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# Some notes on moment inequalities for heavy-tailed distributions

Paul Buterus  Paul Buterus, Germany<EMAIL_ADDRESS>and  Holger
Sambale  Holger Sambale, Faculty of Mathematics, Ruhr University Bochum,
Germany<EMAIL_ADDRESS>
###### Abstract.

We investigate the relation between moments and tails of heavy-tailed (in
particular, Pareto-type) distributions. We also discuss the sharpness of our
results in a number of examples under certain regularity conditions like log-
convexity. Moreover, we derive concentration bounds for polynomial chaos of
any order $d$.

###### Key words and phrases:

concentration of measure, heavy tails, Pareto distribution, moment inequality,
polynomial chaos, Hanson-Wright inequality

###### 1991 Mathematics Subject Classification:

Primary 60E15, 60F10, Secondary 46E30, 46N30

##  1\. Introduction

In classical situations, the family of $L^{p}$ norms of a random variable $X$
contains a wealth of information about the tails of $X$. For sub-Gaussian
random variables, for instance, it is standard knowledge that the growth of
the $L^{p}$ norms characterizes the tail behavior, i. e., the property
$\mathbb{P}(|X|\geq t)\leq 2\mathrm{e}^{-ct^{2}}$ for some $c>0$ and any
$t\geq 0$ is equivalent to

$\lVert X\rVert_{L^{p}}:=(\mathbb{E}[|X|^{p}])^{1/p}\leq Cp^{1/2}$

for any $p\geq 2$, where $C>0$ is some constant only depending on $c$, cf.
e.g. [Ver18, Proposition 2.5.2]. Analogous results remain valid if the random
variable under consideration has slightly heavier tails which are no longer
sub-Gaussian but still decay exponentially, i. e., $\mathbb{P}(|X|\geq t)\leq
2\mathrm{e}^{-ct^{\alpha}}$ for some $\alpha\in(0,2)$ and any $t\geq 0$, cf.
[Sam23, Proposition 5.2].

Against this context, one may ask for similar properties of heavy-tailed
random variables, i. e., random variables with tails of the form

$\mathbb{P}(|X|\geq t)\leq Ct^{-\alpha}$

for some $\alpha>0$, any $t\geq c$ and suitable $C,c>0$. A classical example
are Pareto-type random variables. Recall that a random variable $X$ has Pareto
distribution $\operatorname{Par}(\alpha,b)$, where $\alpha,b>0$, if
$\mathbb{P}(X\geq t)=(b/t)^{\alpha}$ for any $t\geq b$. Equivalently, its
density function $f(x)$ is given by $f(x)=\alpha b^{\alpha}/x^{\alpha+1}$ for
$x\geq b$. We also write $X\sim\operatorname{Par}(\alpha,b)$ and call $X$ a
Pareto random variable. Sometimes it is convenient to study symmetrized Pareto
random variables which we denote by
$\operatorname{Par}_{\mathrm{s}}(\alpha,b)$, and it is obvious how to modify
the definitions.

If $X\sim\operatorname{Par}(\alpha,b)$, then $\lVert X\rVert_{L^{p}}<\infty$
iff $p<\alpha$. More precisely,

$\mathbb{E}[X^{p}]=b^{p}\frac{\alpha}{\alpha-p}$ (1)

for any $p<\alpha$. In particular, applying Chebyshev’s inequality for any
$p<\alpha$ and optimizing in $p$ leads to the bound

$\mathbb{P}(X\geq t)\leq\mathrm{e}\alpha\log(t/b)(b/t)^{\alpha}$ (2)

for any $t\geq b\mathrm{e}^{1/\alpha}$, which is attained at
$p=\alpha-1/\log(t/b)$. Consequently, this approach results in a logarithmic
error factor. Moreover, we cannot recover the moments (1) from (2): indeed, it
is easy to check for a random variable with tails as in (2), it holds that

$\mathbb{E}[X^{p}]=\Theta((\alpha-p)^{-2})$ (3)

as $p\to\alpha$ (where the constant inherent in $\Theta$ depends on $\alpha$
only), which corresponds to the $p$-th moment of the product of two
independent Pareto variables (up to constant). In some sense, this phenomenon
of having access to the moments up to some order $\alpha$ and involving a sort
of “error” in terms of the distribution is also related to classical topics
like the truncated Hamburger moment problem, cf. e. g. [Akh65, Corollary
2.5.4].

