constants (c(R)k),then, for an equivalent norm, an upper decoupling inequality for symmetric SαS (or SαP)chaoses holds, with constants (ac(R)k), where a = a(α, s, ∥· ∥E, p).

(iii) Let E be a Banach lattice, 0 < s < α < 2.If k0 < ∞, and α > k0 or k0 > α, thenE ∈RU(µ, s), and the upper decoupling inequality with exponential constants ck = ak holds.Proof. (i): Let 0 < s < α ≤1, and ∥x∥= 1, x1, .

. .

, xm ∈E. Put t = Pi ∥xi∥.

Then, by thetriangle inequality and [Szu90, Cor. 3.2],∥x + YmXi=1xi∥s ≤∥1 + |Y t|∥s ≤(1 + c1tα)1/αfor some constant c1 = c1(α, s).

On the other hand, since the ℓ1-norm dominates the ℓα-norm, andby Lemma 4.1, we have∥x +mXi=1Yixi∥s ≥(1 + cp2mXi=1∥xi∥α)1/α ≥(1 + C2tα)1/α,(4.3)which yields the assertion of the theorem, by virtue of Lemma 3.7, where all entries of [S, S] areequal 1. (ii): Almost the same argument works for integrable stable chaoses.

Denote now t = (Pi ∥xi∥q)1/q,where q ≥α > 1. By Fubini’s theorem, and the smoothness property (assuming that the norm isalready q-smooth), and by the hypercontractivity of Rademacher random variables, we have∥x + YmXi=1xiεi∥s ≤∥(1 + |c3Y t|q)1/q∥s ≤(1 + c4tα)1/αfor some constants c3 and c4.

Since the right inequality in (4.3) holds also for α > 1, we completethe proof of the second assertion, in view of Lemma 3.7, with Rademacher multipliers. (iii): Assume that k0 < ∞, i.e., E is of Krivine’s type ≤q < ∞.

First, let k0 < α, and choose qsuch that 1 < q < α. By the Figiel-Johnson renorming theorem [FJ74], there exists an equivalentnorm such that the type ≤q-constant is equal 1.By the hypercontractivity of the SαS (or SαP) law, we may use any s-norm, for hα < s < p,with a constant c = cα,q,s(Y ), cf.

[Szu90]). It is important that hα < 1.

Choose s = q.We will check the following inequality in the real case(E|x + XXixiεi|q)1/q ≤(E|x + bXixiXi|q)1/q,(4.4)where b = bα,q. Indeed, assuming that x = 1, we obtain the following upper bounds of the lefthand side, by virtue of the Fubini’s theorem and the hypercontractivity of Rademacher randomvariables (h = hq,α · hα,2 =p(q −1)/(α −1),(E(1 + |Xixiεi|α)q/α)1/q ≤(E(1 + |Xixiεi|α)q/α)1/q ≤(1 + h(Xi|xi|2)α/2)1/α.

INFINITE ORDER DECOUPLING24The right hand side is estimated from below as follows:(E|1 + b Pi xiXi|q)1/q ≥(1 + baα,q(Xi|xi|α)1/αin view of Lemma 4.1. These estimates prove (4.4), with b = h/a.Whence, and also by the Fubini’s theorem and hypercontractivity of SαS (or SαP) law, wehave∥x + YXixiεi∥q ≤∥(E|x + XXixiεi|q)1/q∥≤∥(E|x + bXixiXi|q)1/q∥≤∥E|x + bcq,1XixiXi|∥≤E∥x + bcq,1XixiXi∥≤∥x + bcq,1XixiXi∥qBy applying Lemma 3.7, we complete the proof of assertion (ii) in the case k0 < α.Let now α < k0 (k0 ≤k0).

Choose q ∈(α, k0). This case follows immediately from assertion(ii), since in presence of finite cotype, there is an equivalent q-smooth norm (cf.

[Fig76] or [LT79,Theorem 1.f.1]).Remark 4 Consider SαS (or SαP) symmetric chaoses.That the Krivine’s classification doesnot fully describe the fulfillment of the upper inequality follows from the following observation.Consider the case k0 ≤α ≤k0.Note that an upper decoupling inequality for SαS (or SαP) chaoses, held in Banach spaces(E1, ∥· ∥1) and (E2, ∥· ∥2), with constants (c1k) and (c2k), respectively, holds also in E1 ⊕s E2,1 ≤sα, endowed with the norm ∥· ∥= (∥· ∥s1 + ∥· ∥s2)1/s, with constants ck = max(c1,k, c2,k). Thus,since by assertion (ii) an upper decoupling inequality holds in every Lp, p ̸= α, it will be fulfilledin every Lq ⊕s Lr, q < α < r.Proposition 4.4 Let E be a Banach lattice of finite cotype such that k0 > α.

