CONTRACTION AND DECOUPLING INEQUALITIES FOR MULTILINEAR FORMS AND U-STATISTICS

V.H. de la Peña^{†}

Department of Statistics, Columbia University,

New York, NY 10027.

S.J. Montgomery-Smith^{‡}

Department of Mathematics, University of Missouri,

Columbia, MO 65211.

Jerzy Szulga

Department of Mathematics, Auburn University,

Auburn, AL 36849

We prove decoupling inequalities for random polynomials in independent random variables with coefficients in vector space. We use various means of comparison, including rearrangement invariant norms (e.g., Orlicz and Lorentz norms), tail distributions, tightness, hypercontractivity, etc.

Contents

1 Introduction 2 1.1 Background and scope of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Decoupling for r.i. norms 4 2.1 Extended multilinear forms and U-Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 An example in a certain Lorentz space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 A discourse on probability tails 10

3.1 L^{p}-estimates imply tail estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1.1 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1.3 Contraction for multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Decoupling for tails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Notes 16

Key words and phrases: decoupling principle, symmetric tensor products, random polynomials, multiple random series, multiple stochastic integrals, random multilinear forms, random chaos, tail inequalities, polarization, symmetrization, Banach space, rearrangement invariant, Orlicz space, Lorentz space, Rademacher sequence, Ustatistics

AMS (1980) Classification Primary: 60B11, 46M05 Secondary: 60H07, 46E30, 60E15, 62H05, 62G30 † Research supported in part by N.S.F. Grant DMS 91-08006 and a grant from the CNRS of France.

‡ Research supported in part by N.S.F. Grants DMS 9001796 and DMS 9201357.

1 Introduction

1.1 Background and scope of the paper

Decoupling principles stem from the theory of martingale transforms (cf. [Bur86]). For homogeneous random forms of rank k ≥ 2, decoupling principles were introduced in [MT86a, Kwa87, MT87] (in some special cases, they were known to Pisier, cf. [MT86b]), and subsequently became essential tools in multiple integration (cf. [KS89, Szu91, KW87, RW86, RST91]. One of the most appealing interpretations of such an principle is the reducibility of a study of multiple random series (respectively, of multiple stochastic integrals) to a consecutive treatment of single random series (respectively, of single stochastic integrals which allows one to treat a multiple integral as an Itôtype iterate integral). The concept of a random chaos goes back to N. Wiener [Wie30] (see also [WW43]), who elaborated what we call here a real-valued coupled Gaussian chaos. Decoupling inequalities may be viewed as embedding-projection procedures, since a decoupled random chaos is nothing but a lacunary random chaos. In comparison to the classical L^{2}-theory of multiple summation or integration, decoupling principles make up the lack of L^{2}-isometries.

Since the first publication of the aforementioned decoupling principle, the theory has branched into several directions. For example, comparison of tangent processes (cf. [Hit88, dlPn93, dlPn94],) is akin to the classical decoupling principle. Further contributions can be found, e.g., in [Kwa87, DA87, Hit88, Zin86, NP87, dlPn92]. In some of the aforementioned papers (e.g., [MT86a, MT87, Kwa87, DA87]) the symmetry of random variables is essential for the fulfillment of the decoupling principle, while other papers (e.g., [Hit88, Zin86, KS89]) point out the role of positivity. Norms of L^{p}-spaces, or more general, of Orlicz spaces (basically, subject to growth restriction) have become typical means of comparison of two classes of vector random variables.

In this paper, we prove decoupling inequalities for random variables that are not necessarily symmetric. Theorems 2.1 and 2.3 in Section 2, and Theorem 3.8 in Section 3, are our main results. The decoupling principle by means of probability tails, Theorem 3.8, immediately ensures the parity of tightness of two types of chaoses (that Gaussian decoupled and coupled chaoses are simultaneously tight was proved in [Kwa87]).

A number of decoupling results are obtained for arbitrary rearrangement invariant norms and Orlicz functionals. In particular, we provide one extended example regarding certain Lorentz norms (important in the approximation theory). Another application is the decoupling principle for Ustatistics (a result as in Theorem 2.3 was proven in [dlPn92]).

The utilized techniques are based on ideas, borrowed from [Kwa87], while some are taken from [KW92]. Proofs are straightforward and point out the algebraic nature of decoupling that is fruitfully merged with a widely understood context of convexity. A rule of thumb is that, in the field of random diagonal-free polynomials, a “definable” is “decouplable”. The obtained robust constants are tightly estimated, and are sharper than constants known before.

In the last section, we show tail probability decoupling results for polynomials of symmetric random variables. This section makes use of techniques from [AMS92].

1.2 Notation

Random variables in this paper are defined on a separable probability space (Ω, A, P) that is rich enough to carry independent sequences. A sequence of real random variables is denoted by ξ = (ξ_{1}, ξ_{2}, …), and a matrix of real random variables is denoted by 𝕏 = [ξ_{1}, …, ξ_{n}], where ξ_{j} = (ξ_{j1}, ξ_{j2}, …). We will make particular use of one particular sequence, the Rademacher sequence ε = (ε_{i}), where ε_{i} are independent random variables taking values ±1 with probability 1/2.

Let 𝔼 = (𝔼, ∥·∥) denote a real Banach space. We will be considering 𝔼 valued random variables, that is, strongly measurable mappings from Ω into 𝔼.

Let k be a positive integer and f = (f_{i_{1},...,i_{k}}) be an array of vectors from 𝔼 taking only finitely many non-zero values. Throughout the paper, all such arrays are assumed to vanish on diagonals (We will say “diagonal-free”), i.e., f_{i_{1},...,i_{k}} = 0, if at least two indices i_{j}, i_{j′} are equal.

The main object of our interest will be the following k-homogeneous random polynomial

Q(f; 𝕏) =^{df} Q(f; ξ_{1}, …, ξ_{k}) =^{df} ∑_{i_{1},...,i_{k}} f_{i_{1},...,i_{k}}ξ_{1i_{1}} ⋯ ξ_{ki_{k}}.

We will be desiring to compare this random polynomial with the ‘undecoupled’ version, that is

Q(f; ξ^{k}) =^{df} Q(f; ξ, …, ξ) =^{df} ∑_{i_{1},...,i_{k}} f_{i_{1},...,i_{k}}ξ_{i_{1}} ⋯ ξ_{i_{k}}. (1.1)

The first term in the above definition will be introduced as a notational convenience. We will be quite free to stretch the use of this notation. So for example, we might write

Q(f; ξ^{r}, η^{k−r}) = Q(f; ξ, …, ξ, η, …, η) = ∑_{i_{1},...,i_{k}} f_{i_{1},...,i_{k}}ξ_{i_{1}} ⋯ ξ_{i_{r}}η_{i_{r+1}} ⋯ η_{i_{k}}.

