Random unconditional convergence of vector-valued Dirichlet series

TL;DR

This paper characterizes Banach spaces with type 2 and cotype 2 via the random unconditional convergence or divergence of vector-valued Dirichlet series in Hardy spaces, extending the understanding of unconditionality in these function spaces.

Contribution

It establishes a precise link between Banach space geometry (type and cotype) and the random unconditional behavior of vector-valued Dirichlet series in Hardy spaces, including new examples.

Findings

Banach space $X$ has type 2 iff $(x_n n^{-s})_n$ is RUC in $\\mathcal H_2(X)$ for all sequences.

Banach space $X$ has cotype 2 iff $(x_n n^{-s})_n$ is RUD in $\\mathcal H_2(X)$ for all sequences.

Explicit examples show differences between unconditionality in $\\mathcal H_p(X)$ and $H_p(X)$.

Abstract

We study random unconditionality of Dirichlet series in vector-valued Hardy spaces $\mathcal H_p(X)$. It is shown that a Banach space $X$ has type 2 (respectively, cotype 2) if and only if for every choice $(x_n)_n\subset X$ it follows that $(x_n n^{-s})_n$ is Random unconditionally convergent (respectively, divergent) in $\mathcal H_2(X)$. The analogous question on $\mathcal H_p(X)$ spaces for $p\neq2$ is also explored. We also provide explicit examples exhibiting the differences between the unconditionality of $(x_n n^{-s})_n$ in $\mathcal H_p(X)$ and that of $(x_n z^n)_n$ in $H_p(X)$.

Equations

$$\mathbb{E}\Big{\|}\sum_{n}\varepsilon_{n}a_{n}x_{n}\Big{\|}\lesssim\Big{\|}\sum_{n}a_{n}x_{n}\Big{\|},$$

$$\Big{\|}\sum_{n}a_{n}x_{n}\Big{\|}\lesssim\mathbb{E}\Big{\|}\sum_{n}\varepsilon_{n}a_{n}x_{n}\Big{\|}.$$

$$\mathcal{H}^{rad}_{p}(X):=\big{\{}\textstyle\sum_{n}x_{n}n^{-s}\colon\,\sum_{n}\varepsilon_{n}x_{n}n^{-s}\in\mathcal{H}_{p}(X)\,\text{for a.e. }(\varepsilon_{n})_{n}\in\{-1,+1\}^{\mathbb{N}}\big{\}}\,.$$

$${\mathbb{T}}^{\mathbb{N}}=\{(z_n{)}_{n\in \mathbb{N}}:z_n\in \mathbb{C},|z_n|=1\}.$$

$$\sum _{\alpha \in {\mathbb{Z}}^{(\mathbb{N})}}\hat{f}(\alpha )z^{\alpha },$$

$$\hat{f}(\alpha )={\int }_{{\mathbb{T}}^{\mathbb{N}}}f(w)w^{-\alpha }dw.$$

$$\mathcal{B}\bigg{(}\sum_{\alpha\in\mathbb{N}_{0}^{(\mathbb{N})}}x_{\alpha}z^{\alpha}\bigg{)}=\sum_{n\in\mathbb{N}}x_{\alpha(n)}n^{-s},$$

$$\Vert D{\Vert }_{{\mathcal{H}}_p(X)}=\Vert f{\Vert }_{H_p({\mathbb{T}}^{\mathbb{N}},X)}.$$

$$\Big{\|}\sum_{n\in\mathbb{N}}x_{n}n^{-s}\Big{\|}_{\mathcal{H}_{p}(X)}=\Big{\|}\sum_{\alpha\in\mathbb{N}_{0}^{(\mathbb{N})}}x_{n(\alpha)}z^{\alpha}\Big{\|}_{H_{p}(\mathbb{T}^{\mathbb{N}},X)}.$$

$$\sup_{\varepsilon_{n}\in\{-1,+1\}}\Big{\|}\sum_{n=1}^{m}\varepsilon_{n}a_{n}x_{n}\Big{\|}\leq C\Big{\|}\sum_{n=1}^{m}a_{n}x_{n}\Big{\|}.$$

$$\mathbb{E}\Big{\|}\sum_{n=1}^{m}\varepsilon_{n}y_{n}\Big{\|}=\frac{1}{2^{m}}\sum_{(\varepsilon_{n})\in\{-1,+1\}^{m}}\Big{\|}\sum_{n=1}^{m}\varepsilon_{n}y_{n}\Big{\|}.$$

$$\mathbb{E}\Big{\|}\sum_{n=1}^{m}\varepsilon_{n}a_{n}x_{n}\Big{\|}\leq C\Big{\|}\sum_{n=1}^{m}a_{n}x_{n}\Big{\|}.$$

$$\Big{\|}\sum_{n=1}^{m}a_{n}x_{n}\Big{\|}\leq C\mathbb{E}\Big{\|}\sum_{n=1}^{m}\varepsilon_{n}a_{n}x_{n}\Big{\|}.$$

$${\phi }_{\alpha }(f)={\int }_{{\mathbb{T}}^{\mathbb{N}}}{\gamma }_{\alpha }\left(f(z)\right){\overline{z}}^{\alpha }dz,$$

$${\phi }_{\beta }(x_{\alpha }{z}^{\alpha })={\int }_{{\mathbb{T}}^{\mathbb{N}}}{\gamma }_{\alpha }(x_{\alpha })z^{\beta }{\overline{z}}^{\alpha }dz={\delta }_{\alpha ,\beta }.$$

$$\frac{1}{K_{p}}\bigg{(}\mathbb{E}\Big{\|}\sum_{n=1}^{m}\varepsilon_{n}x_{n}\Big{\|}^{p}\bigg{)}^{\frac{1}{p}}\leq\mathbb{E}\Big{\|}\sum_{n=1}^{m}\varepsilon_{n}x_{n}\Big{\|}\leq K_{p}\bigg{(}\mathbb{E}\Big{\|}\sum_{n=1}^{m}\varepsilon_{n}x_{n}\Big{\|}^{p}\bigg{)}^{\frac{1}{p}}.$$

$$\mathbb{E}\Big{\|}\sum_{n=1}^{m}\varepsilon_{n}a_{n}x_{n}\Big{\|}\leq\frac{\pi}{2}\max|a_{n}|\mathbb{E}\Big{\|}\sum_{n=1}^{m}\varepsilon_{n}x_{n}\Big{\|}.$$

$$\mathbb{E}\Big{\|}\sum_{n=1}^{m}a_{n}x_{n}z_{n}\Big{\|}\leq\frac{\pi}{2}\max|a_{n}|\mathbb{E}\Big{\|}\sum_{n=1}^{m}x_{n}z_{n}\Big{\|}.$$

