# On Tempered Discrete and Lévy White Noises

Julien Fageot111Julien Fageot was supported by the Swiss National Science
Foundation (SNSF) under Grant P400P2_194364.

(École polytechnique fédérale de Lausanne)

###### Abstract

We study the growth properties of the family of i.i.d. random sequences, also
known as discrete white noises, and of their continuous-domain generalization,
the family of Lévy white noises. More precisely, we characterize the members
of both families which are tempered— _i.e._ , whose asymptotic growth is
dominated by some polynomial—in terms of their moment properties. We recover
the characterization of tempered Lévy white noises obtained by Robert Dalang
and Thomas Humeau and provide a new proof of there fundamental result. Our
approach is based on a fruitful connection between the discrete and
continuous-domain white noises.

##  1 Introduction

###  1.1 Lévy White Noises as Generalized Random Processes

The Gaussian white noise is a random process whose sample paths are too rough
to be described as a pointwise random function. For this reason, it has been
considered as a random element in the space
${\mathscr{S}}^{\prime}(\mathbb{R})$ of tempered generalized functions, for
instance in quantum physics [37], stochastic partial differential equations
[42], and white noise analysis [18, 36].

This construction of the Gaussian white noise relies on the theory of
generalized random processes,initiated independently in the 50’s by Israel
Moiseevich Gel’fand [15] and Kiyoshi Itô [21]. This framework is the
stochastic counterpart of the theory of generalized functions of Laurent
Schwartz [35] and has been extensively detailed in the monographs [16, 13,
22]; see also [4] for a recent introduction. The key idea is to endow
generalized function spaces with random structures compatible with their
topology. This allows to define random elements in the space
${\mathscr{S}}^{\prime}(\mathbb{R})$, but also over the larger space
${\mathscr{D}}^{\prime}(\mathbb{R})$ of (not necessarily tempered) generalized
functions. The later is used by Gel’fand and Vilenkin to introduce the
complete family of Lévy white noises [16, Chapter 3]. Those random processes
share the properties of the Gaussian white noise with the exception of
Gaussianity in the same way Lévy processes generalize the Brownian motion [3,
33].

Lévy white noises are therefore constructed as random elements in the space
${\mathscr{D}}^{\prime}(\mathbb{R})$, while the Gaussian white noise is known
to be tempered, _i.e._ located in the space
${\mathscr{S}}^{\prime}(\mathbb{R})$. It is then natural to ask whether a
given Lévy white noise is tempered or not.

###  1.2 The Temperedness of Lévy White Noise: Related Works

Tempered Lévy white noises have several advantages. For instance, as random
elements of ${\mathscr{S}}^{\prime}(\mathbb{R})$, there Fourier transform is
well-defined and they can therefore be used for stochastic partial
differential equations SPDEs. Many works dealing with Lévy white noise
therefore identify sufficient conditions so that the temperedness is achieved.
The authors of [25] consider Lévy white noise with finite first moment, while
finite variance Lévy white noises are used for SPDEs in [28, 29].

In [8], Arash Amini, Michael Unser, and the author of this article have shown
that a Lévy white noise is tempered as soon as it possesses a finite absolute
moment of order $\epsilon>0$ arbitrarily small, what they called the
_$\epsilon$ -condition_. They conjectured that this sufficient condition is
also necessary. This was further proved by Robert Dalang and Thomas Humeau in
[6]. More precisely, these authors proved the following result, exposed in
Theorem  ‣ 1.2, which also reveals that Lévy white noises are almost surely
not tempered when the $\epsilon$-condition does not occur. There result
therefore fully characterize the temperedness of Lévy white noise from simple
moments properties.

######  Theorem 0 (Dalang and Humeau, Theorem 2.5 in [6]).

Let $W$ be a Lévy white noise on ${\mathscr{D}}^{\prime}(\mathbb{R})$. If
$\mathbb{E}[|\langle W,\mathbbm{1}_{[0,1]}\rangle|^{\epsilon}]<\infty$222The
random variable $X_{\varphi}=\langle W,\varphi\rangle$ is a priori well-
defined for $\varphi\in{\mathscr{D}}(\mathbb{R})$, which is not the case for
$\varphi=\mathbbm{1}_{[0,1]}$. However, one can extend the definition of
$X_{\varphi}$ to test functions that are compactly supported and bounded for
any Lévy white noise, as can easily be deduced from [12]. This is implicit in
Theorem  ‣ 1.2. for some $\epsilon>0$, then
$W\in{\mathscr{S}}^{\prime}(\mathbb{R})$ almost surely. Otherwise,
$W\notin{\mathscr{S}}^{\prime}(\mathbb{R})$ almost surely.

Understanding the sample path properties of random processes is a natural
question. The growth properties of Lévy processes has been considered in [30];
see [33, Section 48] for extensive discussions and additional references on
this matter. The control of the growth of Lévy processes has also been
considered by authors studying the regularity of Lévy processes and Lévy white
noises, measured in terms of weighted Besov spaces[34, 1, 9]. Those weighted
Besov spaces being subspaces of ${\mathscr{S}}^{\prime}(\mathbb{R})$, the
temperedness of the considered random processes is established. However, these
works do not prove the necessity of the $\epsilon$-condition.

###  1.3 Contributions and Outline

Our goal in this paper is to revisit the characterization of tempered Lévy
white noise of Dalang and Humeau. In particular, we provide a new proof of
Theorem  ‣ 1.2. The exposition is self-contained and highlights the connection
between Lévy white noises and families of i.i.d. random sequences (_i.e._ ,
discrete white noises). The paper is organized as follows.

  * •

We first focus on discrete white noises in Section 2. Our main result is
exposed in Theorem 1: We characterize the weighted $\ell_{\infty}$-sequence
spaces on which an i.i.d. random sequence belongs in terms of its moment
properties. In particular, we demonstrate that the $\epsilon$-condition is
equivalent to the temperedness of discrete white noises. Even if this result
is not surprising, it has never been observed to the best of our knowledge.

  * •

We then consider the case of (continuous-domain) Lévy white noises in Section
3.2. Our main result is Theorem 2, where the temperedness of Lévy white noise
is characterized from different perspectives. It covers Theorem  ‣ 1.2,
together with known characterizations of the temperedness in terms
characteristic functional, characteristic exponent, and the Lévy measure of
the underlying white noise. The novelty of our result is the connection
between a Lévy white noise and the random sequences $(\langle
W,\varphi(\cdot-n)\rangle)_{n\in\mathbb{Z}}$, and results in a new proof of
Theorem  ‣ 1.2.

##  2 Discrete White Noises

###  2.1 Sequence Spaces

We denote by ${\mathscr{D}}^{\prime}(\mathbb{Z})$ the space of all real
sequences. This notation is somehow unusual and is made in analogy with the
space ${\mathscr{D}}^{\prime}(\mathbb{R})$ of generalized functions (see
Section 3.1). The space ${\mathscr{D}}^{\prime}(\mathbb{Z})$ is the dual of
the space ${\mathscr{D}}(\mathbb{Z})$ of sequences that have finitely many
nonzero elements. Any sequence $u\in{\mathscr{D}}^{\prime}(\mathbb{Z})$ indeed
defines a linear form over the space ${\mathscr{D}}(\mathbb{Z})$ via the
relation

$\langle u,v\rangle=\sum_{n\in\mathbb{Z}}u[n]v[n]$ (1)

for any $v\in{\mathscr{D}}(\mathbb{Z})$. Note that the sum in (1) is actually
finite. One can show that ${\mathscr{D}}^{\prime}(\mathbb{Z})$ is the
topological dual of ${\mathscr{D}}(\mathbb{Z})$ if we endow the later with the
locally convex topology corresponding to the following notion of convergence.
The sequences $v_{k}=(v_{k}[n])_{n\in\mathbb{Z}}\in{\mathscr{D}}(\mathbb{Z})$,
$k\geq 1$, converge to $0$ if they have a common support $K\subset\mathbb{Z}$
outside of which they vanish and if the finite-dimensional vectors
$(v_{k}[n])_{n\in K}$ converge to $0$.

The space $\ell_{\infty}(\mathbb{Z})$ of bounded sequences
$u=(u[n])_{n\in\mathbb{Z}}\in{\mathscr{D}}^{\prime}(\mathbb{Z})$ such that
$\lVert u\rVert_{\infty}=\sup_{n\in\mathbb{Z}}|u[n]|<\infty$ is a Banach space
for the norm $\lVert\cdot\rVert_{\infty}$. More generally, for
$\alpha\in\mathbb{R}$, let $\ell_{\infty,\alpha}(\mathbb{Z})$ be the weighted
Banach space for the norm

$u\mapsto\lVert
u\rVert_{\infty,\alpha}=\sup_{n\in\mathbb{Z}}|n|^{\alpha}|u[n]|<\infty.$ (2)

When $\alpha_{1}\geq\alpha_{2}$, we have the topological embedding
$\ell_{\infty,\alpha_{1}}(\mathbb{Z})\subseteq\ell_{\infty,\alpha_{2}}(\mathbb{Z})$.