We hope to convince the reader of the value of this notation, as it enables us to write many of the proofs in a more compact form, and may ultimately lead to a clearer thinking on the subject. For the unconvinced reader, we hope that we have explained the notation sufficiently that he will be able to rewrite all the proofs and statements below in a more familiar form.

Many of the inequalities that we introduce require the array f to satisfy certain symmetry conditions, and so for this reason we introduce the symmetrized version of f:

f̂_{i_{1},...,i_{k}} =^{df} \frac{1}{k!} ∑_{σ} f_{i_{σ_{1}},...,i_{σ_{k}}}

where the sum is taken over all permutations of the set [1, k] = { 1, …, k }, and let Q̂(f; ·) = Q(f̂, ·). Note that for the undecoupled random polynomial symmetry makes no change: Q̂(f; ξ^{k}) = Q(f; ξ^{k}).

In the sequel, we will occasionally refer to tetrahedral arrays, i.e. f such that f_{i_{1},...,i_{k}} = 0, if indices fail to satisfy i_{1} < … < i_{k}.

We will make frequent use of the following identity, which is known as the Mazur-Orlicz polarization formula [MO35]:

Q̂(f; ξ_{1}, …, ξ_{k}) = \frac{1}{k!} ∑_{δ=(δ_{1},...,δ_{k})∈\{ 0,1 \}^{k}}(−1)^{k−|δ|}Q(f; (δ_{1}ξ_{1} + … + δ_{k}ξ_{k})^{k}), (1.2)

where |δ| = ∑_{i} δ_{i}. Switching to a Rademacher sequence ε, we can write

Q̂(f, ξ_{1}, …, ξ_{k}) = \frac{1}{k!}E ε_{1} ⋯ ε_{k} Q(f, (∑_{i=1}^{k} ε_{i}ξ_{i})^{k}), (1.3)

where the expectation is only over the Rademacher sequence ε.

Rearrangement invariant spaces By (𝕃, ∥ · ∥_{𝕃}) we denote a rearrangement invariant Banach space of integrable random variables (so that the norm of a random variable depends only on its probability distribution), 𝕃 ⊂ L^{1}(P), defined on a separable probability space (Ω, A, P) that is rich enough to carry independent sequences. For more information on rearrangement invariant spaces, we refer the reader to, e.g., [LT79]. The basic examples of rearrangement invariant spaces are 𝕃 = L_{p}, for 1 ≤ p ≤ ∞, (nothing more is needed in many parts of this paper), Orlicz spaces and Lorentz spaces. We will sometimes use the abbreviation ‘r.i.’ for ‘rearrangement invariant.’

The important property of rearrangement invariant spaces that we shall use is the following:

(1.4) Conditional expectations are contractions acting on 𝕃.

The reader unfamiliar with r.i. spaces should note that this is true of L_{p}.

We denote by 𝕃(𝔼) the Banach space of 𝔼-valued random variables (i.e., strongly measurable mappings from Ω into 𝔼) whose norms belong to 𝕃, and let ∥θ∥_{𝕃(𝔼)} = ∥ ∥θ∥_{𝔼} ∥_{𝕃}. Thus if 𝕃 = L_{p}, then ∥θ∥_{𝕃(𝔼)} = ∥θ∥_{L_{p}(𝔼)} = (E∥θ∥p_{𝔼})^{1/p}. In the sequel we sometimes omit the subscript indicating the space, if it causes no ambiguity.

2 Decoupling for r.i. norms

Interchangeability In the sequel, we will use several times the following elementary feature of interchangeable random sequences ξ_{1}, …, ξ_{r} (i.e., such that each permutation has the same distribution). Suppose that each ξ_{k} is itself a sequence of independent random variables. Denote by G_{r} the σ-field spanned by ∑_{j=1}^{r} ξ_{j}. Let f be diagonal-free. Then if j_{1}, …, j_{k} ≤ r, then

E[Q(f, ξ_{j_{1}}, …, ξ_{j_{k}}) | G_{r}] = r^{−k}Q(f, (ξ_{1} + … + ξ_{r})^{k}). (2.1)

We should point out that the last term represents the random polynomial

Q(f, (ξ_{1} + … + ξ_{r})^{k}) = ∑_{i_{1},...,i_{k}} f_{i_{1},...,i_{k}}(ξ_{1i_{1}} + … + ξ_{ri_{1}}) ⋯ (ξ_{1i_{k}} + … + ξ_{ri_{k}}).

Equation (2.1) follows because

E(f_{i_{1},...,i_{k}}ξ_{j_{1}i_{1}} ⋯ ξ_{j_{k}i_{k}}|G_{r}) = f_{i_{1},...,i_{k}}E(ξ_{j_{1}i_{1}}|G_{r}) ⋯ E(ξ_{j_{k}i_{k}}|G_{r})

because f is diagonal free, and hence ξ_{j_{1}i_{1}}, …, ξ_{j_{k}i_{k}} are independent if f_{i_{1},...,i_{k}} ≠ 0, and also because for j ≤ r

E(ξ_{ji}|G_{r}) = r^{−1}(ξ_{1i} + … + ξ_{ri}).

We also point out the following easy consequence of the triangle inequality for 𝕃.

∥Q̂(f, 𝕏)∥_{𝕃(𝔼)} ≤ ∥Q(f, 𝕏)∥_{𝕃(𝔼)}. (2.2)

Now we are ready to present our first decoupling inequality. This result allows us to decouple random polynomials in the rearrangement invariant norm.

Theorem 2.1 Let f = (f_{i_{1},...,i_{k}}) be a diagonal free array of vectors from 𝔼. Let ξ, ξ_{1}, …, ξ_{k} be sequences of integrable independent real random variables. Let 𝕃 be a r.i. space of random variables, containing ξ_{1} ⋯ ξ_{k} (hence, norms of all finitely supported polynomials spanned by ξ_{1}, …, ξ_{k}).

(A) Assume that ξ, ξ_{1}, ξ_{2}, … are independent and identically distributed. Then

∥Q(f, ξ^{k})∥_{𝕃(𝔼)} ≤ A∥Q(f, ξ_{1}, …, ξ_{k})∥_{𝕃(𝔼)}

where A = A_{k} ∼ (2k)^{k}, or, if Eξ = 0, A_{k} = k^{k}.

(B) Assume that ξ, ξ_{1}, …, ξ_{k} are interchangeable. Then

∥Q̂(f, ξ_{1}, …, ξ_{k})∥_{𝕃(𝔼)} ≤ B∥Q(f, ξ^{k})∥_{𝕃(𝔼)},

where B = B_{k} ∼ k^{k}/k!.

Proof.

During this proof, we will suppress the subscript 𝕃(𝔼) on the norms.