$$\mathbb{E}\Big{\|}\sum_{n=1}^{m}\varepsilon_{n}a_{n}x_{n}z^{\alpha(n)}\Big{\|}^{p}_{X}\lesssim\Big{\|}\sum_{n=1}^{m}a_{n}x_{n}z^{\alpha(n)}\Big{\|}^{p}_{X}.$$

$$\mathbb{E}\Big{\|}\sum_{n=1}^{m}\varepsilon_{n}a_{n}x_{n}z^{\alpha(n)}\Big{\|}_{H_{p}(\mathbb{T}^{\mathbb{N}},X)}\lesssim\Big{\|}\sum_{n=1}^{m}a_{n}x_{n}z^{\alpha(n)}\Big{\|}_{H_{p}(\mathbb{T}^{\mathbb{N}},X)}.$$

$$\mathbb{E}\Big{\|}\sum_{n=1}^{m}\varepsilon_{n}a_{n}x_{n}n^{-s}\Big{\|}_{\mathcal{H}_{p}(X)}\lesssim\Big{\|}\sum_{n=1}^{m}a_{n}x_{n}n^{-s}\Big{\|}_{\mathcal{H}_{p}(X)}.$$

$$s_n=\sum _{i=1}^n{e}_{i}n\in \mathbb{N},$$

$$\Big{\|}\sum_{n=1}^{m}a_{n}s_{n}\Big{\|}=\sup_{1\leq k\leq m}\Big{|}\sum_{n=k}^{m}a_{n}\Big{|}.$$

$$\displaystyle\Big{\|}\sum_{n=1}^{m}a_{n}s_{n}n^{-s}\Big{\|}_{\mathcal{H}_{2}(c_{0})}=\bigg{(}\int\sup\limits_{1\leq k\leq m}\Big{|}\sum_{n=k}^{m}a_{n}n^{-s}\Big{|}^{2}ds\bigg{)}^{1/2}=\Big{\|}\sup_{1\leq k\leq m}\Big{|}\sum_{n=k}^{m}a_{n}n^{-s}\Big{|}\Big{\|}_{\mathcal{H}_{2}(\mathbb{C})}.$$

$$\Big{\|}\sup_{1\leq k\leq m}\Big{|}\sum_{n=k}^{m}a_{n}n^{-s}\Big{|}\Big{\|}_{\mathcal{H}_{2}(\mathbb{C})}\geq\Big{\|}\sum_{n=k}^{m}a_{n}n^{-s}\Big{\|}_{\mathcal{H}_{2}(\mathbb{C})}=\bigg{(}\sum_{n=1}^{m}\left|a_{n}\right|^{2}\bigg{)}^{1/2}.$$

$$\Big{\|}\sup_{1\leq k\leq m}\Big{|}\sum_{n=k}^{m}a_{n}n^{-s}\Big{|}\Big{\|}_{\mathcal{H}_{2}(\mathbb{C})}\leq C\Big{\|}\sum_{n=1}^{m}a_{n}n^{-s}\Big{\|}_{\mathcal{H}_{2}(\mathbb{C})}=C\bigg{(}\sum_{n=1}^{m}\left|a_{n}\right|^{2}\bigg{)}^{1/2}.$$

$${\mathcal{H}}_p^{rad}(X):=\left\{\sum x_n{n}^{-s}:\forall \text{ a.e. }{\epsilon }_n=\pm 1,\sum {\epsilon }_n{x}_n{n}^{-s}\in {\mathcal{H}}_p(X)\right\}.$$

$${\phi }_{k,x^{\prime }}(R(\epsilon ))={\epsilon }_k{x}^{\prime }(x_k)\text{ and }\sum _{n=1}^N{\phi }_{k,x^{\prime }}(Y_n)=\sum _{n=1}^N{\delta }_{n,k}{\epsilon }_k{x}^{\prime }(x_k),$$

$${\mathcal{H}}_p^{rad}(X)=\left\{\sum x_n{n}^{-s}:\forall \text{ a.e. }{\epsilon }_n=\pm 1,\sum {\epsilon }_n{x}_n{n}^{-s}\text{ converges in }{\mathcal{H}}_p(X)\right\}.$$

$${\mathcal{H}}_p^{rad}(X)=\left\{\sum x_n{n}^{-s}:\sum r_n{x}_n{n}^{-s}\in L_1\left([0,1],{\mathcal{H}}_p(X)\right)\right\},$$

Full text

1 Introduction

In this article we investigate some basic questions about random unconditionality of Dirichlet series in vector-valued Hardy spaces. Given a complex Banach space $X$, a Dirichlet series in $X$ is a series of the form $D=\sum_{n}x_{n}n^{-s}$, where the coefficients $x_{n}$ are vectors in $X$ and $s$ is a complex variable. The study of functional-analytic aspects of the theory of (vector-valued) Dirichlet series has attracted great attention in the recent years (see, for example, [3, 4, 6, 8, 9], and also [5], where the vector-valued theory is used to study multiple Dirichlet series). The Hardy space $\mathcal{H}_{p}(X)$ of $X$-valued Dirichlet series consists, loosely speaking, of those Dirichlet series whose corresponding Hardy $p$-norm via Bohr’s transform is finite (see next section for the formal definition). It is well known that, even in the scalar case, the standard basis $(n^{-s})_{n}$ of the space $\mathcal{H}_{p}(\mathbb{C})$ is unconditional only when $p=2$ (see [4, Proposition 4]). Since unconditionality is hard to accomplish, we are lead to consider weaker versions, such as random unconditionality. While unconditional convergence of a series $\sum_{n}y_{n}$ is equivalent to the convergence of $\sum_{n}\varepsilon_{n}y_{n}$ for all choice of signs $\varepsilon_{n}=\pm 1$, random unconditional convergence is related to the convergence of $\sum_{n}\varepsilon_{n}x_{n}$ for almost every choice of signs $(\varepsilon_{n})_{n}\in\{-1,+1\}^{\mathbb{N}}$ (with respect to Haar measure).