The space of rapidly decaying sequence is denoted by
${\mathscr{S}}(\mathbb{Z})$. It corresponds to the sequences
$u=(u[n])_{n\in\mathbb{Z}}$ such that
$|n|^{\alpha}|u[n]|\rightarrow_{|n|\rightarrow\infty}0$ for any $\alpha\geq
0$. It is a Fréchet space for the family of norms
$\lVert\cdot\rVert_{\infty,\alpha}$ with $\alpha\geq 0$ and we have the
projective limit

${\mathscr{S}}(\mathbb{Z})=\bigcap_{\alpha\geq
0}\ell_{\infty,\alpha}(\mathbb{Z}).$ (3)

The topological dual of ${\mathscr{S}}(\mathbb{Z})$ is the space
${\mathscr{S}}^{\prime}(\mathbb{Z})$ of tempered sequences, _i.e._ , the space
of sequences which are bounded by a polynomial. It is the inductive limit

${\mathscr{S}}^{\prime}(\mathbb{Z})=\bigcup_{\alpha\geq
0}\ell_{\infty,-\alpha}(\mathbb{Z}).$ (4)

The following simple characterization of tempered sequences will be useful in
Section 3.

######  Proposition 1.

Let $u\in{\mathscr{D}}^{\prime}(\mathbb{Z})$ seen as a linear functional over
${\mathscr{D}}(\mathbb{Z})$ according to (1). Then,
$u\in{\mathscr{S}}^{\prime}(\mathbb{Z})$ if and only if
$u:{\mathscr{D}}(\mathbb{Z})\rightarrow\mathbb{R}$ is continuous for the
topology induced by ${\mathscr{S}}(\mathbb{Z})$. This is equivalent to the
fact that

$\langle u,v_{k}\rangle\underset{k\rightarrow\infty}{\longrightarrow}0$ (5)

for any sequences $v_{k}\in{\mathscr{D}}(\mathbb{Z})$, $k\geq 1$, such that
$\lVert v_{k}\rVert_{\infty,\alpha}\rightarrow 0$ when $k\rightarrow\infty$
for any $\alpha\geq 0$.

###### Proof.

By definition, $u\in{\mathscr{S}}^{\prime}(\mathbb{Z})$ defines a continuous
and linear form over ${\mathscr{S}}(\mathbb{Z})$ as in (1) and its restriction
over ${\mathscr{D}}(\mathbb{Z})$ for the topology induced by
${\mathscr{S}}(\mathbb{Z})$ is clearly continuous. Conversely, any
$u\in{\mathscr{D}}^{\prime}(\mathbb{Z})$ such that
$v\in{\mathscr{D}}(\mathbb{Z})\mapsto\langle u,v\rangle$ is continuous for the
topology of ${\mathscr{S}}(\mathbb{Z})$ and can be uniquely extended as a
linear and continuous form over ${\mathscr{S}}(\mathbb{Z})$ (using that
${\mathscr{D}}(\mathbb{Z})$ is dense in ${\mathscr{S}}(\mathbb{Z})$) and the
extension is therefore in ${\mathscr{S}}^{\prime}(\mathbb{Z})$ by definition
of the dual. This extension corresponds to the sequence $u$ itself, which is
therefore tempered. ∎

###  2.2 Random Sequences and Discrete White Noises

Throughout this paper, we fix a probability space
$(\Omega,\mathcal{F},\mathscr{P})$. A random sequence is a random element in
the space ${\mathscr{D}}^{\prime}(\mathbb{Z})$. More precisely, it is a
mapping $X:\mathbb{Z}\times\Omega\rightarrow\mathbb{R}$ such that
$X(n,\omega)=X_{n}(\omega)\in\mathbb{R}$ for any $\omega\in\Omega$ and
$n\in\mathbb{Z}$. Then, for any fixed $n\in\mathbb{Z}$, $X_{n}$ is a real
random variable. For any $\omega\in\Omega$, we have that
$X(\omega)=(X_{n}(\omega))_{n\in\mathbb{Z}}\in{\mathscr{D}}^{\prime}(\mathbb{Z})$.
The law of $X$ is characterized by the quantities

$\mathscr{P}_{X}(\\{u\in{\mathscr{D}}^{\prime}(\mathbb{Z})|\langle
u,v\rangle\leq a\\})=\mathscr{P}(\langle X,v\rangle\leq a)$ (6)

for any $v\in{\mathscr{D}}(\mathbb{Z})$ and $a\in\mathbb{R}$. Thereafter, it
will be important to characterize the random sequences $X$ that are tempered,
meaning that $X\in{\mathscr{S}}^{\prime}(\mathbb{Z})$ almost surely.

######  Definition 1.

A _discrete white noise_ is a random sequence $X=(X_{n})_{n\in\mathbb{Z}}$ of
independent and identically distributed (i.i.d.) random variables $X_{n}$.

The law of a discrete white noise $X$ is fully characterized by the law of
$X_{0}$. For any sequence $(a_{n})_{n\in\mathbb{Z}}$, we indeed have that

$\mathscr{P}(\forall n\in\mathbb{Z},\ X_{n}\leq
a_{n})=\prod_{n\in\mathbb{Z}}\mathscr{P}(X_{n}\leq
a_{n})=\prod_{n\in\mathbb{Z}}\mathscr{P}(X_{0}\leq a_{n}),$ (7)

where the infinite products are converging in $[0,1]$ (as any infinite product
of terms in $[0,1]$). A discrete white noise is _a priori_ in
${\mathscr{D}}^{\prime}(\mathbb{Z})$. In the next section, we provide
characterizations of discrete white noises to be localized in the sequence
spaces $\ell_{\infty,\alpha}(\mathbb{Z})$ and
${\mathscr{S}}^{\prime}(\mathbb{Z})$.

###  2.3 Characterization of Tempered Discrete White Noises

The goal of this section is to study the growth properties of discrete white
noises. More precisely, we shall prove the following result.

######  Theorem 1.

Let $X=(X_{n})_{n\in\mathbb{Z}}$ be a discrete white noise. Then, we have the
equivalence

$X\in\ell_{\infty,-1/p}(\mathbb{Z})\text{ a.s.
}\quad\Longleftrightarrow\quad\mathbb{E}[|X_{0}|^{p}]<\infty.$ (8)

It this condition is not fulfilled, then
$X\notin\ell_{\infty,-1/p}(\mathbb{Z})$ a.s. We also have the equivalence

$X\in{\mathscr{S}}^{\prime}(\mathbb{Z})\text{ a.s.
}\quad\Longleftrightarrow\quad\exists\ \epsilon>0,\
\mathbb{E}[|X_{0}|^{\epsilon}]<\infty.$ (9)

If this condition is not fulfilled, then
$X\notin{\mathscr{S}}^{\prime}(\mathbb{Z})$ a.s.

Our proof is based on Proposition 2, which establishes $(0-1)$ laws for
discrete white noises.

######  Proposition 2.

For any discrete white noise $X$ and any $\alpha>0$, we have that

$\mathscr{P}(X\in\ell_{\infty,-\alpha}(\mathbb{Z}))\in\\{0,1\\}$ (10)

and $\mathscr{P}(X\in\ell_{\infty,-\alpha}(\mathbb{Z}))=1$ if and only if

$\sum_{n\geq 0}\mathscr{P}(|X_{0}|\geq n^{\alpha})<\infty.$ (11)

Moreover, $\mathscr{P}(X\in{\mathscr{S}}^{\prime}(\mathbb{Z}))\in\\{0,1\\}$,
the value being $1$ if and only if there exists $\alpha\geq 0$ such that (11)
holds.

Remark. The characterization (11) in Proposition 2 is false for $\alpha=0$. In
this case, we have that $\mathscr{P}(X\in\ell_{\infty}(\mathbb{Z}))=1$ if and
only if the common law $\mathscr{P}_{X_{0}}$ of the $X_{n}$ has a finite
support.

###### Proof of Proposition 2.

For $x\geq 0$, we set $F(x)=\mathscr{P}(|X_{0}|\geq x)$. The set
$\ell_{\infty,-\alpha}(\mathbb{Z})$ satisfies

$\ell_{\infty,-\alpha}(\mathbb{Z})=\bigcup_{C>0}\left\\{u=(u_{n})_{n\in\mathbb{Z}}\in{\mathscr{D}}^{\prime}(\mathbb{Z}),\
|u_{n}||n|^{\alpha}\leq C\right\\},$ (12)

where the union over $C>0$ is increasing when $C$ increases. Hence, we deduce
that

$\displaystyle\mathscr{P}(X\in\ell_{\infty,-\alpha}(\mathbb{Z}))$
$\displaystyle=\lim_{C\rightarrow\infty}\mathscr{P}(\forall n\in\mathbb{Z},\
|X_{n}||n|^{-\alpha}\leq
C)=\lim_{C\rightarrow\infty}\prod_{n\in\mathbb{Z}}\mathscr{P}(|X_{0}||n|^{-\alpha}\leq
C)$
$\displaystyle=\lim_{C\rightarrow\infty}\prod_{n\in\mathbb{Z}}\left(1-F(C|n|^{\alpha})\right)=\lim_{C\rightarrow\infty}\exp\left(\sum_{n\in\mathbb{Z}}\log\left(1-F(C|n|^{\alpha})\right)\right),$
(13)

where the infinite product converges in $[0,1]$ and the infinite sum in
$[-\infty,0]$.

If $(u_{n})_{n\in\mathbb{Z}}$ is a sequence such that $0\leq u_{n}<1$ for any
$n\in\mathbb{Z}$, and $u_{n}\rightarrow 0$ when $|n|\rightarrow\infty$, then
$-\log(1-u_{n})\geq 0$ for every $n\in\mathbb{Z}$ and

$\sum_{n\in\mathbb{Z}}u_{n}<\infty\quad\Longleftrightarrow\quad\sum_{n\in\mathbb{Z}}-\log(1-u_{n})<\infty.$
(14)

This follows from the fact that
$-\log(1-u_{n})=\log(1+\frac{u_{n}}{1-u_{n}})\sim\frac{u_{n}}{1-u_{n}}\sim
u_{n}$ when $|n|\rightarrow\infty$. We apply this relation to
$u_{n}=F(C|n|^{\alpha})$ and distinguish two cases.