In this note, we study the connections between (1) and (2) in more detail. We
provide equivalent characterizations of the moment behavior of Pareto-type
distributions. Moreover, we construct examples of random variables which
demonstrate that the logarithmic error which appears in (2) cannot be avoided
in general (in fact, it turns out to be optimal). We also demonstrate that the
error can be avoided if additional assumptions on the tail behaviour are
imposed, which for instance involve log-convexity.

In Section 3, these observations are complemented by moment and concentration
bounds for polynomial chaos in heavy-tailed random variables, i. e.,
functionals of the form
$\sum_{i_{1},\ldots,i_{d}}a_{i_{1},\ldots,i_{d}}X_{i_{1}}\cdots X_{i_{d}}$ for
real-valued coefficients $a_{i_{1},\ldots,i_{d}}$. For $d=2$, we obtain what
could be understood as a sort of preliminary Hanson–Wright type inequality for
heavy-tailed random variables.

##  2\. Moments and tails of heavy-tailed distributions

###  2.1. Moment inequalities

Let $X$ be a random variable such that the growth of its $p$-th (fractional
absolute) moments is at most of Pareto type, i. e., $\mathbb{E}[|X|^{p}]\leq
C(\alpha-p)^{-1}$ for any real $p<\alpha$ and a suitable constant $C>0$. As
mentioned in the introduction, in the sub-Gaussian case the corresponding
property $\mathbb{E}[|X|^{p}]\leq Cp^{p/2}$ has a number of well-known
reformulations involving the tails, the moment-generating function or the sub-
Gaussian norm of $X$. Clearly, and in view of the discussion in the
introduction, none of these properties can have a direct “heavy-tailed”
analogue. Still, it is possible to find a similar collection of equivalent
reformulations of Pareto-type growth of the moments, for instance in terms of
truncations of $X$.

######  Proposition 1.

Let $\alpha>0$ and $X$ be a non-negative random variable. Then, the following
properties are equivalent:

  1. (a)

The moments of $X$ satisfy

$\mathbb{E}[X^{p}]\leq\frac{C_{1}}{\alpha-p}$

for all $p\in(0,\alpha)$.

  2. (b)

The truncated $\alpha$-th moment satisfies

$\mathbb{E}[X^{\alpha}\mathbbm{1}_{\\{X\leq r\\}}]\leq C_{2}\log{r}$

for any $r\geq\mathrm{e}$.

  3. (c)

We have

$\int_{0}^{r}y^{\alpha-1}\mathbb{P}(X\geq y)\,\mathrm{d}y\leq C_{3}\log{r}$

for any $r\geq\mathrm{e}$.

  4. (d)

We have

$\mathbb{E}[X^{\alpha}\exp(-sX)]\leq C_{4}\lvert\log{s}\rvert$

for any $s\in(0,\mathrm{e}^{-1}]$.

Here, the constants $C_{i}>0$, $i=1,\ldots,4$, only differ by
$\alpha$-dependent factors.

Explicit relations between the constants $C_{i}$ can be found in the proofs.
Note that in parts (b) and (c), one may replace the condition
$r\geq\mathrm{e}$ by $r\geq r^{\ast}$ for any fixed $r^{\ast}>1$ at the cost
of modifying the constants. Similarly, (d) can be extended to
$s\in(0,s^{\ast}]$ for any fixed $s^{\ast}<1$.

To prove Proposition 1, we start with a technical lemma, specialized for
$\alpha=2$, which yields characterizations of Pareto-type growth of the
moments of a random variable $X$ in terms of its Laplace transform and its
characteristic function, respectively.

######  Lemma 2.

Let $X$ be a non-negative random variable and $C>0$ a fixed constant which may
depend on $X$. Then the following statements are equivalent.

  1. (a)

For all $p\in(1,2)$ the moments of $X$ satisfy

$\mathbb{E}[X^{p}]\leq\frac{C}{2-p}.$

  2. (b)

$X$ is integrable and the Laplace transform
$L(t):=L_{X}(t):=\mathbb{E}[\mathrm{e}^{-tX}]$ of $X$ satisfies

$\int_{0}^{\infty}\frac{\mathbb{E}[X]+L^{\prime}(u)}{u^{1+s}}\,\mathrm{d}u\leq
C\frac{\Gamma(1-s)}{s(1-s)}$

for all $s\in(0,1)$.