(A):

\underline{Step 1^{o}: Centering procedure}:

Denote \overline{ξ} = ξ − E[ξ], m = (m_{1}, m_{2}, …), where m_{i} = E[ξ_{i}]. For 1 ≤ r ≤ k, if f = (f_{i_{1},...,i_{r}}), then we have ∥Q(f, \overline{ξ}_{1}, …, \overline{ξ}_{r})∥ ≤ 2^{r} ∥Q(f, ξ_{1}, …, ξ_{r})∥. (2.3)

Indeed, by interchangeability

∥Q(f, \overline{ξ}_{1}, …, \overline{ξ}_{r})∥ = ∥Q(f, ξ_{1} − m, …, ξ_{r} − m)∥

= ∥ ∑_{(δ_{1},...,δ_{r})∈{ 0,1 }^{r}} ∑_{i_{1},...,i_{r}} f_{i_{1},...,i_{r}}ξ_{1i_{1}}^{δ_{1}} ⋯ ξ_{ri_{r}}^{δ_{r}}m_{1i_{1}}^{1−δ_{1}} ⋯ m_{ri_{r}}^{1−δ_{r}}∥

≤ ∑_{j=0}^{r} \binom{r}{j}∥ ∑_{i_{1},...,i_{r}} f_{i_{1},...,i_{r}}ξ_{1i_{1}} ⋯ ξ_{ji_{j}}m_{j+1,i_{j+1}} ⋯ m_{ri_{r}}∥

= ∑_{j=0}^{r}\binom{r}{j}∥Q(f, ξ_{1}, …, ξ_{j}, m^{(r−j)})∥.

The latter expression is equal to

∑_{j=0}^{r}\binom{r}{j}∥Q(f, ξ_{1}, …, ξ_{j}, E[ξ_{j+1}|𝕏_{j}], …, E[ξ_{r}|𝕏_{j}])∥ = ∑_{j=0}^{r}\binom{r}{j}∥E[Q(f, ξ_{1}, …, ξ_{j}, ξ_{j+1}, …, ξ_{r})|𝕏_{j}]∥.

where 𝕏_{j} is the matrix [ξ_{1}, …, ξ_{j}]. Using the contractivity property (1.4), we estimate the above term from above by

∑_{j=0}^{r}\binom{r}{j}∥Q(f, ξ_{1}, …, ξ_{r})∥ = 2^{r}∥Q(f, ξ_{1}, …, ξ_{r})∥.

\underline{Step 2^{o}: proving (A)}: Arguing similarly to Step 1^{o} above, we note that

∥Q(f, ξ^{k})∥ = ∥Q(f, (\overline{ξ} + m)^{k})∥ ≤ ∑_{r=0}^{k} \binom{k}{r}∥Q(f, \overline{ξ}^{r}, m^{k−r})∥ := Q_{0}.

Now, using (1.4), and noting that E(\overline{ξ}_{1} + ⋯ + \overline{ξ}_{r}|\overline{ξ}_{1}) = \overline{ξ}_{1}, which is identically distributed to \overline{ξ}, it follows that

Q_{0} ≤ ∑_{r=0}^{k} \binom{k}{r}∥Q(f, (\overline{ξ}_{1} + ⋯ + \overline{ξ}_{r})^{r}, m^{k−r})∥

Then, in virtue of (2.1), the latter expression is equal to

∑_{r=0}^{k} \binom{k}{r}∥ r^{r} E[ Q(f, \overline{ξ_{1}}, …, \overline{ξ_{r}}, m^{k−r}) | G_{r}]∥ =: Q_{1}

Using (1.4) and applying the centering procedure (2.3), the above term gets the following upper bounds:

Q_{1} ≤ ∑_{r=0}^{k} \binom{k}{r}∥ r^{r} Q(f, \overline{ξ_{1}}, …, \overline{ξ_{r}}, m^{k−r})∥

≤ ∑_{r=0}^{k} \binom{k}{r}(2r)^{r}∥ Q(f, ξ_{1}, …, ξ_{r}, m^{k−r})∥ =: Q_{2}

Then, by interchangeability, independence of columns, and (1.4) again, we keep estimating, as follows

Q_{2} = ∑_{r=0}^{k} \binom{k}{r}(2r)^{r} ∥E[ Q(f, ξ_{1}, …, ξ_{r}, ξ_{r+1}, …, ξ_{k}) | 𝕏_{r}]∥

≤ ∑_{i=0}^{k} \binom{k}{r}(2r)^{r} ∥Q(f, ξ_{1}, …, ξ_{r}, ξ_{r+1}, …, ξ_{k})∥

= A_{k}∥ Q(f, ξ_{1}, …, ξ_{k})∥.

Clearly, A_{k} ≤ (2k + 1)^{k} (notice that A_{k} ≥ c(2k)^{k}). If Eξ = 0, the use of centering procedure, and the triangle inequality is superfluous, hence the constant decreases to k^{k}.

(B): By the Mazur-Orlicz polarization formula (1.2), and (2.1), we obtain the following bounds

\begin{matrix}∥Q̂(f, ξ_{1}, …, ξ_{k})∥ = & ∥\frac{1}{k!}Q(f, ∑ ; δ(−1)^{k−|δ|}(δ_{1}ξ_{1} + … + δ_{k}ξ_{k})^{k})∥ \\ ≤ & ∑ ; δ ∥\frac{1}{k!}Q(f, (δ_{1}ξ_{1} + … + δ_{k}ξ_{k})^{k})∥ \\ = & ∑_{r=0}^{k} \binom{k}{r}∥\frac{1}{k!}Q(f, (ξ_{1} + … + ξ_{r})^{k})∥ \\ ≤ & ∑_{r=0}^{k} \binom{k}{r}∥\frac{r^{k}}{k!} Q(f, ξ_{1}^{k})∥ \\ = & B_{k}∥Q(f, ξ_{1}^{k})∥. \end{matrix}

That B_{k} ∼ k^{k}/k! is easy to verify. The proof is completed.

2.1 Extended multilinear forms and U-Statistics

In this section, we show how to extend Theorem 2.1 to the so called U-Statistics. Let k be a positive integer. Let F = (F_{i_{1},...,i_{k}}) be an array of strongly Borel measurable functions F_{i_{1},...,i_{k}} : ℝ^{k} → 𝔼 such that

(F0)  F_{i_{1},...,i_{k}} = 0 if some i_{j} and i_{j′} are identical for j ≠ j′
(F1)  F_{i_{1},...,i_{k}} = 0 for all but finitely many (i_{1}, …, i_{k});

Then we are going to consider U-Statistics, that is, expressions of the form

F (ξ_{1}, …, ξ_{k}) =^{df} ∑_{i_{1},...,i_{k}} F_{i_{1},...,i_{k}}(ξ_{1i_{1}}, …, ξ_{ki_{k}}),

where ξ_{1}, …, ξ_{k} are real valued random variables. (Here, ℝ could be replaced with any other measure space, but there is no loss of generality to take it as ℝ.) We are going to exercise the same notational devices as for the random polynomials, so that the undecoupled U-Statistic is written

F (ξ^{k}) =^{df} ∑_{i_{1},...,i_{k}} F_{i_{1},...,i_{k}}(ξ_{i_{1}}, …, ξ_{i_{k}}).