Then there existsan equivalent renorming such that all lower and upper, symmetric and tetrahedral, decouplingconstants for stable chaoses are equivalent to the corresponding constants in the real line, i.e.,ck(E) = akα,sck(R).Proof.By a result from [FJ74], one can choose an equivalent norm of type ≥q with theconstant equal to 1, q < α. We will use the hypercontractivity of stable (or Pareto) (one may useany s-norm, for hα < s < α (where hα < 1), with a constant aα,s [Szu90]).

Now, denoting by Qand Q′ two type of chaoses under interest, and combining the estimates∥XkQk∥s ≤∥(E|XkQk|s)1/s∥≤∥(E|Xk(ck(R))kQ′k|s)1/s∥and∥(E|XkQ′k|s)1/s∥≤∥E|Xk(aα;s,1)kQ′k| ∥≤E∥Xk(aα;s,1)kQ′k∥≤∥Xk(aα;s,1aα;1,s)kQ′k∥s,we complete the proof.

INFINITE ORDER DECOUPLING255CONCLUDING REMARKSIn this section we display some further features of infinite order decoupling and domination. Someproperties or generalizations can be obtained by well known routines, while other properties, enjoyedby homogeneous chaoses, yield to the dead end.

Yet a number of open problems arise that have nocounterparts for homogeneous chaoses. At this time, the infinite order approach to random chaosesis still in a preliminary stage.5.1Multiple stochastic integralsDecoupling inequalities for infinite order Gaussian or stable polynomials can be carried over toinfinite order multiple stochastic integrals, preserving all constants, the dependence on geometry,and subjection to the presence or lack of symmetry of underlying functions.

These results followby a routine approximation (integrals of simple functions are random chaoses).The real case does not require any comments, since the theory is classical. In the vector case,one needs an appropriate construction of k-tuple stochastic integrals of deterministic functions withrespect to a Gaussian (or more generally, a second order symmetric) process.

One may apply theDunford-Bartle approach, which reduces the integration in Banach space to that with respect toan L2-valued vector measure (cf., e.g. [DU77]).5.2Non-multiplicative functionsIn [Szu92, Theorem 4.1] (and before, in [MT87, dlPn90]), a nonmultiplicative version of the de-coupling principle for homogeneous chaoses was proved.

In such a version, a term f(i1, . .

. , ik) ·X1i1 · · · Xkik was replaced by a term F(i, X1i1 · · · Xkik).

Let us consider a nonhomogeneous analogof such a decoupling principle (as in [Szu92, 4.1]). Let L be an Orlicz space induced by a stronglyconvex function ϕ (3.9).

For the sake of simplicity of formulations, assume that ϕ grows moder-ately. Let F = (Fα) be a function whose components are functions Fα : Nα × Rα →E satisfyingconditions [Szu92](F1)F(i, ·) = 0µk-a.s. for all but finitely many i;(F2)F(i ; Xi1, .

. .

, Xik) ∈Lϕ(E) for every i ∈N k.(5.1)PutF (X⊗) =XαFα(X⊗α).If w = (wα) is a Walsh sequence, write F w = (Fαwα) (i.e. [Fαwα](αi) = Fα(αi)wα).

Thenthe analog of Theorem 3.5 holds, where Fα vanish, unless α = [1, k].Theorem 5.1 Let L be an Orlicz space induced by a strongly convex function ϕ (3.9), F = (Fk)satisfy (F1)-(F2), ∥Fk(X⊗k]) ∈Lϕ, k ≥0, and [X, X] be as in Theorem 3.5. ThenEϕ(∥Xk≥0Fk(X⊗k)∥) ≤Eϕ(∥Xk≥0wk(2ck)kk!Fk(X⊗k)∥),where c depends on the convexity of ϕ.

INFINITE ORDER DECOUPLING26The upper decoupling inequality for functions F shares all deficiencies of the corresponding de-coupling inequality for homogeneous chaoses. But there arise significant difficulties that cannot beremoved by using techniques based on hypercontractivity, since the latter method works efficientlyonly for symmetric random variables.

In the proof of [Szu92, Theorem 4.1], nonsymmetric randomvariables were used, which does not allow one to proceed as in the proof of Theorem 3.13. A verylimited, almost trivial, real line- version of the upper decoupling inequality can be seen as follows.E|XkFk((εX)⊗k)|2 = E|XkckFk((SX)⊗k)|2,where Fk(X⊗k) ∈L2, and ck = 1 for tetrahedral functions, and ck = k!

for symmetric functions.Any non-trivial extension (beyond Hilbert space and L2-norm) would require some intrinsic sym-metry of functions Fk. Therefore, at this stage it is meaningless to look at these kinds of decouplinginequalities from the view point of integration with respect to empirical measures (as in [Szu92]),even though other types of domination might be still of interest.5.3Ces`aro averagesThere exists a variety of operators acting on the entire matrix X.