  3. (c)

The characteristic function
$\varphi(t):=\varphi_{X}(t):=\mathbb{E}[\mathrm{e}^{itX}]$ of $X$ satisfies

$\int_{0}^{\infty}\frac{1-\mathrm{Re}(\varphi(u))}{u^{2+s}}\,\mathrm{d}u\leq
C\frac{\sin(\pi s/2)\Gamma(1-s)}{s(s+1)(1-s)}$

for all $s\in(0,1)$.

We shall need the equivalence of (a) and (c) in the sequel. In view of (3) and
the discussion of polynomial chaos in Section 3, note that similar arguments
as used in Lemma 2 can also be applied to singularities of the form
$(\alpha-s)^{-h}$, where $h>0$. Lemma 2 is essentially an application of
identities for fractional absolute moments of heavy-tailed distributions from
[MP16]. For the sake of completeness, we provide some details in the proof.

###### Proof.

First note that $L$ is continuously differentiable with
$L^{\prime}(t)=-\mathbb{E}[X\exp(-tX)]$ and $L^{\prime}(0)=-\mathbb{E}[X]$ as
long as $X$ is integrable. Therefore, as shown in [MP16, Lemma 1.1], the
identity

$\frac{\Gamma(1-s)}{s}x^{s}=\int_{0}^{\infty}\frac{1-\exp(-xu)}{u^{s+1}}\,\mathrm{d}u,$

valid for $s\in(0,1)$ and $x>0$, implies that

$\mathbb{E}[X^{1+s}]=\frac{s}{\Gamma(1-s)}\int_{0}^{\infty}\frac{L^{\prime}(u)-L^{\prime}(0)}{u^{s+1}}\,\mathrm{d}u$

for any $s\in(0,1)$, so that the equivalence of (a) and (b) directly follows.

Similarly, by formula (7) in [Kaw72, Theorem 11.4.3],

$\int_{0}^{\infty}\frac{1-\cos(xu)}{u^{1+\beta}}\,\mathrm{d}u=\frac{\Gamma(2-\beta)}{\beta}\frac{\sin(\pi(1-\beta)/2)}{1-\beta}|x|^{\beta}$

for $\beta\in(0,2)$ and $x\in\mathbb{R}$. Choosing $\beta=s+1$ and applying
Fubini’s theorem, it was shown in [MP16, Lemma 1.3 (2)] that

$\mathbb{E}[X^{1+s}]=\frac{s(s+1)}{\sin(\pi
s/2)\Gamma(1-s)}\int_{0}^{\infty}\frac{1-\mathrm{Re}(\varphi(u))}{u^{2+s}}\,\mathrm{d}u$

for any $s\in(0,1)$, which establishes the equivalence of (a) and (c). ∎

###### Proof of Proposition 1.

To prove that (a) implies (b), we temporarily assume that $\alpha=2$ by
replacing $X$ by the random variable $\tilde{X}:=X^{\alpha/2}$ and $C_{1}$ by
$\tilde{C}_{1}:=2C_{1}/\alpha$. Using part (c) of Lemma 2, we then obtain

$\tilde{C}_{1}\frac{\sin(\pi
s/2)\Gamma(1-s)}{s(s+1)(1-s)}\geq\int_{0}^{\infty}\frac{1-\mathrm{Re}(\varphi_{\tilde{X}}(u))}{u^{2+s}}\,\mathrm{d}u\geq\int_{0}^{\pi/\tilde{r}}\frac{1-\mathrm{Re}(\varphi_{\tilde{X}}(u))}{u^{2+s}}\,\mathrm{d}u$

for any $\tilde{r}>0$. The last integral can be estimated further by

$\displaystyle\int_{0}^{\pi/\tilde{r}}\frac{1-\mathrm{Re}(\varphi_{\tilde{X}}(u))}{u^{2+s}}\,\mathrm{d}u$
$\displaystyle\geq
2\mathbb{E}\Big{[}\mathbbm{1}_{\\{\tilde{X}\leq\tilde{r}\\}}\int_{0}^{\pi/\tilde{r}}\frac{\sin(\tilde{X}u/2)^{2}}{u^{2+s}}\,\mathrm{d}u\Big{]}$
$\displaystyle\geq\frac{2}{\pi^{2}}\mathbb{E}[\mathbbm{1}_{\\{\tilde{X}\leq\tilde{r}\\}}\tilde{X}^{2}]\int_{0}^{\pi/\tilde{r}}u^{-s}\,\mathrm{d}u$
$\displaystyle\geq\frac{2}{\pi^{1+s}}\frac{\tilde{r}^{s-1}}{1-s}\mathbb{E}[\mathbbm{1}_{\\{\tilde{X}\leq\tilde{r}\\}}\tilde{X}^{2}],$

since $1-\cos(x)=2\sin(x/2)^{2}$ for any $x$ and $\sin(x)\geq 2x/\pi$ on
$x\in[0,\pi/2]$ (use e. g. that $\sin(x)$ is concave on $[0,\pi/2]$).
Combining both inequalities yields