Assume first that $\sum_{n\in\mathbb{Z}}F(C|n|^{\alpha})=\infty$ for any
$C>0$. Then, (14) implies that

$\sum_{n\in\mathbb{Z}}\log\left(1-F(C|n|^{\alpha})\right)=-\infty$ (15)

for any $C>0$ and therefore, due to (2.3),
$\mathscr{P}(X\in\ell_{\infty,-\alpha}(\mathbb{Z}))=\exp(-\infty)=0$.

Assume now that there exists $C_{0}>0$ such that
$\sum_{n\in\mathbb{Z}}F(C_{0}|n|^{\alpha})=\infty$. Then, we have that

$-\log(1-F(C|n|^{\alpha}))\underset{C\rightarrow\infty}{\longrightarrow}-\log(1-F(\infty))=-\log(1-0)=0.$
(16)

Moreover, for $C\geq C_{0}$, using that $F$ is decreasing,

$-\log(1-F(C|n|^{\alpha}))\leq-\log(1-F(C_{0}|n|^{\alpha})),$ (17)

the right term being summable (using (14)) and independent from $C\geq C_{0}$.
We can therefore apply the Lebesgue dominated convergence theorem to deduce
that

$\sum_{n\in\mathbb{Z}}-\log(1-F(C|n|^{\alpha}))\underset{C\rightarrow\infty}{\longrightarrow}0.$
(18)

Finally, this implies with (2.3) that
$\mathscr{P}(X\in\ell_{\infty,-\alpha}(\mathbb{Z}))=\exp(0)=1$.

We have shown that $\mathscr{P}(X\in\ell_{\infty,-\alpha}(\mathbb{Z}))$ can
only take the value $0$ and $1$. Moreover, the value $1$ is taken if and only
if there exists $C_{0}>0$ such that
$\sum_{n\in\mathbb{Z}}F(C_{0}|n|^{\alpha})=\infty$ holds. However, this
condition is then true for any $C_{0}$. To see this, we simply observe that

$\sum_{n\in\mathbb{Z}}F(C_{0}|n|^{\alpha})<\infty\quad\Longleftrightarrow\quad\int_{\mathbb{R}}F(C_{0}|x|^{\alpha})\mathrm{d}x<\infty.$
(19)

Indeed, $F$ is decreasing and therefore, for any $0\leq n\leq x\leq n+1$,
$F(C_{0}(n+1)^{\alpha})\leq F(C_{0}x^{\alpha})\leq F(C_{0}n^{\alpha})$ (with
similar relations for $x\leq 0$), which easily implies (19). Moreover, a
change of variable $x\leftarrow C_{0}^{1/\alpha}x$ gives that
$\int_{\mathbb{R}}F(C_{0}|x|^{\alpha})\mathrm{d}x=C_{0}^{-1/\alpha}\int_{\mathbb{R}}F(|x|^{\alpha})\mathrm{d}x$,
hence the finiteness does not depend on $C_{0}>0$. Finally, (11) follows by
choosing $C_{0}=1$.

For ${\mathscr{S}}^{\prime}(\mathbb{Z})$, we remark that the spaces
$\ell_{\infty,-\alpha}(\mathbb{Z})$ are growning when $\alpha\geq 0$
increases, hence

${\mathscr{S}}^{\prime}(\mathbb{Z})=\bigcup_{\alpha\geq
0}\ell_{\infty,-\alpha}(\mathbb{Z})=\lim_{\alpha\rightarrow\infty}\ell_{\infty,-\alpha}(\mathbb{Z}).$
(20)

This implies that

$\mathscr{P}(X\in{\mathscr{S}}^{\prime}(\mathbb{Z}))=\lim_{\alpha\rightarrow\infty}\mathscr{P}(X\in\ell_{\infty,-\alpha}(\mathbb{Z})).$
(21)

Assume that there exists $\alpha_{0}>0$ such that (11) holds. Then,
$\mathscr{P}(X\in\ell_{\infty,-\alpha_{0}}(\mathbb{Z}))=1$ and since
$\ell_{\infty,-\alpha_{0}}(\mathbb{Z})\subset\ell_{\infty,-\alpha}(\mathbb{Z})$,
we also have that $\mathscr{P}(X\in\ell_{\infty,-\alpha}(\mathbb{Z}))=1$ for
any $\alpha\geq\alpha_{0}$, hence
$\mathscr{P}(X\in{\mathscr{S}}^{\prime}(\mathbb{Z})=1$ due to (21). On the
contrary, assume that $\sum_{n\geq 0}\mathscr{P}(|X_{0}|\geq
n^{\alpha})=\infty$ for any $\alpha$. This implies that
$\mathscr{P}(X\in\ell_{\infty,-\alpha}(\mathbb{Z}))=0$ for every $\alpha\geq
0$ and therefore $\mathscr{P}(X\in{\mathscr{S}}^{\prime}(\mathbb{Z}))=0$,
again due to (21). ∎

###### Proof of Theorem 1.

We first observe that the $(0-1)$ law in Proposition 2 implies that
$X\notin\ell_{\infty,-1/p}(\mathbb{Z})$ a.s. as soon as the condition
$X\in\ell_{\infty,-1/p}(\mathbb{Z})$ is not satisfied (the same holds for the
sequence space ${\mathscr{S}}^{\prime}(\mathbb{Z})$). This covers the last
part of Theorem 1.

Using the same argument as in the proof of Proposition 2, we moreover have
that

$\sum_{n\geq 0}\mathscr{P}(|X_{0}|\geq
n^{1/p})=\sum_{n\in\mathbb{Z}}F(n^{1/p})<\infty\quad\Longleftrightarrow\quad\int_{0}^{\infty}F(x^{1/p})\mathrm{d}x<\infty.$
(22)

Denoting by $\mathscr{P}_{|X_{0}|}$ the probability law of $|X_{0}|$, we also
remark that

$\displaystyle\int_{0}^{\infty}F(x^{1/p})\mathrm{d}x$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}\mathbbm{1}_{t\geq
x^{1/p}}\mathscr{P}_{|X_{0}|}(\mathrm{d}t)\mathrm{d}x\
=\int_{0}^{\infty}\left(\int_{0}^{\infty}\mathbbm{1}_{t^{p}\geq
x}\mathrm{d}x\right)\mathscr{P}_{|X_{0}|}(\mathrm{d}t)$
$\displaystyle=\int_{0}^{\infty}t^{p}\mathscr{P}_{|X_{0}|}(\mathrm{d}t)=\mathbb{E}[|X_{0}|^{p}].$
(23)

Finally, this shows that $\sum_{n\geq 0}\mathscr{P}(|X_{0}|\geq
n^{1/p})<\infty$ if and only if $\mathbb{E}[|X_{0}|^{p}]<\infty$ and Theorem 1
follows from the criterion of Proposition 2. ∎

We illustrate Theorem 1 with some examples. Thereafter,
$X=(X_{n})_{n\in\mathbb{Z}}$ is a discrete white noise.

  * •

Assume that the random variables $X_{n}$, $n\in\mathbb{Z}$, are
symmetric-$\alpha$-stable (S$\alpha$S) [32]. This means that the common
characteristic function of the $X_{n}$ is
$\widehat{\mathscr{P}}_{X_{n}}(\xi)=\exp(-\gamma^{\alpha}|\xi|^{\alpha})$
where $\gamma>0$ and $0<\alpha\leq 2$. The case $\alpha=2$ corresponds to the
Gaussian law, for which all moments are finite. Hence,
$X\in\ell_{\infty,-\epsilon}(\mathbb{Z})$ for any $\epsilon>0$ in this case.

Non-Gaussian S$\alpha$S random variables have an infinite variance: For
$\alpha<2$, we actually have that $\mathbb{E}[|X_{0}|^{p}]<\infty$ if and only
if $p<\alpha$ [32, Property 1.2.16]. Hence, Theorem 1 implies that, for
$0<\alpha<2$,

$X\in\ell_{\infty,-\frac{1}{\alpha}-\epsilon}(\mathbb{Z})$ (24)

almost surely for any $\epsilon>0$. In particular,
$X\in{\mathscr{S}}^{\prime}(\mathbb{Z})$ almost surely. Moreover, the fact
that $\mathbb{E}[|X_{0}|^{\alpha}]=\infty$ implies that
$X\notin\ell_{\infty,-\frac{1}{\alpha}}(\mathbb{Z})$ almost surely.

  * •

As for the Gaussian case, random variables whose moments are all finite are
such that $X\in\ell_{\infty,-\epsilon}(\mathbb{Z})$ almost surely for any
$\epsilon>0$. As such, they only slightly differ from a bounded sequence,
since $\sup_{n\in\mathbb{Z}}\frac{|X_{n}|}{1+|n|^{\delta}}<\infty$ almost
surely for $\delta$ arbitrarily small.