We are interested in identifying Random Unconditional Convergent (in short, RUC, see [2]) and Random Unconditional Divergent (in short, RUD, see [15]) systems of vector-valued Dirichlet series. Namely, a sequence $(x_{n})_{n}$ (usually part of a biorthogonal system) in a Banach space is called RUC when there is a uniform estimate of the form

[TABLE]

for every choice of scalars $(a_{n})_{n}$, where $\mathbb{E}$ denotes the expectation with respect to i.i.d. Rademacher random variables $\varepsilon_{n}$; analogously, $(x_{n})_{n}$ is called RUD when the converse estimate holds

[TABLE]

With this terminology, one can deduce from [4, Proposition 4] that the canonical basis $(n^{-s})_{n}\subset\mathcal{H}_{p}(\mathbb{C})$ is RUC, for $p\geq 2$, while it is RUD for $p\leq 2$. The main question we want to address here corresponds to the vector-valued version of this phenomenon: When is $(x_{n}n^{-s})_{n}$ a RUC system in $\mathcal{H}_{p}(X)$ (respectively, RUD), for every choice of $(x_{n})_{n}\subset X$?

These questions have the following equivalent formulations. Suppose a Dirichlet series $D=\sum_{n}x_{n}n^{-s}$ belongs to $\mathcal{H}_{p}(X)$; does $\sum_{n}\varepsilon_{n}x_{n}n^{-s}$ also belong to $\mathcal{H}_{p}(X)$ for almost every choice of signs $(\varepsilon_{n})_{n}\in\{-1,+1\}^{\mathbb{N}}$? In the opposite direction, if $\sum_{n}\varepsilon_{n}x_{n}n^{-s}$ belongs to $\mathcal{H}_{p}(X)$ for almost every choice of signs $(\varepsilon_{n})_{n}\in\{-1,+1\}^{\mathbb{N}}$, does $\sum_{n}x_{n}n^{-s}$ necessarily belong to $\mathcal{H}_{p}(X)$? This notion of almost sure sign convergence of Dirichlet series has been the object of recent research in [4], where the following space was introduced

[TABLE]

The previous questions can be reformulated in terms of inclusion relations between $\mathcal{H}_{p}(X)$ and $\mathcal{H}^{rad}_{p}(X)$. One of our main results in this direction is the following characterization (see Theorem 4.1):

Theorem A For a Banach space $X$ the following statements are equivalent:

1. (a)

$(x_{n}n^{-s})_{n}$ is RUC in $\mathcal{H}_{2}(X)$, for every choice of $(x_{n})_{n}\subset X$. 2. (b)

$\mathcal{H}_{2}(X)\subset\mathcal{H}^{rad}_{2}(X)$. 3. (c)

$X$ has type 2.

In an analogous way, one can prove that the spaces $X$ where $\mathcal{H}^{rad}_{2}(X)\subset\mathcal{H}_{2}(X)$, or where $(x_{n}n^{-s})_{n}$ is always RUD, are precisely those with cotype 2. The case when $p\neq 2$ will also be considered in Section 4.

The paper is structured as follows: in the next section we introduce preliminaries and notation on Dirichlet series and random unconditionality. Section 3 is devoted to provide equivalent reformulations of the property that a Banach space $X$ satisfies that $(x_{n}n^{-s})_{n}$ is a RUC system in $\mathcal{H}_{p}(X)$ (respectively, RUD), for every choice of $(x_{n})_{n}\subset X$. As an application of Green-Tao’s theorem on arithmetic progressions in the set of primes [12], we will provide examples that show how differently Dirichlet series and power series behave in terms of random unconditionality. In particular, we will show that for a fixed sequence of vectors $(x_{n})_{n}$, one could have that $(x_{n}n^{-s})_{n}$ is RUC in $\mathcal{H}_{p}(X)$, while $(x_{n}z^{n})_{n}$ is not RUC in $H_{p}(X)$, and vice versa. Finally, in Section 4, we provide the connection between type (or cotype) and the above questions.

2 Definitions and general results

We refer to the books [7] and [19] for the general theory of Dirichlet series. We denote by $\mathbb{T}^{\mathbb{N}}$ the infinite complex polytorus

[TABLE]

Given $1\leq p<\infty$ and a Banach space $X$, let $L_{p}(\mathbb{T}^{\mathbb{N}},X)$ be the space of $p$-Bochner integrable functions $f:\mathbb{T}^{\mathbb{N}}\rightarrow X$ with respect to the Haar measure. Let us write $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$ and denote by $\mathbb{Z}^{(\mathbb{N})}$ (respectively $\mathbb{N}_{0}^{(\mathbb{N})}$ ) the set of eventually null sequences of integer numbers (respectively, non-negative integer numbers). Also, for any sequence of scalars $z=(z_{n})_{n}$ and $\alpha=(\alpha_{1},\ldots,\alpha_{m},0\ldots)\in\mathbb{Z}^{(\mathbb{N})}$, let us denote $z^{\alpha}=z_{1}^{\alpha_{1}}\cdot\ldots\cdot z_{m}^{\alpha_{m}}$.

Recall that every $f\in L_{p}(\mathbb{T}^{\mathbb{N}},X)$ is uniquely determined by its (formal) Fourier series

[TABLE]

where

[TABLE]

The space $H_{p}(\mathbb{T}^{\mathbb{N}},X)$ is the closed subspace of $L_{p}(\mathbb{T}^{\mathbb{N}},X)$ consisting of those functions $f$ with $\hat{f}(\alpha)=0$ whenever $\alpha\in\mathbb{Z}^{(\mathbb{N})}\backslash\mathbb{N}_{0}^{(\mathbb{N})}$.

Given $\alpha=(\alpha_{1},\ldots,\alpha_{m},0\ldots)\in\mathbb{N}_{0}^{(\mathbb{N})}$, let us denote $n(\alpha)=p_{1}^{\alpha_{1}}\cdot\dots\cdot p_{m}^{\alpha_{m}}\in\mathbb{N}$, where $p_{1}<p_{2}<\dots$ is the ordered sequence of prime numbers. Similarly, given a natural number $n=p_{1}^{\alpha_{1}}\cdot\dots\cdot p_{m}^{\alpha_{m}}$, define $\alpha(n)=(\alpha_{1},\ldots,\alpha_{m},0\ldots)\in\mathbb{N}_{0}^{(\mathbb{N})}$.

We can formally consider Bohr’s transform

[TABLE]

and define $\mathcal{H}_{p}(X)$ as the image of $H_{p}(\mathbb{T}^{\mathbb{N}},X)$ equipped with the norm that turns this mapping into an isometry. To be more precise, a Dirichlet series $D=\sum_{n}x_{n}n^{-s}$ is in $\mathcal{H}_{p}(X)$ if there is a function $f\in H_{p}(\mathbb{T}^{\mathbb{N}},X)$ such that $\hat{f}(\alpha)=x_{n(\alpha)}$ and in that case

[TABLE]

In particular, if $(x_{n})_{n}\subseteq X$ has finitely many non zero elements, we have

[TABLE]

Due to this isometry some properties are translated from the power series to the Dirichlet series setting.