$\frac{\pi^{1+s}}{2}\tilde{C}_{1}\frac{\sin(\pi
s/2)\Gamma(1-s)}{s(s+1)}\tilde{r}^{1-s}\geq\mathbb{E}[\mathbbm{1}_{\\{\tilde{X}\leq\tilde{r}\\}}\tilde{X}^{2}].$

Next, we optimize this expression in $s$ by taking
$s=1-\log(\tilde{r})^{-1}\in[0,1)$ if $\tilde{r}\geq\mathrm{e}$. Using that
$\sin(x)\leq x$ for any $x\geq 0$ as well as $\Gamma(x)=x^{-1}\Gamma(1+x)\leq
x^{-1}$ on $x\in(0,1]$, we find that

$\mathbb{E}[\mathbbm{1}_{\\{\tilde{X}\leq\tilde{r}\\}}\tilde{X}^{2}]\leq\frac{\mathrm{e}\,\pi^{3}}{4}\,\tilde{C}_{1}\log(\tilde{r}).$

Now setting $\tilde{r}:=r^{\alpha/2}$, it follows that

$\mathbb{E}[\mathbbm{1}_{\\{X\leq
r\\}}X^{\alpha}]=\mathbb{E}[\mathbbm{1}_{\\{\tilde{X}\leq\tilde{r}\\}}\tilde{X}^{2}]\leq\frac{\mathrm{e}\,\pi^{3}}{4}\,\tilde{C}_{1}\log(\tilde{r})=\frac{\mathrm{e}\,\pi^{3}}{4}\,C_{1}\log(r)$

provided that $r\geq\mathrm{e}^{2/\alpha}$. If $\alpha\geq 2$, this yields
(b), while if $\alpha<2$, it remains to note that

$\mathbb{E}[\mathbbm{1}_{\\{X\leq
r\\}}X^{\alpha}]\leq\mathbb{E}[\mathbbm{1}_{\\{X\leq\mathrm{e}^{2/\alpha}\\}}X^{\alpha}]\leq\frac{\mathrm{e}\,\pi^{3}}{2\alpha}\,C_{1}\leq\frac{\mathrm{e}\,\pi^{3}}{2\alpha}\,C_{1}\log(r)$

for any $r\in[\mathrm{e},\mathrm{e}^{2/\alpha}]$.

To show that that (b) implies (a), we replace $X$ by
$\hat{X}:=X\mathbbm{1}_{\\{X\geq\mathrm{e}\\}}$. We may then write

$\mathbb{E}[\hat{X}^{p}]=(\alpha-p)\int_{1}^{\infty}u^{-(\alpha-p)-1}\mathbb{E}[\hat{X}^{\alpha}\mathbbm{1}_{\\{\hat{X}\leq
u\\}}]\,\mathrm{d}u$

and thus conclude that

$\mathbb{E}[\hat{X}^{p}]\leq
C_{2}(\alpha-p)\int_{1}^{\infty}u^{-(\alpha-p)-1}\log{u}\,\mathrm{d}u=\frac{C_{2}}{\alpha-p}$

and hence

$\mathbb{E}[X^{p}]\leq\mathrm{e}^{\alpha}+\frac{C_{2}}{\alpha-p}\leq\frac{C_{2}+\alpha\mathrm{e}^{\alpha}}{\alpha-p}.$

Moreover, noting that

$\mathbb{E}[X^{\alpha}\mathbbm{1}_{\\{X\leq
r\\}}]=\alpha\int_{0}^{r}y^{\alpha-1}\mathbb{P}(y\leq X\leq
r)\,\mathrm{d}y\leq\alpha\int_{0}^{r}y^{\alpha-1}\mathbb{P}(X\geq
y)\,\mathrm{d}y,$

(c) trivially implies (b) with $C_{2}:=\alpha C_{3}$. On the other hand, using
that

$\int_{0}^{r}y^{\alpha-1}\mathbb{P}(X\geq
y)\,\mathrm{d}y=\int_{0}^{r}y^{\alpha-1}\mathbb{P}(y\leq X\leq
r)\,\mathrm{d}y+\frac{r^{\alpha}}{\alpha}\mathbb{P}(X\geq r),$

the first term is bounded $C_{2}\alpha^{-1}\log(r)$ due to (b), and by (a) and
Chebyshev’s inequality similarly as in (2), the second term is bounded by
$C_{1}\mathrm{e}\alpha^{-1}\log(r)$. In particular, (c) follows with
$C_{3}:=(C_{1}\mathrm{e}+C_{2})/\alpha$.