  * •

Assume that the $X_{n}$ have a common probability density function given by
$f(x)=\frac{c}{(|x|+1)\log^{2}(|x|+2)}$ where $c>0$ is such that
$\int_{\mathbb{R}}f(x)\mathrm{d}x=1$. Note that $f$ is indeed integrable as a
Bertrand integral. However, we have for any $p>0$ that

$\mathbb{E}[|X_{0}|^{p}]=c\int_{\mathbb{R}}\frac{|x|^{p}}{(|x|+1)\log^{2}(|x|+2)}\mathrm{d}x=\infty$
(25)

due to the fact the integrand behaves asymptotically like
$|x|^{p-1}/\log^{2}(|x|)$ and is therefore non-integrable for $p>0$. Hence,
$X\notin{\mathscr{S}}^{\prime}(\mathbb{Z})$ almost surely.

##  3 Lévy White Noises

###  3.1 Spaces of Generalized Functions

We briefly introduce the relevant function spaces that will be used
thereafter. More details can be found, for instance, in [14]. We refer the
interested reader to [38] for a comprehensive exposition on the topological
structures of the proposed function spaces.

Let ${\mathscr{D}}(\mathbb{R})$ be the space of compactly supported and
infinitely smooth functions. It is equipped with its natural inductive
topology as the union of the spaces ${\mathscr{D}}([-n,n])$ of infinitely
smooth functions whose support is included in $[-n,n]$ for $n\geq 1$. In
particular, a family of functions $(\varphi_{k})_{k\in\mathbb{Z}}$ in
${\mathscr{D}}(\mathbb{R})$ converges to $0$ for this topology if there exists
$n\geq 1$ such that $\mathrm{Supp}(\varphi_{k})\subset[-n,n]$ for every $k\geq
1$ and $\varphi_{k}$ converges uniformly to $0$ on $[-n,n]$ together with all
its derivatives.

The topological dual ${\mathscr{D}}^{\prime}(\mathbb{R})$ of
${\mathscr{D}}(\mathbb{R})$ is the space of generalized functions. A locally
integrable function $f$ defines a linear and continuous functional over
${\mathscr{D}}(\mathbb{R})$ via the relation

$f:\varphi\mapsto\langle
f,\varphi\rangle=\int_{\mathbb{R}}\varphi(x)f(x)\mathrm{d}x.$ (26)

The framework of generalized functions allows to define functions with no
pointwise interpretation, such as the Dirac impulse $\delta$ and all its
derivatives. We will define Lévy white noise are random elements in
${\mathscr{D}}^{\prime}(\mathbb{R})$ in Section 3.2.

The Schwartz space of rapidly decaying and infinitely smooth functions is
denoted by ${\mathscr{S}}(\mathbb{R})$. A function $f$ is in
${\mathscr{S}}(\mathbb{R})$ if it vanishes faster than any polynomial together
with all its derivatives. Then, ${\mathscr{S}}(\mathbb{R})$ is a Fréchet space
for the family of semi-norms

$\lVert\varphi\rVert_{\infty,\alpha,\beta}=\sup_{x\in\mathbb{R}}|x|^{\alpha}|\mathrm{D}^{\beta}\varphi(x)|$
(27)

where $\alpha\geq 0$ and $\beta\in\mathbb{N}$. In particular, a sequence
$(\varphi_{k})_{k\geq 1}$ of functions in ${\mathscr{S}}(\mathbb{R})$
converges to $0$ if $\lVert\varphi_{k}\rVert_{\infty,\alpha,\beta}\rightarrow
0$ as $k\rightarrow\infty$ for any $\alpha\geq 0$ and $\beta\in\mathbb{N}$.

The topological dual of ${\mathscr{S}}(\mathbb{R})$ is the space
${\mathscr{S}}^{\prime}(\mathbb{R})$ of tempered generalized functions. It is
a subspace of ${\mathscr{D}}^{\prime}(\mathbb{R})$ and is informally made of
generalized functions that are slowly growing, _i.e._ growing slower than some
polynomial.

###  3.2 Lévy White Noises as Generalized Random Processes

A generalized random process is a random element of the space
${\mathscr{D}}^{\prime}(\mathbb{R})$ of generalized functions. The theory of
generalized random processes was initiated independently by Gel’fand [15] and
Itô [21], it offers a broad framework for defining random processes, including
some with no pointwise interpretation such as the Lévy white noises.
Thereafter, we mostly rely on the expositions in [16] and [22].

A generalized random process $S$ is a collection of random variable $(\langle
S,\varphi\rangle)_{\varphi\in{\mathscr{D}}(\mathbb{R})}$ such that,

  * •

Linearity: For any $\varphi,\psi\in{\mathscr{D}}(\mathbb{R})$ and
$a,b\in\mathbb{R}$, $\langle S,a\varphi+b\psi\rangle=a\langle
S,\varphi\rangle+b\langle S,\psi\rangle$ almost surely.

  * •

Continuity: For any sequence $(\varphi_{k})_{k\geq 1}$ of test functions
$\varphi_{k}\in{\mathscr{D}}(\mathbb{R})$ such that
$\varphi_{k}\rightarrow\varphi$ in ${\mathscr{D}}(\mathbb{R})$ when
$k\rightarrow\infty$, the random variables $\langle S,\varphi_{k}\rangle$
converges in probability towards $\langle S,\varphi\rangle$.

A generalized random process is therefore a linear and continuous functional
$S:{\mathscr{D}}(\mathbb{R})\rightarrow L_{0}(\Omega)$ where $L_{0}(\Omega)$
is the space of real random variables endowed with the convergence in
probability [22]. We associate to ${\mathscr{D}}^{\prime}(\mathbb{R})$ its
cylindrical $\sigma$-field, _i.e._ the $\sigma$-field generated by the sets

$\\{f\in{\mathscr{D}}^{\prime}(\mathbb{R}),\ \langle f,\varphi\rangle<a\\}$
(28)

where $\varphi\in{\mathscr{D}}(\mathbb{R})$ and $a\in\mathbb{R}$. The
probability law of $S$ is then the probability measure $\mathscr{P}_{S}$ on
${\mathscr{D}}^{\prime}(\mathbb{R})$ such that
$\mathscr{P}_{S}(\\{f\in{\mathscr{D}}^{\prime}(\mathbb{R}),\ \langle
f,\varphi\rangle<a\\})=\mathscr{P}(\langle S,\varphi\rangle<a)$.

######  Definition 2.

The _characteristic functional_
$\widehat{\mathscr{P}}_{S}:{\mathscr{D}}(\mathbb{R})\rightarrow\mathbb{C}$ of
a generalized random process $S$ is defined for
$\varphi\in{\mathscr{D}}(\mathbb{R})$ by

$\widehat{\mathscr{P}}_{S}(\varphi)=\mathbb{E}\left[\mathrm{e}^{\mathrm{i}\langle
S,\varphi\rangle}\right]=\int_{{\mathscr{D}}^{\prime}(\mathbb{R})}\mathrm{e}^{\mathrm{i}\langle
f,\varphi\rangle}\mathscr{P}_{S}(\mathrm{d}f).$ (29)

The characteristic functional of $S$ is the (infinite dimensional) Fourier
transform of its probability law $\mathscr{P}_{S}$. It generalizes the
characteristic function of random variables to the case of generalized random
processes. In particular, two generalized random processes have the same law
if and only if they share the same characteristic functional.

######  Definition 3.

We say that $\Psi:\mathbb{R}\rightarrow\mathbb{C}$ is a _characteristic
exponent_ if it can be written as

$\Psi(\xi)=\mathrm{i}\mu\xi-\frac{\sigma^{2}\xi^{2}}{2}+\int_{\mathbb{R}\backslash\\{0\\}}(\mathrm{e}^{\mathrm{i}\xi
t}-1-\mathrm{i}\xi t\mathbbm{1}_{|t|\geq 1})\nu(\mathrm{d}t),$ (30)

with $\mu\in\mathbb{R}$, $\sigma^{2}\geq 0$, and $\nu$ a Lévy–measure, _i.e._
, a Radon measure over $\mathbb{R}\backslash\\{0\\}$ such that

$\int_{\mathbb{R}\backslash\\{0\\}}\inf(1,t^{2})\nu(\mathrm{d}t)<\infty.$ (31)

A random variable $X$ is infinitely divisible if it can be decomposed as a sum
$X=X_{1}+\cdots+X_{N}$ of $N$ i.i.d. random variables for any $N\geq 1$.
Infinitely divisible random variables are closely related to continuous-domain
random processes. The Lévy–Khintchine theorem [33, Theorem 8.1] ensures that a
random variable $X$ is infinitely divisible if and only if its characteristic
function can be written as $\widehat{\mathscr{P}}_{X}(\xi)=\exp(\Psi(\xi))$
with $\Psi$ a characteristic exponent defined in Definition 3. Following [16,
Chapter III], we now define Lévy white noise from their characteristic
functional and exponent.

######  Definition 4.

A _Lévy white noise_ is a generalized random process $W$ in
${\mathscr{D}}^{\prime}(\mathbb{R})$ whose characteristic functional is given
by

$\widehat{\mathscr{P}}_{W}(\varphi)=\exp\left(\int_{\mathbb{R}}\Psi(\varphi(x))\mathrm{d}x\right),$
(32)

where $\Psi$ is a characteristic exponent, called by extension the
characteristic exponent of $W$. If moreover $W$ is almost surely in
${\mathscr{S}}^{\prime}(\mathbb{R})$, then $W$ is a _tempered Lévy white
noise_.

The fact that $\widehat{\mathscr{P}}_{W}$ specifies the characteristic
functional of some generalized random process in ${\mathscr{D}}(\mathbb{R})$
is proved in [16, Theorem 5, Section III.4.3]. It relies on the Bochner–Minlos
theorem, which characterizes the characteristic functionals as the continuous
and positive-definite functionals
$\widehat{\mathscr{P}}:{\mathscr{D}}(\mathbb{R})\rightarrow\mathbb{C}$ such
that $\widehat{\mathscr{P}}(0)=1$ [13, 22].