As before, in order to prove the results, we require certain symmetry properties to hold for F. So we defined the symmetrized version of F as follows:

F̂(x_{1}, …, x_{k}) =^{df} \frac{1}{k!} ∑_{σ} F_{i_{σ_{1}},...,i_{σ_{k}}} (x_{i_{σ_{1}}} , …, xi_{i_{σ_{k}}}),

where the sum runs over all permutations of [1, k], and we set

F̂(ξ_{1}, …, ξ_{k}) =^{df} ∑_{i_{1},...,i_{k}} F̂_{i_{1},...,i_{k}}(ξ_{1i_{1}}, …, ξ_{ki_{k}}).

Decoupling results were proved by [dlPn92] for Orlicz modulars (and so by Note 8 below, one can obtain results for all rearrangement invariant spaces). We will prove similar decoupling results, weakening some of the hypotheses.

More interestingly, we are going to prove the decoupling results for U-statistics as a corollary of Theorem 2.1 which decouples random polynomials. The technique is to approximate the Ustatistic as a sum of random polynomials. That is, let D be an integer, and for 1 ≤ d ≤ D, let f^{d} = (f_{i_{1},...,i_{k}}^{d}) be a diagonal free array of vectors in 𝔼 taking only finitely many non-zero values, and let (ξ_{1}^{d} : 1 ≤ d ≤ D), …, (ξ_{k}^{d} : 1 ≤ d ≤ D) be sequences of independent random variables. Then we set

R(f; ξ_{1}, …, ξ_{k}) =^{df} ∑_{d=1}^{D} Q(f^{d}; ξ_{1}^{d}, …, ξ_{k}^{d}). (2.4)

Then the remarkable thing is that the proof of Theorem 2.1 works for R(f; ξ_{1}, …, ξ_{k}) exactly as it does for Q(f; ξ_{1}, …, ξ_{k}), that is, we have the following result.

Theorem 2.2 Theorem 2.1 is valid, for the multilinear form (2.4).

A version of the following result for Orlicz modulars Eϕ(·), where ϕ was a moderately increasing function, was proved in [MT87]. In that paper, terms of the underlying sums were sign-randomized, i.e., each F (i, ·) was multiplied by Walsh functions ε_{i_{1}} ⋯ ε_{i_{k}}. More precisely, the decoupling was proved for

(F ∘ ε)(ξ_{1}, …, ξ_{k}) =^{df} ∑ ; i ε_{i_{1}} ⋯ ε_{i_{k}}F_{i_{1},...,i_{k}}(ξ_{1i_{1}}, …, ξ_{ki_{k}}), (2.5)

where ε is independent of 𝕏. That the presence of Walsh functions is not necessary in the context of Orlicz modulars, was shown in [dlPn92]. We observe that the following result, generalizing theorems in the mentioned papers, is implicit in the main decoupling principle. Moreover, constants remain the same. For the sake of completeness we give the full proof.

In the proof we will use the fact that any inequality involving norms of functions of discrete r.v.’s, that converge to some limits, is preserved for these limits. Fix i, say i = (1, …, k). Consider X = F (ξ_{1}, ⋯, ξ_{k}). We may assume that the probability space (Ω, F, P) and F is spanned by ξ_{1}, ξ_{2}, . . .. Also, we may assume that it is separable, i.e., F_{n} = σ{⋃_{n} F_{n}}, where F_{n} are finite σ-fields. Put ξ_{i}^{n} = E[ξ|F_{n}]. Thus E[X|F_{n}] → E[X|F] = X a.s. and in 𝕃.

Theorem 2.3 Let F : ℕ^{k} × ℝ^{k} → 𝔼 satisfy (F0) – (F1), and also and additional condition:

(F2)  F (i; ξ_{i_{1}}, …, ξ_{i_{k}}) ∈ 𝕃 for every i ∈ ℕ^{k}.

Let ξ_{1}, ξ_{2}, … be sequences of independent random variables.

(A′) Let ∥ · ∥ be a r.i. norm. Let ξ_{1}, ξ_{2}, … be independent and identically distributed. Then

∥ F (ξ^{k}) ∥ ≤ A ∥ F (𝕏) ∥ ,

where A is the constant from Theorem 2.1.(A).

(B′) Let ξ_{1}, …, ξ_{k} be interchangeable (in particular, i.i.d.) and ∥ · ∥ be a r.i. norm. Then

∥ F̂(𝕏) ∥ ≤ B_{k} ∥ F (ξ^{k}) ∥ .

where B is the constant from Theorem 2.1.(B).

Proof. By Note 8, we may assume that the rearrangement space 𝕃 is separable. In that case, we may assume without loss of generality that ξ_{1}, ξ_{2}, … are real discrete random variable.

Thus we may assume that the random variables are defined on a product probability space

(∏_{ij} Ω_{ij}, (F^{⊗N})^{⊗k}, (P^{⊗ℕ})^{⊗k}),

where Ω_{ij} are equal, Ω_{i} = ∏_{j} Ω_{ij}, and the superscript ⊗ indicates the product σ-field and the product probability, respectively. So, let

ξ_{i} = ∑_{m} x_{im}1I_{A_{im}},

where A_{i1}, A_{i2}, … ⊂ Ω_{i} are bases of rectangular sets that form a disjoint finite partition of Ω, and let (A_{sim}), s = 1, …, k, be independent copies of (A_{im}). Put I_{sim} = 1I_{A_{sim}}, hence

ξ_{si} = ∑_{m} x_{im}I_{sim}, s = 1, …, k.

Then

∑_{i_{1},...,i_{k}} F_{ik}(ξ_{1i_{1}}, …, ξ_{ki_{k}}) = ∑_{m_{1},...,m_{k}} ∑_{i_{1},...,i_{k}} F_{i_{1},...,i_{k}}(x_{i_{1}m_{1}}, …, x_{i_{k}m_{k}})I_{1i_{1}m_{1}} ⋯ I_{ki_{k}m_{k}}.

Now, we can apply Theorem 2.2, and the proof is complete.

2.2 An example in a certain Lorentz space

Motivated by the results in [dlPn93] where the problem as to when expectation results imply tail probability results is treated, we obtained the following asymptotic tail probability comparison.

Proposition 2.4 Let ξ or η be the norm of F̂(ξ^{k}) or F̂(𝕏), and let W : [0, ∞) → [0, ∞) be an increasing function such that there exists constants p > 1 and c > 0 such that W (st) ≤ cs^{p}W (t) for all 0 < s < 1 and all t > 0. Then there is a constant C, depending only on p and c, such that

lim sup_{t→∞} W (t)P(ξ ≥ t) ≤ lim sup_{t→∞} W (t)P(η ≥ Ct).