1. (i)

For a Dirichlet series $D\in\mathcal{H}_{p}(X)$ the coefficients of D are bounded by $\|D\|_{\mathcal{H}_{p}(X)}$. More precisely, the operator $c_{n}$ that takes the $n-$th coefficient is contractive. As a consequence, if a sequence of Dirichlet series $(D_{N})_{N}$ converges in $\mathcal{H}_{p}(X)$ to some $D$, the coefficients $c_{n}(D_{N})$ converge to $c_{n}(D)$ for all $n\in\mathbb{N}$. 2. (ii)

The set of Dirichlet polynomials $\sum_{n=1}^{N}x_{n}n^{-s}$ is dense in $\mathcal{H}_{p}(X)$ for $1\leq p<\infty$ since the analytic polynomials are dense in $H_{p}(\mathbb{T}^{\mathbb{N}},X)$ [7, Proposition 24.6].

Recall that a basis $(x_{n})_{n}$ of a Banach space $X$ is unconditional if for every $x\in X$, its expansion $\sum_{n}a_{n}x_{n}$ converges unconditionally. Equivalently, there is a constant $C>0$ such that for every $m\in\mathbb{N}$ and every sequence of scalars $(a_{n})_{n=1}^{m}$, we have

[TABLE]

Banach spaces with unconditional bases have a nice structure, including a wealth of operators acting on them. However, in the landmark paper [11], Banach spaces which do not have any subspace with an unconditional basis are constructed. Therefore, weaker versions of unconditionality have to be considered. In this direction, we will next discuss two notions of random unconditionality that were introduced in [2] and [15].

Recall, a series $\sum_{n}y_{n}$ in a Banach space is random unconditionally convergent when $\sum_{n}\varepsilon_{n}y_{n}$ converges almost surely on signs $(\varepsilon_{n})\in\{-1,+1\}^{\mathbb{N}}$ with respect to Haar probability measure on $\{-1,+1\}^{\mathbb{N}}$. A basis $(x_{n})_{n}$ of a Banach space $X$ is of Random Unconditional convergence (in short, RUC), if every convergent series $\sum_{n}\varepsilon_{n}x_{n}$ is random unconditionally convergent. Analogously, we say $(x_{n})_{n}$ is a basis of Random Unconditional divergence (in short, RUD), if every random unconditionally convergent series $\sum_{n}a_{n}x_{n}$ must be convergent.

An equivalent formulation of these notions can be given in terms of the expectation

[TABLE]

Indeed, $(x_{n})_{n}$ is RUC if and only if there is a constant $C$ such that for every $m\in\mathbb{N}$ and every sequence of scalars $(a_{n})_{n=1}^{m}$ one has that

[TABLE]

In this case, we will say that $(x_{n})_{n}$ is $C$-RUC. Similarly, $(x_{n})_{n}$ is RUD if and only if there is a constant $C$ such that for every $m\in\mathbb{N}$ and every sequence of scalars $(a_{n})_{n=1}^{m}$ one has that

[TABLE]

In this case, we will say that $(x_{n})_{n}$ is $C$-RUD. It is immediate to see that a basis $(x_{n})_{n}$ is unconditional if and only if it is both RUC and RUD (see [15, Proposition 2.3]).

Moreover, the notions of RUC and RUD make also sense in the more general context of biorthogonal systems. Notice that for every sequence $(x_{n})_{n}\subseteq X$, the non-zero elements of $(x_{n}n^{-s})_{n}$ can be considered as part of a biorthogonal system. Indeed, write $\left(x_{n}n^{-s}\right)_{n\in A}\in\mathcal{H}_{p}(X)$ where $A\subseteq\mathbb{N}$ is the set of indexes $n$ for which $x_{n}\neq 0$. Applying the Bohr transform we may regard this sequence as $\left(x_{\alpha}z^{\alpha}\right)_{\alpha\subseteq\Lambda}\in H_{p}(X)$ for a suitable $\Lambda$ corresponding to $A$. For each $\alpha\in\Lambda$ choose $\gamma_{\alpha}\in X^{*}$ such that $\gamma_{\alpha}(x_{\alpha})=1$. Notice that if we define $\varphi_{\alpha}\in H_{p}(X)^{*}$ by

[TABLE]

then $\left(x_{\alpha}z^{\alpha},\varphi_{\alpha}\right)_{\alpha\in\Lambda}$ is a biorthogonal system, since

[TABLE]

From now on, when we say that $(x_{n}n^{-s})_{n}$ is RUC or RUD we mean that the nonzero elements are RUC or RUD as part of the biorthogonal system just defined. Note that the conditions (3) and (4) which define RUC and RUD can be checked for the whole sequence (it is not necessary to omit the zero elements).

When dealing with expectations of the form $\mathbb{E}\|\sum_{n}\varepsilon_{n}x_{n}\|$, we will repeatedly make use of Kahane’s inequality (cf. [10, 11.1]): For any $0<p<\infty$, there is $K_{p}>0$ such that for every $(x_{n})_{n=1}^{m}$ in a Banach space

[TABLE]

Another fundamental property that will be used throughout is the Contraction principle (cf. [10, 12.2], see also [21] for the sharp version for complex scalars). For any scalars $(a_{n})_{n=1}^{m}\subset\mathbb{C}$ and any $(x_{n})_{n=1}^{m}$ in a Banach space

[TABLE]

Moreover, since Steinhaus variables are symmetric $(z_{n})_{n=1}^{m}$ and $(\varepsilon_{n}z_{n})_{n=1}^{m}$ are identically distributed. Therefore, we also have

[TABLE]

Remark 2.1**.**

If $(x_{n})_{n}$ is $RUC$ in $X$ then the sequence $(x_{n}n^{-s})_{n}$ is $RUC$ in $\mathcal{H}_{p}(X)$ for every $p\geq 1$.

Proof.

By Kahane’s inequality (5), and applying the $RUC$ condition to $(x_{n})_{n}$ with coefficients $a_{n}z^{\alpha(n)}$ (for a fixed $z\in\mathbb{T}^{\mathbb{N}}$), we obtain

[TABLE]

Integrating with respect to $z$ and switching the order of integration, again by Kahane’s inequality, the last expression becomes

[TABLE]

Equivalently applying Bohr’s transform we get

[TABLE]

∎

It is interesting to observe that the converse is not true. A simple example of a sequence which is neither $RUC$ nor $RUD$ is the summing basis in $c_{0}$ (see [15]), defined by

[TABLE]

where $(e_{i})$ denotes the canonical basis of $c_{0}$. However, the sequence $(s_{n}n^{-s})_{n}$ is actually equivalent to the $\ell_{2}$ basis and thus unconditional. To see this, we recall the following property of the summing basis:

[TABLE]

Hence, it follows that

[TABLE]

On the one hand, we obviously have

[TABLE]

On the other hand, a version of Carleson-Hunt’s theorem for Dirichlet series (see the proof of Theorem 1.5 in [13]) provides us with $C>0$ such that

[TABLE]

Joining this with (8) yields the desired result.