The characteristic exponent of the centered Gaussian white noise $W$ is given
by $\Psi(\xi)=-\sigma^{2}\xi^{2}/2$, which corresponds to (30) with $\mu=0$
and $\nu=0$. The characteristic functional of the Gaussian white noise is
therefore
$\widehat{\mathscr{P}}_{W}(\varphi)=\exp(-\sigma^{2}\lVert\varphi\rVert_{2}^{2}/2)$.
The family of Lévy white noises include the S$\alpha$S white noises and the
compound Poisson white noises. These examples, among others, will be
considered below.

We say that a generalized random process $S$ is stationary if $S$ has the same
law than its shifted version $S(\cdot+x_{0})$ for any $x_{0}\in\mathbb{R}$. We
say that $S$ is stationary at every point if the random variables $\langle
W,\varphi\rangle$ and $\langle W,\psi\rangle$ are independent as soon as
$\varphi,\psi\in{\mathscr{D}}(\mathbb{R})$ have disjoint support (or
equivalently, $\varphi(x).\psi(x)=0$ for any $x\in\mathbb{R}$).

######  Proposition 3.

A Lévy white noise $W$ is stationary and independent at every point. In
particular, the random variables $\langle W,\varphi\rangle$ and $\langle
W,\varphi(\cdot-x_{0})\rangle$ are identically distributed for any
$x_{0}\in\mathbb{R}$ and are independent for $x_{0}\in\mathbb{R}$ such that
$\varphi(x)\varphi(x-x_{0})=0$ for any $x\in\mathbb{R}$.

The stationarity follows from the fact that the characteristic functionals
$\widehat{\mathscr{P}}_{W}$ and $\widehat{\mathscr{P}}_{W(\cdot-x_{0})}$ are
equal for any $x_{0}$. The independence at every point is proved in [16,
Section III.4]. Let $X=\langle W,\varphi\rangle$ with $W$ a Lévy white noise
and $\varphi\in{\mathscr{D}}(\mathbb{R})$. We denote by $\Psi$ and $\nu$ the
characteristic exponent and Lévy measure of $W$, respectively. Then, the
characteristic function of $X$ is given for any $\xi\in\mathbb{R}$ by

$\widehat{\mathscr{P}}_{X}(\xi)=\mathbb{E}\left[\mathrm{e}^{\mathrm{i}\xi
X}\right]=\exp\left(\int_{\mathbb{R}}\Psi(\xi\varphi(x))\mathrm{d}x\right).$
(33)

This simply follows from the fact that
$\widehat{\mathscr{P}}_{X}(\xi)=\mathbb{E}\left[\mathrm{e}^{\mathrm{i}\xi\langle
W,\varphi\rangle}\right]=\mathbb{E}\left[\mathrm{e}^{\mathrm{i}\langle
W,\xi\varphi\rangle}\right]=\widehat{\mathscr{P}}_{W}(\xi\varphi)$. Then, $X$
is infinitely divisible and its Lévy measure $\nu_{X}$ can be deduced from the
one of $W$ according to the relation

$\int_{\mathbb{R}\backslash\\{0\\}}f(t)\nu_{X}(\mathrm{d}t)=\int_{\mathrm{Supp}(\varphi)}\int_{\mathbb{R}\backslash\\{0\\}}f(t\varphi(x))\nu(\mathrm{d}t)\mathrm{d}x\in[0,\infty]$
(34)

for any measurable and positive function
$f:\mathbb{R}\backslash\\{0\\}\rightarrow\mathbb{R}^{+}$. The relation (34) is
proved in [41, Theorem 9.1] and can be recovered from the Lévy–Khintchine
expansion (30) injected in (33).

###  3.3 Tempered Lévy White Noises

The goal of this section is to characterize tempered Lévy white noises among
the complete family in ${\mathscr{D}}^{\prime}(\mathbb{R})$.

######  Theorem 2.

Let $W$ be a Lévy white noise in ${\mathscr{D}}^{\prime}(\mathbb{R})$. Then,
the following conditions are equivalent.

  1. 1.

The Lévy white noise $W$ is tempered; _i.e._ ,
$w\in{\mathscr{S}}^{\prime}(\mathbb{R})$ almost surely.

  2. 2.

The characteristic functional $\widehat{\mathscr{P}}_{W}$ of $W$ is continuous
over ${\mathscr{D}}(\mathbb{R})$ for the topology induced by
${\mathscr{S}}(\mathbb{R})$.

  3. 3.

Let $\Psi$ be the characteristic exponent of $W$. Then, there exist $C>0$ and
$\epsilon>0$ such that, for any $\xi\in\mathbb{R}$,

$\left\lvert\Psi(\xi)\right\rvert\leq
C\left(\lvert\xi\rvert^{\epsilon}+\lvert\xi\rvert^{2}\right).$ (35)

  4. 4.

For any $\varphi\in{\mathscr{D}}(\mathbb{R})$, we have that
$X_{W,\varphi}=\left(\langle
W,\varphi(\cdot-n)\rangle\right)_{n\in\mathbb{Z}}\in{\mathscr{S}}^{\prime}(\mathbb{Z})$
almost surely.

  5. 5.

The Lévy measure $\nu$ of $W$ satisfies that $\int_{|t|\geq
1}|t|^{\epsilon}\nu(dt)<\infty$ for some $\epsilon>0$.

  6. 6.

There exists $\epsilon>0$ such that, for any
$\varphi\in{\mathscr{D}}(\mathbb{R})$, $\mathbb{E}[|\langle
W,\varphi\rangle|^{\epsilon}]<\infty$.

  7. 7.

There exists $\epsilon>0$ and
$\varphi\in{\mathscr{D}}(\mathbb{R})\backslash\\{0\\}$ such that
$\mathbb{E}[|\langle W,\varphi\rangle|^{\epsilon}]<\infty$.

Moreover, if these conditions are not satisfied, then
$W\notin{\mathscr{S}}^{\prime}(\mathbb{Z})$ almost surely.

Remarks. Theorem 2 provides characterizations of the temperedness of a Lévy
white noise $W$ on its characteristic functional (2.), its characteristic
exponent (3.), and its Lévy measure (5.). It also recovers the
characterization 6. in terms of positive absolute moments that have been
demonstrated in [6, 8]. More precisely, the implications $6.\Rightarrow
3.\Rightarrow 1.$ were shown in [8], revealing that the existence of positive
absolute moment is sufficient for $W$ to be tempered. The later contribution
[6] then proved that (i) this condition is also necessary and (ii) that $W$ is
almost surely not tempered when 6. do not hold.

The characterization 7. is a slight variation of 6.: it shows that the moment
property is independent from the test function
$\varphi\in{\mathscr{D}}(\mathbb{R})\backslash\\{0\\}$. The key novelty is the
characterization 4., which allows us to use the characterization of discrete
white noises obtained in Theorem 1. In particular, this provides a new proof
for a Lévy white noise to be tempered or not depending on the existence of
positive absolute moments, following an approach different from [6].

###### Proof of Theorem 2.

We will prove the theorem thanks to the relations:

$1.\Rightarrow 4.\Rightarrow 6.\Rightarrow 7.\Rightarrow 5.\Rightarrow
3.\Rightarrow 2.\Rightarrow 1.$ (36)

The equivalence between 1. and 2. is true for any generalized random process
and is not restricted to Lévy white noises. It is established in [13, Section
III.6.1, Corollary 2] and will not be covered in this paper. On this matter,
see also [22, Section 2.5]. For alternative proofs and discussions restricted
to Lévy white noises, see [26, Theorem 4.1] and [8, Theorem 4].

1\. $\Rightarrow$ 4. Let $\varphi\in{\mathscr{D}}(\mathbb{R})$. We fix
$(v_{k})_{k\geq 1}$ such that
$v_{k}=(v_{k}[n])_{n\in\mathbb{Z}}\in{\mathscr{D}}(\mathbb{Z})$ for any $k\geq
1$ and $v_{k}\rightarrow 0$ in ${\mathscr{S}}(\mathbb{Z})$ when
$k\rightarrow\infty$. This means that $\lVert
v_{k}\rVert_{\infty,\alpha}\rightarrow 0$ when $k\rightarrow\infty$ for any
$\alpha\geq 0$. We observe that

$\langle X_{W,\varphi},v_{k}\rangle=\sum_{n\in\mathbb{Z}}\langle
W,\varphi(\cdot-n)\rangle v_{k}[n]=\left\langle
W,\sum_{n\in\mathbb{Z}}v_{k}[n]\varphi(\cdot-n)\right\rangle.$ (37)

Note that the sums in (37) are finite and
$\varphi_{k}=\sum_{n\in\mathbb{Z}}v_{k}[n]\varphi(\cdot-n)\in{\mathscr{D}}(\mathbb{R})$
for $v_{k}\in{\mathscr{D}}(\mathbb{Z})$ and
$\varphi\in{\mathscr{D}}(\mathbb{R})$. We will prove that

$\varphi_{k}\underset{k\rightarrow\infty}{\longrightarrow}0\text{ in
}{\mathscr{S}}(\mathbb{R}).$ (38)

Assume first that (38) is established. Since
$W\in{\mathscr{S}}^{\prime}(\mathbb{Z}^{d})$ a.s., we therefore have that,
a.s., $\langle X_{W,\varphi},v_{k}\rangle=\langle
W,\varphi_{k}\rangle\rightarrow 0$ as $k\rightarrow\infty$. Thanks to
Proposition 1, this implies that
$X_{W,\varphi}\in{\mathscr{S}}^{\prime}(\mathbb{Z})$ almost surely and 4. is
proved.