This result is a consequence of Theorem 2.3, and follows by arguments from the theory of Lorentz-Zygmund spaces. If ξ is a random variable, let F (t) = P(|ξ| ≥ t), and define the decreasing rearrangement of ξ to be the function ξ^{∗}(t) =^{df} sup{s : F (s) > t} (i.e. the right-continuous inverse of F). Obviously, |ξ| and ξ^{∗} are equidistributed. When ξ is integrable, an average operator is often considered

ξ^{∗∗}(t) = \frac{1}{t} ∫_{0}^{t} ξ^{∗}(u) du,

which corresponds to a rearrangement invariant norm for every t > 0. Therefore, our decoupling inequalities for U-statistics hold for Φ(X) = (∥X∥)^{∗∗}. That is, denoting by ξ or η the norm of F̂(ξ^{k}) or F̂(𝕏), we have

ξ^{∗∗}(t) ≤ Cη^{∗∗}(t)

for some constant C > 0.

Now consider the Lorentz-Zygmund space defined by the quasi-norm |||f||| = sup_{x} w(x)ξ^{∗}(x), where w : [0, 1] → [0, ∞) is an increasing function. Note that |||f||| ≤ 1 if and only if sup_{t} W (t)P(|f| ≥ t) ≤ 1, where w(t) = \frac{1}{W^{−1}(1/t)}. If W satisfies the relation given in Proposition 2.4, then for some

constant c, the function w satisfies the relation w(x) ≤ ca^{−1/p}w(xa) for a ≤ 1. Then it is possible to show that |||f||| ≤ |||f^{∗∗}||| ≤ C|||f|||. Indeed, the first inequality is obvious, and for the second:

w(x)f^{∗∗}(x) = w(x) ∫_{0}^{1} f^{∗}(xa) da ≤ c ∫_{0}^{1} a^{−1/p}w(xa)f^{∗}(xa) da ≤ \frac{cp}{p − 1}|||f|||.

Thus, to show Proposition 2.4, let F (t) = P(ξ ≥ t) and G(t) = P(η ≥ t). Then

sup_{t} W (t)F (t) ≤ sup_{t} W (t)G(Ct).

If we now set w(x) = 0, for x ≥ x_{0}, the same argument applies, and letting x_{0} → ∞, we obtain

lim sup_{t→∞} W (t)F (t) ≤ lim sup_{t→∞} W (t)G(Ct).

3 A discourse on probability tails

3.1 L^{p}-estimates imply tail estimates

3.1.1 Auxiliary results

The following result can be found in [AMS92].

Lemma 3.1 Let {X; X_{i}} be a sequence of positive i.i.d. random variables. Then, for all positive integers n, all α > 0 and all 0 ≤ θ ≤ n

P(X ≥ α) ≥ \frac{θ}{n}  ⇒ P(sup_{1≤i≤n} X_{i} ≥ α) ≥ \frac{θ}{1+θ} ,
P(X ≥ α) ≤ \frac{θ}{n}  ⇒ P(sup_{1≤i≤n} X_{i} ≥ α) ≤ θ.

Proof. To show the first inequality, observe first that for θ > 0,

(1 − \frac{θ}{n})^{n} ≤ \frac{1}{(1 + θ)}.

Hence, by independence assumption,

P(sup_{j} X_{j} ≥ α) = 1 − P(sup_{j} X_{j} < α)

= 1 − ∏_{j=1}^{n} P(X_{j} < α)r

≥ 1 − (1 − \frac{θ}{n})^{n} ≥ 1 − \frac{1}{(1 + θ)} = \frac{θ}{(1 + θ)}.

The second inequality is easy: from the imposed condition, one gets

P( sup_{1≤j≤n} X_{j} ≥ α) ≤ ∑_{i=1}^{n} P(X_{i} ≥ α) ≤ θ.

The proof is completed.

The following result can be found in [LT91, Chap. 4].

Lemma 3.2 Consider a positive random variable Z such that ∥Z∥_{q} ≤ C∥Z∥_{p} for q > p > 0. Then,

P(Z > t) ≤ (2C^{p})^{\frac{q}{p−q}}  ⇒  ∥Z∥_{p} ≤ 2^{1/p}t and ∥Z∥_{q} ≤ 2^{1/p}Ct.

Putting together Lemmas 3.1 and 3.2 we get the following.

Lemma 3.3 Let {X; X_{i}} be a sequence of positive i.i.d. random variables. Assume that there exists a constant c such that for 0 < p < q < ∞

∥ sup_{1≤i≤n} ∥X_{i}∥∥_{q} ≤ c∥ sup_{1≤i≤n} ∥X_{i}∥∥_{p}.

Then, letting θ = (2c^{p})^{\frac{q}{p−q}},

P(X ≥ t) ≤ \frac{θ}{n} ⇒ ∥ sup_{1≤i≤n} X∥_{p} ≤ 2^{\frac{1}{p}} t.

For later reference, we also include the next lemma.

Lemma 3.4 Let {X; X_{i}} be a sequence of positive i.i.d. random variables. Then,

∥ sup_{1≤i≤n} X_{i}∥_{p} ≤ t ⇒ P(X ≥ 2^{\frac{1}{p}} t) ≤ \frac{1}{n}.

Proof. Use Chebychev’s inequality and Lemma 3.1 with θ = 1.

3.1.2 Main result

Now we are ready to prove an extension of a result from [AMS92] that deals with strict tail probability comparisons for pairs of random variables.

Theorem 3.5 Let (X, X_{i}) and (Y, Y_{i}) be sequences of positive i.i.d. random variables. For some 0 < p < q and all positive integers n assume that

∥ sup_{1≤i≤n} X_{i}∥_{q} ≤ c_{1}∥ sup_{1≤i≤n} X_{i}∥_{p}, (3.1)

and

∥ sup_{1≤i≤n} Y_{i}∥_{p} ≤ c_{2}∥ sup_{1≤i≤n} X_{i}∥_{p}.

Then there exists c_{3}, depending only on p, q, c_{1} and c_{2} such that for all t ≥ 0

P(Y ≥ c_{3}t) ≤ c_{3}P(X ≥ t).

Proof. Given an arbitrary α = α_{1} > 0 with

P(Y ≥ α_{1}) > 0, (3.2)

choose μ to be the smallest positive integer satisfying

\frac{1}{2μ} ≤ P(Y ≥ α_{1}) ≤ \frac{1}{μ}. (3.3)

¿From Lemma 3.1 it follows that

P( sup_{1≤j≤μ} Y_{j} ≥ α_{1}) ≥ \frac{1}{3}.