2.1 The Banach space $\mathcal{H}^{rad}_{p}(X)$

In order to study the almost sure convergence of random Dirichlet series, we recall the space defined in [4]:

Definition 2.2**.**

Given $1\leq p\leq\infty$ and a Banach space $X$, we define

[TABLE]

A Dirichlet series $\sum_{n}x_{n}n^{-s}$ should be regarded as a formal expression. As it was mentioned before, when we say $D=\sum_{n}x_{n}n^{-s}\in\mathcal{H}_{p}(X)$ we mean that there is a function $f\in H_{p}(\mathbb{T}^{\mathbb{N}},X)$ with Fourier coefficients $\hat{f}_{\alpha}=x_{n(\alpha)}$. Therefore, the fact that $D\in\mathcal{H}_{p}(X)$ does not necessarily imply that the partial sums $D_{N}=\sum_{n=1}^{N}x_{n}n^{-s}$ converge to $D$ in $\mathcal{H}_{p}(X)$. Luckily, for random Dirichlet series we have the following result.

Proposition 2.3**.**

For every Banach space $X$ and every $1\leq p<\infty$. If $(x_{n})_{n}\subseteq X$ satisfies that $\sum_{n}\varepsilon_{n}x_{n}n^{-s}\in\mathcal{H}_{p}(X)$ for almost every choice of signs $\varepsilon_{n}$ then $\sum_{n}\varepsilon_{n}x_{n}n^{-s}$ converges a.e.

To accomplish this we need the following classical theorem which can be found in [14, Theorem 2.1.1].

Theorem 2.4** (Itô-Nisio).**

Let $Y_{1},Y_{2},\ldots$ be a sequence of independent symmetric random variables with values in a separable Banach space $Y$. Then the following conditions are equivalent:

1. (a)

$\sum_{n=1}^{\infty}Y_{n}$* converges a.e.;* 2. (b)

There exists a random variable $R$ with values in $Y$ and a family $\mathcal{F}\subseteq Y^{\prime}$ separating points in $Y$, such that for each $y^{\prime}$ in $\mathcal{F}$ the series $\sum_{n=1}^{\infty}y^{\prime}(Y_{n})$ converges a.e. to $y^{\prime}(R)$.

Proof of Proposition 2.3.

Let $(x_{n})_{n}\subseteq X$ be as stated and for $\varepsilon=(\varepsilon_{n})_{n}\in\{-1,+1\}^{\mathbb{N}}$ set $R(\varepsilon)=\sum_{n}\varepsilon_{n}x_{n}n^{-s}$. We start by showing that $R$ is a random variable (i.e. a measurable function). For $\sigma>0$ and a Dirichlet series $D=\sum_{n}a_{n}n^{-s}\in\mathcal{H}_{p}(X)$ define $D_{\sigma}=\sum_{n}\left(a_{n}/n^{\sigma}\right)n^{-s}$. In [8, Proposition 2.3] it is shown that $D_{\sigma}$ converges to $D$ in $\mathcal{H}_{p}(X)$ as $\sigma\to 0$, and the partial sums of $D_{\sigma}$ converge to $D_{\sigma}$ uniformly. Applying this to $R$ we may construct a sequence of measurable functions converging almost everywhere to $R$. Furthermore, the same argument shows that $R(\varepsilon)$ belongs to ${Y=\overline{\langle x_{n}n^{-s}\rangle_{n}}\subseteq\mathcal{H}_{p}(X)}$ for almost every $\varepsilon$. Set $Y_{n}=\varepsilon_{n}x_{n}n^{-s}\in\mathcal{H}_{p}(X)$ and for every $k\in\mathbb{N}$ and $x^{\prime}\in X^{\prime}$ define $\varphi_{k,x^{\prime}}=x^{\prime}\circ c_{k}$ where $c_{k}$ (as defined in (i)) returns the $k-$th coefficient of a Dirichlet series. Notice that the family $\mathcal{F}=\{\varphi_{k,x^{\prime}}\}_{k,x^{\prime}}$ separates points of $Y$. Furthermore, we have

[TABLE]

and both coincide when $N\geq k$. This means that assertion $(b)$ of the theorem holds. Thus, the partial sums of $R(\varepsilon)$ converge for almost every choice of signs $\varepsilon_{n}$. ∎

Corollary 2.5**.**

For every Banach space $X$ and every $1\leq p<\infty$

[TABLE]

Next we are going to endow $\mathcal{H}^{rad}_{p}(X)$ with a norm. To do this we apply [10, Proposition 12.3] to obtain the following reformulation

[TABLE]

where $r_{n}$ denote Rademacher random variables. Hence it is natural to define

[TABLE]

which turns $\mathcal{H}^{rad}_{p}(X)$ into a Banach space (see [4]).

Recall that the space of unconditionally summable sequences $(x_{n})_{n}$ in a Banach space is denoted

[TABLE]

and becomes a Banach space under the norm

[TABLE]

(cf. [10, Chapter 12]). Note that by Kahane’s inequalities (5) we may replace the right term by any $p$-norm with $1\leq p<\infty$ to get an equivalent norm.

The next proposition provides an easy way to estimate $\mathcal{H}_{p}^{rad}$ norms without computing Dirichlet norms.

Proposition 2.6**.**

For $1\leq p<\infty$ there is $C_{p}>0$ such that if $X$ is a Banach space and $(x_{n})_{n=1}^{N}\subset X$ we have

[TABLE]

where we set $x_{n}=0$ for $n>N$. In particular, we have that $\mathcal{H}^{rad}_{p}(X)=Rad(X)$, up to an equivalent norm.

Proof.

Given arbitrary $(x_{n})_{n=1}^{N}\subset X$, using Kahane’s inequality (5) and the Contraction principle (6), we have

[TABLE]

Clearly, the equivalence constants above only depend on $1\leq p<\infty$. ∎

3 The $\mathcal{H}_{p}$ random convergence property

In this section we will focus on characterizing those Banach spaces $X$ such that every sequence $(x_{n})_{n}\subset X$ satisfies that $(x_{n}n^{-s})_{n}$ is $RUC$ in $\mathcal{H}_{p}(X)$. By the definition of a $RUC$ system, this means that there exists a constant (depending on the sequence) such that the inequality (3) is satisfied. Next proposition shows that a uniform constant (not depending on the sequences) can be chosen. In fact, this condition is also equivalent to the inclusion $\mathcal{H}_{p}(X)\subseteq\mathcal{H}_{p}^{rad}(X)$.