Let $\alpha\geq 0$ and $\beta\in\mathbb{N}$. Then, for any $x\in\mathbb{R}$,
we have

$\displaystyle|x|^{\alpha}|\mathrm{D}^{\beta}\varphi_{k}(x)|$
$\displaystyle=|x|^{\alpha}\left|\sum_{n\in\mathbb{Z}}v_{k}[n](\mathrm{D}^{\beta}\varphi_{k})(x-n)\right|\leq\sum_{n\in\mathbb{Z}}\left|v_{k}[n]|x|^{\alpha}(\mathrm{D}^{\beta}\varphi_{k})(x-n)\right|$
$\displaystyle\leq\lVert
v_{k}\rVert_{\infty,\alpha+2}\left(\sum_{n\in\mathbb{Z}}\frac{(1+|x|)^{\alpha}}{(1+|n|)^{\alpha+2}}\left|(\mathrm{D}^{\beta}\varphi_{k})(x-n)\right|\right)$
$\displaystyle\leq\lVert
v_{k}\rVert_{\infty,\alpha+2}\lVert(1+|\cdot|)^{\alpha}\mathrm{D}^{\beta}\varphi\rVert_{\infty}\left(\sum_{n\in\mathbb{Z}}\frac{(1+|x|)^{\alpha}}{(1+|n|)^{\alpha+2}(1+|x-n|)^{\alpha}}\right).$
(39)

We define, for $x,y\in\mathbb{R}$,

$F(x,y)=\frac{(1+|y-x|)^{\alpha}}{(1+|x|)^{\alpha}(1+|y|)^{\alpha}}.$ (40)

If $x$ and $y$ have the same sign, then $|x-y|\leq\max(|x|,|y|)$ and therefore
$1+|x-y|\leq 1+\max(|x|,|y|)\leq(1+|x|)(1+|y|)$. If now $x$ and $y$ have
opposite signs, for instance $y\leq 0\leq x$, then $1+|x-y|=1+x-y\leq 1+x-y-
xy=(1+x)(1-y)=(1+|x|)(1+|y|)$. This shows that, $0<F(x,y)\leq 1$ for anx
$x,y\in\mathbb{R}$. Finally, this implies that

$\sum_{n\in\mathbb{Z}}\frac{(1+|x|)^{\alpha}}{(1+|n|)^{\alpha+2}(1+|x-n|)^{\alpha}}=\sum_{n\in\mathbb{Z}}\frac{F(n,x-n)}{(1+|n|)^{2}}\leq\sum_{n\in\mathbb{Z}}\frac{1}{(1+|n|)^{2}}=\frac{\pi^{2}}{3}<\infty.$
(41)

Combining (3.3) and (41), we just proved that

$\lVert\varphi_{k}\rVert_{\infty,\alpha,\beta}=\sup_{x\in\mathbb{R}}|x|^{\alpha}|\mathrm{D}^{\beta}\varphi_{k}(x)|\leq
C\lVert v_{k}\rVert_{\infty,\alpha+2}$ (42)

for some constant $C>0$ independent from $k\geq 1$. Since $\lVert
v_{k}\rVert_{\infty,\alpha+2}$ vanishes for any $\alpha\geq 0$, we deduce that
$\lVert\varphi_{k}\rVert_{\infty,\alpha,\beta}\rightarrow 0$ as
$k\rightarrow\infty$ for any $\alpha\geq 0$ and $\beta\in\mathbb{N}$.
Therefore $\varphi_{k}\rightarrow 0$ in ${\mathscr{S}}(\mathbb{R})$ when
$k\rightarrow\infty$ and $X_{W,\varphi}\in{\mathscr{S}}^{\prime}(\mathbb{Z})$
almost surely.

4\. $\Rightarrow$ 6. We first assume that
$\varphi\in{\mathscr{D}}(\mathbb{R})$ has its support included in $[0,1]$, so
that the test functions $\varphi(\cdot-n)$ have disjoint supports for
$n\in\mathbb{Z}$. Hence, the random variables $\langle
W,\varphi(\cdot-n)\rangle$ are independent and identically distributed for
$n\in\mathbb{Z}$ (see Proposition 3). We apply Theorem 1 and deduce that
$\mathbb{E}[|\langle W,\varphi\rangle|^{\epsilon}]<\infty$ for some
$\epsilon>0$.

For a general $\varphi\in{\mathscr{D}}(\mathbb{R})$ with no support
restriction, the random variables $\langle W,\varphi(\cdot-n)\rangle$ are not
independent in general. However, the function $\varphi$ being compactly
supported, there exists $K_{0}\geq 1$ such that the functions
$\varphi(\cdot+K_{0}n)$, $n\in\mathbb{Z}$, have disjoint supports. We can
apply the same argument to the sequence $(\langle W,\varphi(\cdot-
K_{0}n)\rangle)_{n\in\mathbb{Z}}$ and 6. is proved.

6\. $\Rightarrow$ 7\. $\Rightarrow$ 5. The implication 6. $\Rightarrow$ 7\. is
obvious. We assume that $\varphi\neq 0$ and $\epsilon>0$ are such that
$\mathbb{E}[|\langle W,\varphi\rangle|^{\epsilon}]<\infty$. The key is the
connection between the moments of an infinitely divisible random variable $X$
and the moments of its Lévy measure $\nu_{X}$. According to [33, Theorem
25.3], we indeed have that, for any $p>0$,

$\mathbb{E}[|X|^{p}]<\infty\Longleftrightarrow\int_{|t|\geq
1}|t|^{p}\nu_{X}(\mathrm{d}t)<\infty.$ (43)

We apply this relation to $X=\langle W,\varphi\rangle$ and $p=\epsilon$ to
deduce that $\int_{|t|\geq 1}|t|^{\epsilon}\nu_{\langle
W,\varphi\rangle}(\mathrm{d}t)<\infty$. Applying (34) with
$f(t)=|t|^{\epsilon}\mathbbm{1}_{|t|\geq 1}$ then reveales that

$\int_{|t|\geq 1}|t|^{\epsilon}\nu_{\langle
W,\varphi\rangle}(\mathrm{d}t)=\int_{\mathbb{R}\backslash\\{0\\}}\left(\int_{\\{x\in\mathbb{R},\
|t\varphi(x)|\geq
1\\}}|t\varphi(x)|^{\epsilon}\mathrm{d}x\right)\nu(\mathrm{d}t).$ (44)

The function $\varphi$ being non identically zero, there exists an interval
$[a,b]$ and $m>0$ such that $|\varphi(x)|\geq m$ for $a\leq x\leq b$. We then
deduce that

$\displaystyle\int_{|t|\geq 1}|t|^{\epsilon}\nu_{\langle
W,\varphi\rangle}(\mathrm{d}t)$
$\displaystyle=\int_{\mathbb{R}\backslash\\{0\\}}\left(\int_{a}^{b}\mathbbm{1}_{|\varphi(x)||t|\geq
1}|t|^{\epsilon}\mathrm{d}x\right)\nu(\mathrm{d}t)\geq
m^{\epsilon}\int_{\mathbb{R}\backslash\\{0\\}}\int_{a}^{b}\mathbbm{1}_{m|t|\geq
1}|t|^{\epsilon}\nu(\mathrm{d}t)\mathrm{d}x$
$\displaystyle=m^{\epsilon}(b-a)\int_{|t|\geq
1/m}|t|^{\epsilon}\nu(\mathrm{d}t)$ (45)

where we used that $\mathbbm{1}_{|t\varphi(x)|\geq 1}\geq\mathbbm{1}_{m|t|\geq
1}$ for any $x\in[a,b]$. The finiteness of $\int_{|t|\geq
1}|t|^{\epsilon}\nu_{\langle W,\varphi\rangle}(\mathrm{d}t)$ therefore implies
that $\int_{|t|\geq 1/m}|t|^{\epsilon}\nu(\mathrm{d}t)<\infty$. Finally, since
$\int_{|t|\geq 1}|t|^{\epsilon}\nu(\mathrm{d}t)$ and $\int_{|t|\geq
1/m}|t|^{\epsilon}\nu(\mathrm{d}t)$ differ by a finite constant (given by
$c=\int_{\min(1,1/m)\leq|t|\leq\max(1,1/m)}|t|^{p}\nu(\mathrm{d}t)$), we
deduce 5.

5\. $\Rightarrow$ 3. This is a reformulation of [7, Proposition 2.4]. We
repeat the proof for the sake of self-completeness. Note that if $\epsilon>1$,
then $\int_{|t|\geq 1}|t|\nu(\mathrm{d}t)\leq\int_{|t|\geq
1}|t|^{\epsilon}\nu(\mathrm{d}t)<\infty$. Hence, we can assume that
$\epsilon\leq 1$ without loss of generality. We use the Lévy–Khintchine
expansion (30) of $\Psi$. The function
$\xi\mapsto\mathrm{i}\mu\xi-\frac{\sigma^{2}\xi^{2}}{2}$ is clearly dominated
by $\xi\mapsto|\xi|^{\epsilon}+|\xi|^{2}$ (since $\epsilon\leq 1$). It then
suffices to bound the last term in (30).