Hence, by Chebychev’s inequality,

\frac{1}{α_{1}^{p}} ∥ sup_{1≤j≤μ} Y_{j}∥_{p}^{p} ≥ \frac{1}{3},

which, by assumptions, yields

\frac{c_{2}^{p}}{α_{1}^{p}} ∥ sup_{1≤j≤μ} X_{j}∥_{p}^{p} ≥ \frac{1}{3},

and, consequently, for any α_{2} > 0,

\frac{1}{α_{2}^{p}} ∥ sup_{1≤j≤μ} X_{j}∥_{p}^{p} ≥ \frac{α_{1}^{p}}{3α_{2}^{p}c_{2}^{p}} .

In particular, if α_{2}^{p} = \frac{α_{1}^{p}}{6c_{2}^{p}}, we get from the latter inequality that

\frac{1}{α_{2}^{p}} ∥ sup_{1≤j≤μ} X_{j}∥_{p}^{p} ≥ 2.

Now, Lemma 2.3 implies that

P(X ≥ \frac{α_{1}}{(6^{\frac{1}{p}} c_{2})}) = P(X ≥ α_{2}) ≥ (2c_{1}^{p})^{\frac{q}{p−q}} \frac{1}{μ}.

Finally, (3.3) gives,

P(X ≥ \frac{α_{1}}{(6^{\frac{1}{p}} c_{2})}) ≥ (2c_{1}^{p})^{\frac{q}{p−q}} P(Y ≥ α_{1}). (3.4)

Note that (3.4) holds for all α_{1} for which (3.2) holds. For any other α_{1} > 0, (3.4) holds trivially.

3.1.3 Contraction for multipliers

Condition (3.1) yields an example of a class of random variables with the so called Marcinkiewicz-Paley-Zygmund property (MPZ in short). The concept was studied in [KS88], and can be traced back to [PZ32] and [MZ37]. A family Z ∈ L_{+}^{q} of random variables is said to be in the class MP Z(q) (in short: have MPZ), if one of the following equivalent conditions is satisfied:

(3.5) ∃ p < q (equivalently, ∀ q ≤ p) m_{q,p} =^{df} sup_{Z∈Z} \frac{∥Z∥_{q}}{∥Z∥_{p}} < ∞

∃ δ > 0 inf_{Z∈Z} P(Z > δ∥Z∥_{q}) > δ (3.6)

That is, (3.1) involves Z = { sup_{1≤i≤n} ∥X_{i}∥ : n ∈ ℕ }. Also, in [KS88] it was shown that the space of diagonal-free random polynomials of finite degree, spanned by symmetric random variables with, so called, semi-regular distribution, has MPZ. A random variable ξ is said to have the semiregular distribution, if its tail G(t) = P(|ξ| > t) satisfies the relation,

V (a) = lim sup_{t→∞} G(at)/G(t) < 1

for some (or all) a > 1 (by convention, 0/0 = 0). For example, any bounded random variable has semi-regular distribution. In particular, (the norm of) any normed space-valued Rademacher polynomial of degree d has MPZ with the constant m_{qp} = [2(q − 1)/(p − 1)]^{d} ([KS88, Corollary 2.7]).

Now, the essence of Theorem 3.5 is that the continuity of a certain operator, once is fulfilled by means of L^{p}-norms, will be also fulfilled by means of probability tails. We will illustrate this concept by the following result.

Theorem 3.6 Let f_{i_{1},...,i_{k}} be a, finitely supported, diagonal-free array, taking values in a Banach space 𝔼. Let ξ = (ξ_{1}, …) be a sequence of symmetric independent random variables.

(i) (Contraction Inequality) There is a constant c > 0 such that

P(∥Q(f, (sξ)^{k})∥ > ct) ≤ cP(∥Q(f, ξ)∥ > t), t > 0,

where sξ = (s_{i}ξ_{i}) and ∥s∥_{∞} = sup_{i} |s_{i}| ≤ 1.

(ii) (Maximal Inequality) There is a constant C > 0 such that

P( sup_{m_{1},...,m_{k}} ∥T_{m_{1},...,m_{k}}Q(f, ξ^{k})∥ > Ct) ≤ CP(∥Q(f, ξ^{k})∥ > t), t > 0,

where

T_{m_{1},...,m_{k}}Q(f; 𝕏) = ∑_{i_{1}≤m_{1},...,i_{k}≤m_{k}} f_{i_{1},...,i_{k}}ξ_{1i_{1}} ⋯ ξ_{ki_{k}}.

Proof. Let us first prove (i). In virtue of symmetry assumption and Fubini’s theorem, it suffices to give the proof for the case when 𝕏 is a matrix of Rademacher random variables. Let Q_{1}, …, Q_{n} be independent copies of Q(f,). Then the vector (Q_{1}, …, Q_{n}) is a Rademacher homogeneous polynomial of degree k taking values in ℓ_{n}^{∞}(𝔼). Similarly, let R_{1}, …, R_{n} be independent copies of Q(f, sξ). ¿From the contraction principle for L^{p}-norms, which may be found in [KS88, Remark 2.9] (essentially, it is due to [Kwa87]), it follows that for all p ≥ 1

∥∥(R_{1}, …, R_{n})∥_{ℓ_{n}^{∞}(𝔼)}∥_{p} ≤ c∥∥(Q_{1}, …, Q_{n})∥_{ℓ_{n}^{∞}(𝔼)}∥_{p}.

¿From the observation that ∥(Q_{1}, …, Q_{n})∥_{ℓ_{n}^{∞}(𝔼)} = sup_{1≤i≤n} ∥Q_{i}∥_{𝔼}, and using the fact that Rademacher polynomials are MPZ, and also citing Theorem 3.6 above, the result follows.

The proof of part (ii) is the same, using the corresponding result for L^{p}-norms of polynomials for the Rademacher random variables, which follows easily from [MT86a] and Lévy’s inequality.

Remark 1 In fact, Theorem 3.6 is also valid for the sign randomized U-Statistics as in equation (2.5). The proof is identical.

Theorem 3.7 [ Comparison Inequality] Let (f_{i_{1},...,i_{k}}) be a diagonal-free, finitely supported, diagonal free array. Let ξ = (ξ_{i}) and η = (η_{i}) be sequences of symmetric independent random variables. such that, for some constant A > 0,

P(|ξ_{i}| > t) ≤ AP(|η_{i}| > t), t > 0, i ∈ ℕ.

Then, for some constant K = K(c, d, A),

P(∥Q(f, ξ^{k})∥ > t) ≤ KP(K∥Q(f, η^{k})∥ > t), t > 0.

Proof. We have

Q(f, ξ^{k}) =^{D} Q(f, (ε|ξ|)^{k}),

and

Q(f, η^{k}) =^{D} Q(f, (ε|η|)^{k}),

where ε is a Rademacher sequence independent of ξ and η.