Proposition 3.1**.**

Let $X$ be a Banach space and $p\geq 2$. The following statements are equivalent:

1. (a)

$(x_{n}n^{-s})_{n}$* is RUC in $\mathcal{H}_{p}(X)$ for every $(x_{n})_{n}\subset X$.* 2. (b)

There is $C\geq 1$ such that for every $N\in\mathbb{N}$ and $(x_{n})^{N}_{n=1}$ we have

[TABLE] 3. (c)

The following inclusion holds:

[TABLE] 4. (d)

There is $C\geq 1$ such that for every $N\in\mathbb{N}$ and $(x_{n})^{N}_{n=1}$ we have

[TABLE]

Definition 3.2**.**

Given $p\geq 2$, we will say that a Banach space $X$ has the $\mathcal{H}_{p}$ random convergence property (or, in short, $X$ has $\mathcal{H}_{p}-RCP$ ) if $X$ satisfies any (and all) of the conditions in Proposition 3.1

In order to prove Proposition 3.1, we need the following lemma.

Lemma 3.3**.**

Assume that there exist $M\in\mathbb{N}$ and $K>0$ such that $(y_{n}n^{-s})_{n=M+1}^{\infty}$ is $K-$RUC for every vector sequence $(y_{n})_{n=M+1}^{\infty}$. Then $(y_{n}n^{-s})_{n\in\mathbb{N}}$ is $(M+K+KM)-$RUC for every vector sequence $(y_{n})_{n\in\mathbb{N}}\subseteq X$.

Proof.

Let $(x_{n})_{n}$ be a vector sequence in $X$. We start by proving that $(x_{n}n^{-s})_{n}$ is $(M+K+KM)-$RUC. Given $N\in\mathbb{N}$ and and $(a_{n})_{n=1}^{N}\subseteq\mathbb{C}$, we have

[TABLE]

Applying (i) for $1\leq k\leq N$ we get

[TABLE]

Therefore, we may deduce

[TABLE]

On the other hand, since $(x_{n}n^{-s})_{n=M+1}^{\infty}$ is $K-$RUC by hypothesis, we get

[TABLE]

Joining the last two inequalities concludes the argument. ∎

Proof of Proposition 3.1.

For convenience we will prove first the equivalence $(b)\Leftrightarrow(d)$, and then we will prove $(a)\Rightarrow(b)\Rightarrow(c)\Rightarrow(a)$.

$(b)\Rightarrow(d)$: Given $(x_{n})_{n}\subseteq X$ we define

[TABLE]

Using Proposition 2.6 and (9) we have

[TABLE]

by definition of the norm in $\mathcal{H}_{p}(X)$, the last term is equal to

[TABLE]

$(d)\Rightarrow(b)$: We proceed as in [4, Proposition 2.4]. Recall that by definition of the $\mathcal{H}_{p}(X)$ norm we have

[TABLE]

Let $m$ be the maximum of all $\alpha_{i}$ such that $x_{n(\alpha)}$ is not zero. For each $z_{1}\in\mathbb{T}$ fixed, using a change of variables with $z_{i}^{\prime}=z_{i}\cdot z_{1}^{(m+1)^{i-1}}$ for $2\leq i\leq N$, we have

[TABLE]

Changing the order of integration we get

[TABLE]

Since all the numbers of the form $\sum_{i=1}^{N}\alpha_{i}(m+1)^{i-1}$ appearing as powers of $z_{1}$ are different, we can apply $(d)$ and the contraction principle (6) in the inner integral with $z_{2},\ldots,z_{N}$ fixed. We get

[TABLE]

where in the last step we used Proposition 2.6.

$(a)\Rightarrow(b)$: Assume $(b)$ does not hold for any $C\geq 1$. Using Proposition 2.6, Lemma 3.3 tells us that for any $M\in\mathbb{N}$ and any $K>0$, there is a vector sequence $(x_{n})_{n\in\mathbb{N}}\subseteq X$ such that $(x_{n}n^{-s})_{n\geq M}$ is not $K-$RUC. Using this fact repeatedly for different values of $M$ and $K$, we will construct a vector sequence which contradicts $(a)$. Taking $M=M_{0}=0$ and $K=1$ we may deduce that there are $M_{1}\in\mathbb{N}$, $(x_{n})_{n=1}^{M_{1}}\subseteq X$ and $(a_{n})_{n=1}^{M_{1}}\subseteq\mathbb{C}$ such that

[TABLE]

Proceeding inductively suppose we have defined $M_{k}\in\mathbb{N}$, $(x_{n})_{n=1}^{M_{k}}\subseteq X$ and $(a_{n})_{n=1}^{M_{k}}\subseteq\mathbb{C}$ so that

[TABLE]

for every $1\leq j\leq k$. Taking $M=M_{k}$ and $K=k+1$ we may deduce that there are $M_{k+1}\in\mathbb{N}$, $(x_{n})_{n=M_{k}+1}^{M_{k+1}}\subseteq X$ and $(a_{n})_{n=M_{k}+1}^{M_{k+1}}\subseteq\mathbb{C}$ such that

[TABLE]

Note that for the sequence $(x_{n})_{n}\subseteq X$ thus defined it follows that $(x_{n}n^{-s})_{n}$ fails to be RUC. Hence, $(a)$ does not hold.

$(b)\Rightarrow(c)$: Assuming there is a constant $C>0$ such that (9) holds for Dirichlet polynomials, we prove that the same inequality is valid for every Dirichlet series in $\mathcal{H}_{p}(X)$. Fix $D\in\mathcal{H}_{p}(X)$ and $M\in\mathbb{N}$. As mentioned in (i), since Dirichlet polynomials are dense in $\mathcal{H}_{p}(X)$ there is a sequence of polynomials $(D_{N})_{N}\subseteq\mathcal{H}_{p}(X)$ converging to $D$. In particular, we have that the coefficients $c_{n}(D_{N})$ of $D_{N}$ converge to those of $D$ for every $n\in\mathbb{N}$. Thus, given $\varepsilon>0$ we may choose $N\in\mathbb{N}$ sufficiently large so that

[TABLE]

Using (11) and the contraction principle (6) we get

[TABLE]

As $\varepsilon$ was arbitrary we have proven that

[TABLE]

for every $M\in\mathbb{N}$. Finally invoking Corollary 2.5, we conclude that

[TABLE]

for every $D\in\mathcal{H}_{p}(X)$.