Applying the inequality $|\mathrm{e}^{\mathrm{i}\xi t}-1-\mathrm{i}\xi
t|\leq\xi^{2}t^{2}$, we deduce that

$\int_{0<|t|<1}\left|\mathrm{e}^{\mathrm{i}\xi t}-1-\mathrm{i}\xi
t\right|\nu(\mathrm{d}t)\leq\xi^{2}\int_{0<|t|<1}t^{2}\nu(\mathrm{d}t).$ (46)

The quantity $\int_{0<|t|<1}t^{2}\nu(\mathrm{d}t)$ is finite for any Lévy
measure, hence this term is also dominated as expected.

We now use the inequality $|\mathrm{e}^{\mathrm{i}\xi t}-1|^{2}=2(1-\cos(\xi
t))\leq 2\min(2,\xi^{2}t^{2})\leq 4\min(1,\xi^{2}t^{2})\leq 4|\xi
t|^{2\epsilon}$ (using that $2\epsilon\leq 2$). This implies that

$\int_{|t|\geq 1}\left|\mathrm{e}^{\mathrm{i}\xi
t}-1\right|\nu(\mathrm{d}t)\leq 2|\xi|^{\epsilon}\int_{|t|\geq
1}|t|^{\epsilon}\nu(\mathrm{d}t).$ (47)

Combining the different bounds, we obtain 3.

3\. $\Rightarrow$ 2. The characteristic functional is positive-definite and
normalized such that $\widehat{\mathscr{P}}_{W}(0)=1$. Hence, we deduce that
[27, Section 13.4.B]

$\left\rvert\widehat{\mathscr{P}}_{W}(\varphi)-\widehat{\mathscr{P}}_{W}(\psi)\right\rvert\leq
2\left(1-\Re\left\\{\widehat{\mathscr{P}}_{W}(\varphi-\psi)\right\\}\right)$
(48)

for any $\varphi,\psi\in{\mathscr{D}}(\mathbb{R})$, where $\Re\\{z\\}$ stands
for the real part of $z\in\mathbb{C}$. In particular,
$\widehat{\mathscr{P}}_{W}$ is continuous if and only if it is continuous at
$\varphi=0$, which we will prove now. The condition 3. implies that

$\left\lvert\int_{\mathbb{R}}\Psi(\varphi(x))\mathrm{d}x\right\rvert\leq\int_{\mathbb{R}}|\Psi(\varphi(x))|\mathrm{d}x\leq
C(\lVert\varphi\rVert_{\epsilon}^{\epsilon}+\lVert\varphi\rVert_{2}^{2}).$
(49)

Assume that $\varphi_{k}\rightarrow 0$ as $k\rightarrow\infty$ for the
topology of ${\mathscr{S}}(\mathbb{R})$. This readily implies that
$\lVert\varphi_{k}\rVert_{\epsilon}^{\epsilon}+\lVert\varphi_{k}\rVert_{2}^{2}\rightarrow
0$, and therefore
$\left\lvert\int_{\mathbb{R}}\Psi(\varphi_{k}(x))\mathrm{d}x\right\rvert\rightarrow
0$. Finally,

$\left\rvert\widehat{\mathscr{P}}_{W}(\varphi_{k})-\widehat{\mathscr{P}}_{W}(0)\right\rvert=\left\rvert\widehat{\mathscr{P}}_{W}(\varphi_{k})-1\right\rvert=\left\rvert\mathrm{e}^{\int_{\mathbb{R}}\Psi(\varphi_{k}(x))\mathrm{d}x}-1\right\rvert\underset{k\rightarrow\infty}{\longrightarrow}|\mathrm{e}^{0}-1|=0,$
(50)

proving 2.

We assume finally that the equivalent conditions 1. to 7. are not satisfied.
Set $\Omega_{W}=\\{\omega\in\Omega,\
W(\omega)\in{\mathscr{S}}^{\prime}(\mathbb{R})\\}$. Then, the proof above for
the implication $1.\Rightarrow 4.$ implies that the sequence
$X_{W(\omega),\varphi}\in{\mathscr{S}}^{\prime}(\mathbb{Z})$ for
$\omega\in\Omega_{W}$. In other terms, we have that
$\mathscr{P}(\Omega_{W})\leq\mathscr{P}(X_{W,\varphi}\in{\mathscr{S}}^{\prime}(\mathbb{Z})\\}$.
However, Theorem 1 implies that
$X_{W,\varphi}\notin{\mathscr{S}}^{\prime}(\mathbb{Z})$ a.s. as soon as it is
not a.s. tempered, which is the case here. Hence,

$\mathscr{P}(\Omega_{W})\leq\mathscr{P}(X_{W,\varphi}\in{\mathscr{S}}^{\prime}(\mathbb{Z})\\}=0$
(51)

and $\mathscr{P}(\Omega_{W})=0$; _i.e._ ,
$W\notin{\mathscr{S}}^{\prime}(\mathbb{R})$ a.s. ∎

As we have seen, the first characterization of tempered Lévy white noises in
terms of moment properties has been obtained in [6]. The proof’s strategy of
Dalang and Humeau is based on (i) the Lévy–Itô decomposition of a Lévy
process, (ii) the reduction of the problem to compound Poisson processes, and
(iii) the deduction of the corresponding result for Lévy white noises, which
are the weak derivatives of Lévy processes).

Our proof rather relies on the random sequences $X_{W,\varphi}=\left(\langle
W,\varphi(\cdot-n)\rangle\right)_{n\in\mathbb{Z}}$ and on the fact that
tempered i.i.d. random sequences have been characterized in Theorem 1. This
new proof highlights the strong connection between discrete and Lévy white
noises.

We illustrate Theorem 2 on classical examples of Lévy white noises. The
terminology for the noise $W$ is given by the corresponding infinitely
divisible random variable $X$ with identical characteristic exponent.

_S $\alpha$S white noises._ The S$\alpha$S case has been studied in [11, 20]
in the framework of tempered generalized random processes. A S$\alpha$S white
noise $W_{\alpha}$ corresponds to the characteristic exponent
$\Psi(\xi)=-\gamma^{\alpha}|\xi|^{\alpha}$ with $0<\alpha\leq 2$ and
$\gamma>0$. The characteristic functional of $W_{\alpha}$ is then

$\widehat{\mathscr{P}}_{W_{\alpha}}(\varphi)=\exp(-\gamma^{\alpha}\lVert\varphi\rVert_{\alpha}^{\alpha})$
(52)

for any $\varphi\in{\mathscr{D}}(\mathbb{R})$. The criterion 3. of Theorem 2
directly recovers that $W_{\alpha}\in{\mathscr{S}}^{\prime}(\mathbb{R})$
almost surely. The Gaussian white noise corresponds to $\alpha=2$.

_Compound Poisson white noises._ A compound Poisson white noise $W$ is a point
process such that

$W=\sum_{n\in\mathbb{Z}}A_{n}\delta(\cdot-Y_{n})$ (53)

where $A=(A_{n})_{n\in\mathbb{Z}}$ is a sequence of i.i.d. random variables
with common law $\mathscr{P}_{A}$ and where the $Y_{n}$ are such that
$N=\mathrm{Card}(n\in\mathbb{Z},\ a\leq Y_{n}\leq b)$ is a Poisson random
variable of parameter $\lambda(b-a)$ for some $\lambda>0$ and any $a<b$. We
call $\mathscr{P}_{A}$ the law of the jumps of $W$, which can be any
probability law with support in $\mathbb{R}\backslash\\{0\\}$. The compound
Poisson white noises can be used as the approximation in law of any Lévy white
noises [10]. This principle has been used to generate Lévy white noises and
their extensions such as Lévy processes [5]. The characteristic exponent of
$W$ is given by [40, Theorem 1]

$\Psi(\xi)=\exp\left(\lambda(\widehat{\mathscr{P}}_{A}(\xi)-1)\right)$ (54)

where $\widehat{\mathscr{P}}_{A}$ is the characteristic function of the
$A_{n}$.

Let $X$ be a compound Poisson random variable with characteristic exponent
$\Psi$. Then, we have the equivalence

$\mathbb{E}[|X|^{p}]<\infty\quad\Longleftrightarrow\quad\mathbb{E}[|Y_{0}|^{p}]<\infty$
(55)

for any $p>0$. Indeed, the Lévy measure of $X$ is $\lambda\mathscr{P}_{A}$ and
therefore (55) readily follows from (43). This implies the following result
for compound Poisson white noise.

######  Proposition 4.

Let $W$ be a compound Poisson white noise in
${\mathscr{D}}^{\prime}(\mathbb{R})$ with parameter $\lambda>0$ and law of
jump $\mathscr{P}_{A}$. Then, $W$ is tempered if and only if there exists
$\epsilon>0$ such that

$\int_{\mathbb{R}\backslash\\{0\\}}|x|^{\epsilon}\mathscr{P}_{A}(\mathrm{d}x)<\infty.$
(56)

Otherwise, $W\notin{\mathscr{S}}^{\prime}(\mathbb{R})$ almost surely.

_Some tempered Lévy white noises._ Any infinitely divisible law specifies a
Lévy white noise with identical characteristic exponent. Student’s
t-distributions and Laplace laws are infinitely divisible. We can therefore
consider Student’s t-white noises [43], Laplace white noises [24], which are
all known to admits to positive absolute moments and are therefore tempered.
This is also the case for layered stable white noises [19], tempered stable
white noises [17, 31], and inverse Gaussian white noises [2].