If A = 1, then we may replace each |ξ_{i}| and |η_{i}| by their decreasing rearrangements |ξ_{i}|^{∗} and |η_{i}|^{∗}, respectively. The assumption yields |ξ_{i}|^{∗} ≤ |η_{i}|^{∗} a.s. Hence, by Theorem 3.6(i), the inequality follows.

Let A > 1. Then there exist a sequence α = (α_{i}) of i.i.d. random variables, independent of ξ, such that P(α_{i} = 1) = 1/K, P(α_{i} = 0) = 1−1/K, so that P(α_{i}|ξ_{i}| > t) = P(|ξ_{i}| > t)/K. Therefore, by the first part of the proof,

cP(∥Q(f, (εα|ξ|)^{k})∥ > t) ≤ P(K∥Q(f, (ε|η|)^{k})∥ > t), t > 0.

Conditioning on ξ, it remains to prove that, for every polynomial Q, and every diagonal free array f_{i_{1},...,i_{k}}, P(∥Q(f, ε^{k})∥ > mt) ≤ mP(∥Q(f, (εα)^{k})∥ > t) (3.7)

for some constant m = m_{k}. Let β = εα. Then it is clear that β is semiregular, as defined earlier, and hence homogeneous random polynomials of degree k over β have MPZ. Furthermore, the comparison inequality is true for L^{p} for p ≥ 1 (see, for example, [KS88, Theorem 2.13]) Hence arguing as in the proof of Theorem 3.6, we obtain (3.7).

3.2 Decoupling for tails

In order to prove any tail inequality of the type P(ξ > t) ≤ KP(η > t), where ξ, η are real random variables, it is enough to prove it for an arbitrarily chosen conditional probability

P [ ξ > t | G] ≤ KP [ η > t | G] .

This observation was used in proving the inequality [KW92, (6.9.5)]. Denote by G the σ-field spanned by all random variables of the form ∑_{j=1}^{i} h(ξ_{j}) (in other words, by the random point measure

∑_{j=1}^{i} δ_{ξ_{j}}

on (ℝ^{ℕ})^{k} (cf. [KW92, p. 182]). Then (ξ_{1}, …, ξ_{k}) is concentrated on a finite permutation invariant subset of (ℝ^{ℕ})^{k}. Now, (2.1) can be rewritten, as follows (recall the notation, preceding (2.1)).

E[Q(f, ξ_{j_{1}}, …, ξ_{j_{k}}) | G] = k^{−k}Q(f, (ξ_{1} + … + ξ_{k})^{k}), (3.8)

Theorem 3.8 Let f, ξ, ξ_{1}, …, ξ_{k} be as in Theorem 2.1 (but \underline{we do not assume integrability}).

(A′′) Let ξ, ξ_{1}, ξ_{2}, … be independent and symmetric. Then, there exists a constant A′′, depending only on k, such that, for all t ≥ 0,

P(∥Q(f, ξ^{k})∥∥ ≥ A′′t) ≤ A′′P(∥Q(f, ξ_{1}, …, ξ_{k})∥ ≥ t).

(B′′) Let ξ, ξ_{1}, …, ξ_{k} be interchangeable. Then, there exists some constant B′′, depending only on k, such that, for all t ≥ 0,

P(∥Q̂(f, ξ_{1}, …, ξ_{k})∥ ≥ B′′t) ≤ B′′P(∥Q(f, ξ^{k})∥ ≥ t).

Proof.

(A′′): By symmetry, using Theorem 3.6.(ii), with η = ξ_{1} + … + ξ_{k} and A = k, we obtain that

P(Q(f, ξ^{k})∥ ≥ tK) ≤ KP[k^{k}Q(f, (ξ_{1} + … + ξ_{k})^{k})∥ ≥ tK)

By (3.8), and inequality [KW92, (6.9.5)], the latter quantity can be estimated from below by

c_{k} K P(k^{2k}k^{k}Q(f, ξ_{1}, …, ξ_{k})∥ ≥ t),

which completes the proof of (A′′).

(B′′): By the Mazur-Orlicz polarization formula (1.2), and (2.1), we obtain the following estimates

\begin{matrix}P(∥Q̂(f, ξ_{1}, …, ξ_{k})∥ ≥ t) = & P(∥\frac{1}{k!}Q(f, ∑ ; δ(−1)^{k−|δ|}(δ_{1}ξ_{1} + … + δ_{k}ξ_{k})^{k})∥ ≥ t) \\ ≤ & ∑ ; δ P(2^{k}∥\frac{1}{k!}Q(f, (δ_{1}ξ_{1} + … + δ_{k}ξ_{k})^{k})∥ ≥ t) \\ = & ∑_{i=0}^{k} \binom{k}{i}P(2^{k}∥\frac{1}{k!}Q(f, (ξ_{1} + … + ξ_{i})^{k})∥ ≥ t). \end{matrix}

By (3.8), with j_{1} = … = j_{k}, and the inequality [KW92, (6.9.5)], we estimate the above expression from above by

\begin{aligned}& c_{k}^{−1} ∑_{i=0}^{k} \binom{k}{i}∥P(\frac{(2i)^{k}}{k!} Q(f, ξ_{1}^{k})∥∥ ≥ t) \\ ≤ & c_{k}^{−1}2^{k}P(\frac{(2k)^{k}}{k!} ∥Q(f, ξ_{1}^{k})∥ ≥ t), \end{aligned}

which completes the proof.

Remark 2 While the symmetry assumption is irrelevant in condition (B′′) (or in (B), before), the symmetrization procedure used in the proof of (A) fails. The reason is, that we use the conditioning on G, which destroys the independence, which is essential in applications of (2.3).

4 Notes

1. The inverse estimate in (2.3) is not true, in general, even if k = 1. For example, let ξ_{1}, …, ξ_{n} be Bernoulli random variables with p = P(ξ_{1} = 1) = 1/2, and f(i) = 1. Then E|f\overline{ξ}|^{2} = n/4 and E|fξ|^{2} = (n + n^{2})/4.

2. The symmetry of functions f is essential in Theorem 2.1.(B) and its analogs, as was pointed out in [MT87]. The Bourgain’s counterexample, given there, involves 𝔼 = ℓ^{2}⊗ℓ^{2} endowed with the projective norm ∥a∥ = inf { ∑_{i,j} ∥a_{i}^{1}∥ · ∥a_{j}^{2}∥ : a = ∑_{i,j} a_{i}^{1} ⊗ a_{j}^{2} }, Rademacher chaoses, and tetrahedral functions f. However, the inequalities (B) of both Theorems 2.1 and 2.3 hold for tetrahedral Rademacher chaoses induced by ξ and 𝕏 (with independent columns), whenever 𝔼 is (a) a Banach lattices with no subspace isomorphic to c_{0}, or (b) a UMD-space.