$(c)\Rightarrow(a)$: This implication is straightforward, so the proof is complete.

∎

Theorem 3.1 brings up a natural question: given a fixed sequence $(x_{n})_{n}$, are conditions (9) and (10) equivalent? In other words, is the sequence $(x_{n}n^{-s})_{n}$ $RUC$ in $\mathcal{H}_{p}(X)$ if and only if $(x_{n}z^{n})_{n}$ is $RUC$ in $H_{p}(X)$? In the following examples we provide a negative answer to this question. In fact, we will see that none of the implications hold.

Example 3.4**.**

A sequence $(x_{n})_{n}\subseteq X$ such that $(x_{n}n^{-s})_{n}$ is $RUC$ in $\mathcal{H}_{2}(X)$ but $(x_{n}z^{n})_{n}$ fails to be $RUC$ in $H_{2}(X)$.

Let $X$ be the Banach space $L_{1}(\mathbb{T}^{2})$ and consider the sequence $(x_{n})_{n}$ defined by

[TABLE]

Proof.

First, we see that $(x_{n}n^{-s})_{n}$ is $RUC$ in $\mathcal{H}_{2}(X)$. For $N\in\mathbb{N}$, let $m\in\mathbb{N}$ such that $p_{m}\leq N<p_{m+1}$, in other words, $\{p_{1},\ldots,p_{m}\}$ is the set of prime numbers less than or equal to $N$. Set $M_{N}=\{1,\ldots,N\}\smallsetminus\{p_{1},\ldots,p_{m}\}$. We have

[TABLE]

We analyze both terms in the right hand side separately. First observe that the Bohr transform maps the terms $(p_{i}^{-s})_{i=1}^{m}$ to $m$ independent Steinhaus random variables $(z_{i})_{i=1}^{m}$. Therefore, using Kahane’s inequality (5), the definition of the norm in $\mathcal{H}_{2}(X)$ and (7), we get

[TABLE]

Now, observe that

[TABLE]

Hence, by Jensen’s inequality we have

[TABLE]

Therefore, it follows that

[TABLE]

Thus, we have shown that

[TABLE]

For the second term in (13), observe that the variables $w_{2}^{2^{n}}$ behave as if they were independent Rademacher variables, since $2^{n}$ is a lacunary sequence [18, Theorem 2.1]. As it was done before, an unconditionality argument yields

[TABLE]

Combining (14) and (15) with (13) we get the desired result.

It remains to see that $(x_{n}z^{n})_{n}$ fails to be $RUC$ in $H_{2}(X)$. The proof of this fact is inspired in that of [20, Proposition 12.8], and uses Green-Tao’s theorem which states that the sequence of prime numbers contains arbitrarily long arithmetic progressions [12].

Assume that $(x_{n}z^{n})_{n}$ is $RUC$. Given $N\in\mathbb{N}$ there exists an arithmetic progression $A_{N}$ of length $N$ contained in the prime numbers. Consider the coefficients

[TABLE]

Since $(x_{n}z^{n})_{n}$ is $RUC$ we get

[TABLE]

where the $\log N$ term comes from the classical estimation of the $L_{1}-$norm of the Dirichlet kernel (see for example [16, pp. 59-60]). This leads to a contradiction since the inequality cannot hold for arbitrarily large $N$. ∎

Example 3.5**.**

A sequence $(x_{n})_{n}\subseteq X$ such that $(x_{n}z^{n})_{n}$ is $RUC$ in $H_{2}(X)$ but $(x_{n}n^{-s})_{n}$ fails to be $RUC$ in $\mathcal{H}_{2}(X)$.

Let $X$ be the Banach space $L_{1}(\mathbb{T}^{2})$. Define $(x_{n})_{n}$ by

[TABLE]

Proof.

We omit the proof of the first assertion since it is similar to the previous example. Assume that $(x_{n}n^{-s})_{n}$ is $RUC$. In particular, the $RUC$ inequality holds for sequences supported in the powers of two. Recall that the Bohr transform maps $\left(2^{k}\right)^{-s}$ to $z_{1}^{k}$. In other words, we have

[TABLE]

A quick computation leads to a contradiction since

[TABLE]

cannot hold. ∎

An analogous result to Proposition 3.1 holds replacing the $RUC$ property by $RUD$, leading to the corresponding definition of $\mathcal{H}_{p}-RDP$. We state this result without proof, as it follows the same arguments.

Proposition 3.6**.**

Let $X$ be a Banach space and $p\leq 2$. The following statements are equivalent:

1. (a)

$(x_{n}n^{-s})_{n}$* is RUD in $\mathcal{H}_{p}(X)$ for every $(x_{n})_{n}\subset X$.* 2. (b)

There is $C\geq 1$ such that for every $N\in\mathbb{N}$ and $(x_{n})^{N}_{n=1}$ we have

[TABLE] 3. (c)

The following inclusion holds:

[TABLE] 4. (d)

There is $C\geq 1$ such that for every $N\in\mathbb{N}$ and $(x_{n})^{N}_{n=1}$ we have

[TABLE]

Definition 3.7**.**

Given $p\leq 2$, we will say that a Banach space $X$ has the $\mathcal{H}_{p}$ random divergence property (or, in short, $X$ has $\mathcal{H}_{p}-RDP$ ) if $X$ satisfies any (and all) of the conditions in Proposition 3.6

Remark 3.8**.**

Both examples 3.4 and 3.5 actually work if we consider $x_{n}\in L_{r}(\mathbb{T}^{2})$ for $1<r<2$ instead of $L_{1}(\mathbb{T}^{2})$. Also, the dual statements involving $RUD$ may be proven with the same tools and taking $2<r<\infty$.

4 The role of type and cotype

In this section we show that $\mathcal{H}_{2}-RCP$ and $\mathcal{H}_{2}-RDP$ are equivalent, respectively, to type 2 and cotype 2. The proof is based on the work of Arendt and Bu [1, Theorem 1.5]. We also consider the case $p\neq 2$. For the definition and general properties of type and cotype we refer to [10, Chapter 11].

Theorem 4.1**.**

Given a Banach space $X$ the following statements hold:

1. (i)

$X$* has type $2$ if and only if it has the $\mathcal{H}_{2}-RCP$;* 2. (ii)

$X$* has cotype $2$ if and only if it has the $\mathcal{H}_{2}-RDP$.*

Recalling Kwapień’s characterization of Hilbert spaces and joining both theorems we arrive at the following conclusion.