_Non-tempered Lévy white noises._ In order to construct Lévy white noises
which are almost surely not tempered, it suffices to consider Lévy measures
which do not satisfy the condition 5. of Theorem 2. An example is given by the
measure

$\nu(\mathrm{d}t)=\frac{\mathrm{d}t}{(1+|t|)\log^{2}(2+|t|)},$ (57)

which satisfies
$\int_{\mathbb{R}\backslash\\{0\\}}\min(1,t^{2})\nu(\mathrm{d}t)<\infty$ and
$\int_{\mathbb{R}\backslash\\{0\\}}|t|^{\epsilon}\nu(\mathrm{d}t)=\infty$ for
any $\epsilon>0$. Alternatively, any compound Poisson white noise whose law of
jump $\mathscr{P}_{A}$ has no positive absolute moment is almost surely not
tempered according to Proposition 4.

_Connection with the Pruitt index._ In [30], W.E. Pruitt studied the growth
properties of random walks and Lévy processes. His result can be reinterpreted
by considering the _Pruitt index_ of the Lévy white noise $W$ with
characteristic exponent $\Psi$, defined as333The usual definition of the
Pruitt index is different, but its equivalence with (58) is for instance
proved in [34, Setion 5].

$\beta_{0}=\sup\left\\{p\geq 0,\ \underset{|\xi|\rightarrow
0}{\lim\sup}\frac{|\Psi(\xi)|}{|\xi|^{p}}<\infty\right\\}.$ (58)

According to [33, Proposition 48.10], denoting by $\mu$ the Lévy measure of
$W$, we have that $\beta_{0}=\sup\left\\{0\leq p\leq 2,\
\int_{\mathbb{R}\backslash\\{0\\}}|t|^{p}\nu(\mathrm{d}t)<\infty\right\\}.$ We
therefore see with Theorem 2 that

$W\in{\mathscr{S}}^{\prime}(\mathbb{R})\text{ a.s. }\Longleftrightarrow\
\beta_{0}>0\quad\text{and}\quad
W\notin{\mathscr{S}}^{\prime}(\mathbb{R})\text{ a.s. }\Longleftrightarrow\
\beta_{0}=0.$ (59)

##  4 Conclusion

We provided a new and self-contained proof characterizing the temperedness of
Lévy white noises. We recovered the results of previous works, including [6,
7, 8]. Our analysis highlights the connection between the Lévy white noises
and their discrete counterparts, which are infinite sequences of independent
and identically distributed random variables. In particular, we characterized
the location of discrete white noises in polynomially weighted
$\ell_{\infty}$-spaces in terms of their absolute moments in Theorem 1. We
finally used the discrete characterization in our analysis of the tempered
Lévy white noises in Theorem 2. In future works, we plan to extend the
proposed considerations to new spaces of generalized functions, including
tempered ultra-distributions [39] and Fourier hyperfunctions [23].

## References

  * [1] S. Aziznejad and J. Fageot.  Wavelet analysis of the Besov regularity of Lévy white noise.  Electronic Journal of Probability, 25, 2020.
  * [2] O.E. Barndorff-Nielsen.  Processes of normal inverse Gaussian type.  Finance and Stochastics, 2(1):41–68, 1997.
  * [3] J. Bertoin.  Lévy Processes, volume 121.  Cambridge University Press, 1998.
  * [4] H. Biermé, O. Durieu, and Y. Wang.  Generalized random fields and Lévy’s continuity theorem on the space of tempered distributions.  Commun. Stoch. Anal., 12(4):Article 4, 427–445, 2018.
  * [5] L. Dadi, S. Aziznejad, and M. Unser.  Generating sparse stochastic processes using matched splines.  IEEE Transactions on Signal Processing, 68:4397–4406, 2020.
  * [6] R.C. Dalang and T. Humeau.  Lévy processes and Lévy white noise as tempered distributions.  The Annals of Probability, 45(6b):4389–4418, 2017.
  * [7] J. Fageot.  Gaussian versus Sparse Stochastic Processes: Construction, Regularity, Compressibility.  PhD thesis, Ecole Polytechnique Fédérale de Lausanne, 2017.
  * [8] J. Fageot, A. Amini, and M. Unser.  On the continuity of characteristic functionals and sparse stochastic modeling.  Journal of Fourier Analysis and Applications, 20:1179–1211, 2014\.
  * [9] J. Fageot, A. Fallah, and M. Unser.  Multidimensional Lévy white noise in weighted Besov spaces.  Stochastic Processes and Their Applications, 127(5):1599–1621, 2017\.
  * [10] J. Fageot, V. Uhlmann, and M. Unser.  Gaussian and sparse processes are limits of generalized Poisson processes.  Applied and Computational Harmonic Analysis, in press, 2018.
  * [11] J. Fageot, M. Unser, and J.P. Ward.  On the Besov regularity of periodic Lévy noises.  Applied and Computational Harmonic Analysis, 42(1):21 – 36, 2017\.
  * [12] Julien Fageot and Thomas Humeau.  The domain of definition of the lévy white noise.  Stochastic Processes and their Applications, 135:75–102, 2021.
  * [13] X. Fernique.  Processus linéaires, processus généralisés.  Annales de l’Institut Fourier, 17:1–92, 1967.
  * [14] F.G. Friedlander and M. Joshi.  Introduction to the Theory of Distributions.  Cambridge University Press, 1998.
  * [15] I.M. Gel’fand.  Generalized random processes.  Doklady Akademii Nauk SSSR, 100:853–856, 1955.
  * [16] I.M. Gel’fand and N.Y. Vilenkin.  Generalized Functions. Vol. 4: Applications of Harmonic Analysis.  Academic Press, New York-London, 1964.
  * [17] M. Grabchak.  Tempered Stable Distributions: Stochastic Models for Multiscale Processess.  Springer, 2016.
  * [18] T. Hida.  Analysis of Brownian functionals.  Stochastic Systems: Modeling, Identification and Optimization, I, pages 53–59, 1975.
  * [19] C. Houdré and R. Kawai.  On layered stable processes.  Bernoulli, 13(1):252–278, 2007.
  * [20] Z. Huang and C. Li.  On fractional stable processes and sheets: White noise approach.  Journal of Mathematical Analysis and Applications, 325(1):624–635, 2007.
  * [21] K. Itô.  Stationary random distributions.  Kyoto Journal of Mathematics, 28(3):209–223, 1954.
  * [22] K. Itô.  Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces, volume 47.  SIAM, 1984.
  * [23] A. Kaneko.  Introduction to the Theory of Hyperfunctions, volume 3.  Springer Science & Business Media, 1989.
  * [24] S. Koltz, T.J. Kozubowski, and K. Podgorski.  The Laplace Distribution and Generalizations.  Boston, MA: Birkhauser, 2001.
  * [25] Y.-J. Lee and H.-H. Shih.  The Segal–Bargmann transform for Lévy functionals.  Journal of functional analysis, 168(1):46–83, 1999.
  * [26] Y.-J. Lee and H.-H. Shih.  Lévy white noise measures on infinite-dimensional spaces: Existence and characterization of the measurable support.  Journal of Functional Analysis, 237(2):617–633, 2006.
  * [27] M. Loève.  Probability theory I.  Springer, 1977.
  * [28] G. Di Nunno, B. Øksendal, and F. Proske.  White noise analysis for Lévy processes.  Journal of Functional Analysis, 206(1):109–148, 2004.
  * [29] B. Øksendal.  Stochastic partial differential equations driven by multi-parameter white noise of lévy processes.  Quarterly of applied mathematics, 66(3):521–537, 2008.
  * [30] W.E. Pruitt.  The growth of random walks and Lévy processes.  The Annals of Probability, 9(6):948–956, 1981.
  * [31] J. Rosinski.  Tempering stable processes.  Stochastic processes and Their applications, 117(6):677–707, 2007\.
  * [32] G. Samorodnitsky and M.S. Taqqu.  Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance.  Stochastic Modeling. Chapman & Hall, New York, 1994.
  * [33] K. Sato.  Lévy Processes and Infinitely Divisible Distributions, volume 68.  Cambridge University Press, Cambridge, 2013.
  * [34] R.L. Schilling.  Growth and Hölder conditions for the sample paths of Feller processes.  Probability Theory and Related Fields, 112(4):565–611, 1998.
  * [35] L. Schwartz.  Théorie des distributions.  Hermann, 1966.
  * [36] Si-Si.  Introduction to Hida distributions.  World scientific, 2012.
  * [37] B. Simon.  Functional integration and quantum physics, volume 86.  Academic press, 1979.
  * [38] F. Trèves.  Topological Vector Spaces, Distributions and Kernels.  Academic Press, New York-London, 1967.
  * [39] H. Triebel.  Ultra-distributions and weighted spaces of entire analytic functions.  In Theory of Function Spaces, pages 245–253. Springer, 1983.
  * [40] M. Unser and P. D. Tafti.  Stochastic models for sparse and piecewise-smooth signals.  IEEE Transactions on Signal Processing, 59(3):989–1006, 2011\.
  * [41] M. Unser and P. D. Tafti.  An Introduction to Sparse Stochastic Processes.  Cambridge University Press, 2014.
  * [42] J.B. Walsh.  An introduction to stochastic partial differential equations.  In École d’Été de Probabilités de Saint Flour XIV-1984, pages 265–439. Springer, 1986.
  * [43] H. Zhang and F.H. Nieto.  TAR modeling with missing data when the white noise process follows a student’s t-distribution.  Revista Colombiana de Estadística, 38(1):239–266, 2015.