For example, one may use theoperator D, which nullifies diagonal values of functions fα. For the sake of consistency, denotethe basic symmetrizator by S, S(f) = bf.

Many an operator do not have meaning for a singlehomogeneous polynomial. We will consider a certain multilinear analog (one of many) of Ces`aroaverages.

Let us confine ourselves to subsets α ⊂[1, n], and functions f = (fba : α ⊂[1, n]). Weintroduce the “index average” operator A = (Aα), which unifies values of functions fα along setsα with the same cardinality.First, we define the symmetrizator A′ = (A′k), which transforms f into a function g = (gk : k =0, 1, .

. .

, n), where gk : N k = N [0,k] →E.Let |α| = k.Denote by sα the “stretching map” which embeds N k = N [0,k] into Nα bymoving the elements of a sequence ik = (i1, . .

. , ik) = (i1, .

. .

, ik, 0, . .

.) into the places marked bythe consecutive ones of the sequence α = (α1, α2, .

. .

), and filling up the remaining places by zeros.Put, for ik = (i1, . .

. , ik),A′k(f) (ik) = 1nkX|α|=kfα(sαik).Clearly,DDfEE=nXk=0 nk!DA′k(f)E.Denote by cα the “contracting” mapping from Nα onto N k, which just cancels all elements markedby zeros of the sequence α.Now, we define the “inverse” mapping A′′ = (A′′α) transformingfunctions g = (gk) into functions f = (fα), according to the formulaA′′α(g) (αi) = gk(cα(αi)),|α| = k.Define A = A′′A′.

The operators D, S, and A are idempotent and commute with each other.

INFINITE ORDER DECOUPLING27Let E1, E2 be additive abelian groups. Denote by x1x2 a bi-additive mapping from E1 × E2into E. Use the same notation f 1f 2 for functions taking values in E1 and E2, respectively.

IfU and V are compositions of selected symmetrizators D, A, S, then the following symmetrizationformulas hold:DDU(f 1)V(f 2)EE=DDf 1UV(f 2)EE=DDUV(f 1)f 2EE=DDV(f 1)U(f 2)EE. (5.2)Note that X⊗is A-symmetric.

By Ak = Ak(X) denote the σ-field generated by the family ofrandom variablesnh(Xα : |α| = k) : h = ˆh, h : (Rk)(nk) →Ro.Notice that the symmetry assumption is applied to h as to a function ofnk vector variables, andthat Ak are ascending σ-fields. The symmetrizator A can be expressed as a conditional expectation.Note the following equalities:DDfXA(⊗) EE=DDfE[X⊗|A(X)]EE= E[DDfX⊗EE|A(X),(5.3)EhD(X1 + .

. .

+ Xk)⊗k | A(X)i= DAα(( αX ))⊗α;(5.4)or equivalently,ED(X1 + . .

. + Xkk)⊗k | A(X)= DAααX|α|⊗α.

(5.5)Now, Theorem 3.5 holds for A-convex functions. That is,Eϕ(∥Xα⊂[1,n]wαDfαX⊗α E∥) ≤Eϕ(∥Xα⊂[1,n]wαhαDfαX⊗α E∥),(5.6)for every D−, A, &S-symmetric function f, where, for |α| = k, hα = hk = (2ck)k/k!, and c = cϕ.However, the A-symmetry is too strong for the upper inequality of arbitrary order to be fulfilled.For integrable symmetric random variables, by examining just polynomials of the first degree, wewould obtain the inequalityEϕ(∥x +Xi≤KXixi∥) ≤Eϕ(∥x + c1Xi≤KPnj=1 Xjinxi∥)which is impossible, as can be seen by applying the strong law of large numbers and Fatou’s lemma.Yet, the above observations open a new, even in the real case, direction in a study of A-symmetricchaoses.

Clearly, any domination “constant” is expected to depend on n, which makes the conceptof infinite order much more difficult.Problem. Describe the closure in L2 and limit distributions of real Gaussian (or Rademacher)A-symmetric decoupled chaosesXα⊂[1,n]DfαXA(α) E: n ∈N.Note that the metric (L2-) problem is easy for S-symmetric or tetrahedral functions (cf.

the firstsubsection of this section). For S-symmetric functions, a related limit theorem for coupled Gaussianrandom chaoses, obtained in [DM83], brought up infinite order Wiener integrals.

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In A. Beck et al., editor, Probabilityon Banach spaces, Lecture Notes in Math.1153:453–457. Springer .Jerzy SzulgaDepartment of Mathematics, Mathematics Annex 120Auburn University, Auburn, AL 36849-3501phone(205) 844-3649emailszulga@auducvax.bitnetszulga@ducvax.auburn.edu

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