Corollary 4.2**.**

For a Banach space $X$ we have that $\mathcal{H}_{p}(X)=\mathcal{H}_{p}^{rad}(X)$ if and only if $p=2$ and $X$ is a Hilbert space.

Proof.

We may deduce from the scalar case that $p$ must be equal to $2$. Furthermore, from Theorem 4.1 we obtain that $X$ has type and cotype $2$ and therefore it is a Hilbert space. The converse is straightforward. ∎

Regarding the case where $p>2$, we can apply Theorem 4.1 to get that type $2$ implies $\mathcal{H}_{p}-RCP$. Whether or not the converse holds still eludes us. However, a slightly weaker result can be established.

Theorem 4.3**.**

If $X$ has the $\mathcal{H}_{p}$ random convergence property for some $2\leq p<\infty$, then

[TABLE]

For convenience we start by analyzing the $\mathcal{H}_{p}-RCP$ for the spaces $L_{r}(\mathbb{T}^{\mathbb{N}})$.

Proposition 4.4**.**

If $2\leq r<\infty$, the space $L_{r}(\mathbb{T}^{\mathbb{N}})$ has the $\mathcal{H}_{p}-RCP$ for every $2\leq p<\infty$. On the other hand if $1\leq r<2$, the space $L_{r}(\mathbb{T}^{\mathbb{N}})$ does not have the $\mathcal{H}_{p}-RCP$ for any $2\leq p<\infty$.

Proof.

Assume first that $2\leq r<\infty$. It suffices to show that the space $L_{r}(\mathbb{T}^{\mathbb{N}})$ enjoys the $\mathcal{H}_{2}-RCP$. Indeed, given $(f_{n})_{n}\subseteq L_{r}(\mathbb{T}^{\mathbb{N}})$ by Kahane’s inequality (5), we have that

[TABLE]

where in the last step we use Minkowski’s integral inequality (regarding the $\mathcal{H}_{2}$ norm as an integral via Bohr’s transform).

It remains to check that for $1\leq r<2$, the space $L_{r}(\mathbb{T}^{\mathbb{N}})$ does not have $\mathcal{H}_{p}-RCP$ for any $p\geq 2$. Let $m\in\mathbb{N}$ and for every $1\leq n\leq m$ let $f_{n}\in L_{r}(\mathbb{T}^{\mathbb{N}})$ be the function defined by $f_{n}(w)=w^{\alpha(n)}$. Fix scalars $(a_{n})_{n=1}^{m}$. Using Proposition 2.6, the contraction principle (6) and Khintchine’s inequality, we get

[TABLE]

On the other hand, we have

[TABLE]

Since $r<2$ and $\mathcal{H}_{p}(\mathbb{C})$ norms are pairwise comparable like $L_{p}(\mathbb{C})$, there is no constant $C>0$ independent of $m$ such that for every choice of scalars $(a_{n})_{n=1}^{m}$

[TABLE]

This completes the proof. ∎

Now the proof of Theorem 4.3 is simple.

Proof of Theorem 4.3.

We prove this statement by contraposition. Assume

[TABLE]

By the Maurey-Pisier Theorem [17] (see also [10, Chapter 14]), $\ell_{s}$ is finitely representable in $X$. Consequently, the space $L_{s}(\mathbb{T}^{\mathbb{N}})$ is also finitely representable in $X$ since $L_{s}$ spaces are finitely representable in $\ell_{s}$. As $L_{s}(\mathbb{T}^{\mathbb{N}})$ fails to have the $\mathcal{H}_{p}$ random convergence property for $2\leq p<\infty$ and this is clearly a local property, the result follows. ∎

Finally, we show Theorem 4.1 holds.

Proof of Theorem 4.1.

We prove only the first assertion. The second one is omitted since the proof is very similar. Assume that $X$ has type $2$. Let $(\gamma_{n})_{n=1}^{m}$ denote independent identically distributed gaussian random variables. Given $(x_{n})_{n=1}^{m}\subset X$, let us consider the following operators:

[TABLE]

Using the Proposition 2.6, the comparison between Rademacher and gaussian variables (cf. [22, (4.2)]) and [22, Theorem 12.2], we have

[TABLE]

where $\pi_{2}$ is the 2-summing operator norm (see [10, Chapter 2] for the definition and basic properties). From the definition of 2 summing operators, it is easy to check that

[TABLE]

which together with (20) proves $X$ has $\mathcal{H}_{2}-RCP$.

For the converse we follow [1]. Given $x_{1},\ldots,x_{N}\in X$, we have to prove that

[TABLE]

Fix $f_{n}\in L^{2}(\mathbb{T})$ with $\lVert f_{n}\rVert_{2}=1$ and disjoint support. An easy calculation gives us

[TABLE]

Fix $\varepsilon>0$. Since trigonometric polynomials are dense in $L_{2}(\mathbb{T})$, there is a polynomial $h_{n}=\sum_{j=-N_{n}}^{N_{n}}a_{n,j}z^{j}$ such that

[TABLE]

There is no loss of generality in assuming that $\lVert h_{n}\rVert_{2}=1$. Therefore, we have

[TABLE]

Notice that we can choose $M_{n}\in\mathbb{N}$ sufficiently large so that the powers of $z$ appearing in $h_{n}z^{M_{n}}$ are positive and do not overlap. More precisely, we have

[TABLE]

where $b_{n,j}=a_{n,j-M_{n}}$ and $J_{n}=\{M_{n}-N_{n},\ldots,M_{n}+N_{n}\}$ are pairwise disjoint. By the contraction principle (6) and Proposition 3.1 we deduce

[TABLE]

where $\delta_{n},j$ are independent Bernoulli random variables. Theorem 4.3 tells us that $X$ has non-trivial type and therefore finite cotype. Thus, the variables $\delta_{n,j}$ may be replaced by independent Gaussian variables $\gamma_{n,j}$. Define $\gamma_{n}=\sum_{j\in J_{n}}b_{n,j}\gamma_{n,j}$ and observe that they are independent Gaussian variables of variance 1 since $\sum_{j\in J_{n}}|b_{n,j}|^{2}=\lVert h_{n}z^{M_{n}}\rVert_{2}=\lVert h_{n}\rVert_{2}=1$. Pushing inequality (4) a little further we get

[TABLE]

Gathering (4) and (4) together leads to the conclusion. ∎

Acknowledgements

The first three authors were partially supported by CONICET-PIP 11220130100329CO and ANPCyT PICT 2015-2299. The second and third authors are also supported by a CONICET doctoral fellowship. Fourth author gratefully acknowledges support of Spanish Ministerio de Economía, Industria y Competitividad through grants MTM2016-76808-P, MTM2016-75196-P, and the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554).