3. The full analog of Theorem 2.1 is valid in locally convex spaces.

4. The decoupling results from Section 2 can be carried over to linear spaces over the field of complex numbers. To obtain similar results for Section 3 is more difficult. One approach is to show that if ε denotes a sequence of independent Rademacher random variables, and if σ denotes a sequence of independent Steinhaus random variables (that is, σ_{i} is uniformly distributed over the complex unit circle), then ∥Q(f; ε^{k})∥ ≈ ∥Q(f; σ^{k})∥. We omit the details of the development.

5. In the case when the tail decoupling holds, i.e., in Theorems 3.8, 2.3.(A′′′) and (B′′′), we obtain the comparison of tightness. That is, for a family of functions { f : f ∈ F }, we have that, if one type of chaos { Q_{d}(f) : f ∈ F } is tight, so is the other, { Q̃_{d}(f) : f ∈ F }, subject to restrictions listed in the above theorems. That remark also applies to functions f taking values in a locally convex space.

6. In the context discussed above, we immediately obtain the comparison of generalized Orlicz modulars, i.e., functionals of the form Φ(·) = Eϕ(∥ · ∥), where ϕ is a nondecreasing function on the positive half-line, ϕ(0) = 0.

7. Multiple stochastic integrals of deterministic multivariate functions (cf., e.g., [KS89]) can be seen as limits of multilinear random forms. Therefore if ξ, ξ_{1}, ξ_{k} are stochastic processes with independent increments, and the symbols ⟨ f ξ_{1} ⊗ ⋯ ⊗ ξ_{d} ⟩ and ⟨ f ξ^{⊗k} ⟩ are understood as such integrals, then all decoupling inequalities carry over word-for-word.

8. Our decoupling inequalities involve a certain means of domination. Essentially, we show that the domination by means of L^{p}-norms yields the same for probability tails. The passing from one to another type of domination may be of an intrinsic interest. Recall the definition of f^{∗∗} mentioned in Section 2.2. Let us note the following result, which can be applied in a wider context than ours. Suppose that ξ and η are two given non-negative random variables, and define quantities c_{1}, …, c_{5} below.

(i) Let c_{1} be the smallest constant such that for every Orlicz function, ∥ξ∥_{ϕ} ≤ c_{1}∥η∥_{ϕ}; (ii) Let c_{2} be the smallest constant such that for all t > 0, if ϕ_{t}(x) = (x − 1)_{+}/t, then

∥ξ∥_{ϕ_{t}} ≤ c_{2}∥η∥_{ϕ_{t}}; (iii) Let c_{3} be the smallest constant such that ξ^{∗∗} ≤ c_{3}η^{∗∗}; (iv) Let c_{4} be the smallest constant such that for every r.i. norm, ∥ξ∥ ≤ c_{4}∥η∥;

(v) Let c_{5} be the smallest constant such that for every separable r.i. norm, ∥ξ∥ ≤ c_{5}∥η∥.

Then c_{1} = c_{2} ≤ c_{3} = c_{4} = c_{5} ≤ 2c_{1}. Indeed, inequalities c_{2} ≤ c_{1} ≤ c_{4} and c_{3} ≤ c_{5} ≤ c_{4} are obvious. That c_{1} ≤ c_{2} follows immediately from the formula

ϕ(x) = ∫_{0}^{∞} ϕ_{t}(x) d(ϕ′(t)).

That c_{4} ≤ c_{3} was proved in [LT79, Proposition 2.a.8]. That c_{3} ≤ 2c_{2} follows from the formula ∥ξ∥_{ϕ_{t}} ≤ ξ^{∗∗}(t) ≤ 2∥ξ∥_{ϕ_{t}}. To show the left hand side, suppose that ξ^{∗∗}(t) ≤ 1. Then

∫_{0}^{t} ξ^{∗}(s) ds ≤ t.

Thus we have that ξ^{∗}(t) ≤ 1, and hence

Eϕ_{t}(ξ) = \frac{1}{t} ∫_{0}^{1}(ξ^{∗}(s) − 1)_{+} ds = \frac{1}{t} ∫_{0}^{t}(ξ^{∗}(s) − 1)_{+} ds ≤ \frac{1}{t} ∫_{0}^{t} ξ^{∗}(s) ds ≤ 1.

To show the right hand side, suppose that ∥ξ∥_{ϕ_{t}} ≤ 1. Thus

∫_{0}^{t_{0}}(ξ^{∗}(s) − 1) ds ≤ t,

where t_{0} = P(ξ > 1). If t_{0} ≥ t, then it follows that

∫_{0}^{t}(ξ^{∗}(s) − 1) ds ≤ t,

from whence it follows that

∫_{0}^{t} ξ^{∗}(s) ds ≤ 2t.

If t_{0} < t, then

∫_{0}^{t} ξ^{∗}(s) ds = ∫_{0}^{t_{0}}(ξ^{∗}(s) − 1) ds + ∫_{0}^{t_{0}} ds + ∫_{t_{0}}^{t} ξ^{∗}(s) ds ≤ 2t,

because ξ^{∗}(s) ≤ 1 if s > t_{0}.

9. A decoupling principle for multivalued functions (proved in [dlPn92]) also follows from our basic decoupling inequalities. Suppose that F (·, ξ) is a countably multivalued function, i.e., a countable family of functions Fi is associated with each i. In equivalent terms, one may think of a decision function τ : D × ℕ^{k} → ∏_{i∈ℕ^{k}} Fi (D is countable). Then the statements of Theorem 2.3 hold uniformly with respect to τ, that is, the norm ∥F (·)∥ is replaced by sup_{τ} sup_{d} ∥τ(d, ·)(·)∥. The theorem follows for a finite collection of decision functions { τ_{1}, …, τ_{n} }, since this means the replacement of the underlying Banach space 𝔼 by another Banach space ℓ_{n}^{∞}(𝔼). In the full statement we need the banach lattice 𝕃(ℓ^{∞}) to satisfy the property “sup_{n} ∥x_{n}∥ = ∥ sup_{n} x_{n}∥, for an increasing sequence of nonnegative vectors”. In view of the preceding note, we may choose a family of Orlicz spaces, and the required property holds. Other sequential functionals on ℝ^{D}, e.g. ℓ^{p}, Orlicz ℓ^{ψ}, etc., yield numerous variations of Theorem 2.3.

Remark 3 This paper represents the combination of the papers [dlPnMS92] and [Szu92]

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Victor H. de la Peña Department of Statistics, Columbia University, New York, NY 10027

phone (212) 854-5360
email vp@stat.columbia.edu

Stephen J. Montgomery-Smith Department of Mathematics, University of Missouri, Columbia, MO 65211

phone (314) 882-7492
email stephen@mont.cs.missouri.edu
email szulgje@mail.auburn.edu

Jerzy Szulga Department of Mathematics, Auburn University, Auburn, AL 36849-3501 phone (205) 844-6569