To prove the equivalence of (b) and (d), we have

$\displaystyle\mathbb{E}[X^{\alpha}\exp(-sX)]$
$\displaystyle=s\mathbb{E}[X^{\alpha}\int_{X}^{\infty}\exp(-st)\,\mathrm{d}t]=s\int_{0}^{\infty}\mathrm{e}^{-st}\mathbb{E}[X^{\alpha}\mathbbm{1}_{\\{X\leq
t\\}}]\,\mathrm{d}t$
$\displaystyle\leq\mathrm{e}^{\alpha}+C_{2}s\int_{1}^{\infty}\mathrm{e}^{-st}\log(t)\,\mathrm{d}t=\mathrm{e}^{\alpha}+C_{2}\int_{s}^{\infty}\frac{\exp(-u)}{u}\mathrm{d}u$

by splitting the integral into the parts $t\leq\mathrm{e}$ and
$t\geq\mathrm{e}\geq 1$. Splitting the integral on the right-hand side at
$u=1$ then yields

$\mathbb{E}[X^{\alpha}\exp(-sX)]\leq\mathrm{e}^{\alpha}+C_{2}(\mathrm{e}^{-1}+|\log{s}|)\leq(\mathrm{e}^{\alpha}+C_{2}(\mathrm{e}^{-1}+1))|\log{s}|$

for any $s\leq\mathrm{e}^{-1}$. On the other hand, we have

$\mathbb{E}[X^{\alpha}\exp(-sX)]\geq\mathrm{e}^{-1}\mathbb{E}[X^{\alpha}\mathbbm{1}_{\\{X\leq
1/s\\}}],$

which implies (b) with $C_{2}:=\mathrm{e}C_{4}$. ∎

In view of the discussion in the introduction, and as we will confirm in the
course of this note, the properties listed in Proposition 1 cannot be
complemented by a statement about the tails in general. However, it turns out
that they are equivalent to a Pareto-type tail behaviour if we impose
additional “structural” assumptions on the tails.

To this end, recall that a function $f\colon[b,\infty)\to\mathbb{R}$ for some
$b\geq 0$ is called log-convex if $\log(f)$ is convex on $[b,\infty)$. A non-
negative random variable $X$ is called log-convex if its tails
$\mathbb{P}(X\geq t)$ are a log-convex function of $t\in[b,\infty)$ with
$b:=\mathrm{ess}\inf X$. More generally, a real-valued random variable $X$ is
called log-convex if $|X|$ is log-convex. In the same way, one may also define
log-concave functions and random variables.

Clearly, Pareto random variables are log-convex. Note that sometimes, log-
convexity of a (say, non-negative) random variable $X$ is defined by requiring
the tails of $X$ to be log-convex on the full non-negative half-axis
$[0,\infty)$, i. e. not only restricted to $t\geq b=\mathrm{ess}\inf X$. To
see that the two definitions essentially lead to the same concept of log-
convexity, one may replace $X$ by $X-b$ if necessary. For instance, if
$X\sim\operatorname{Par}(\alpha,b)$, it holds that
$2^{-\alpha}\mathbb{P}(X\geq t)\leq\mathbb{P}(X-b\geq t)\leq\mathbb{P}(X\geq
t)$ for all $t\geq 0$. In particular, similar relations also hold for the
$L^{p}$ norms of $X$ and $X-b$, noting that
$2^{-\alpha}\mathbb{E}[X^{p}]\leq\mathbb{E}[(X-b)^{p}]\leq\mathbb{E}[X^{p}]$
for any $p>0$.

######  Proposition 3.

If $t^{\alpha}\mathbb{P}(X\geq t)$ is $\log$-convex on $t\in[b,\infty)$ with
$b:=\mathrm{ess}\inf X$, then the characterizations of Proposition 1 are also
equivalent to the tail bound

$\mathbb{P}(X\geq t)\leq\frac{C_{5}}{t^{\alpha}}$

for $t\geq\mathrm{e}$, where the constant $C_{5}>0$ only differs from the
constants $C_{1}$ to $C_{4}$ by $\alpha$-dependent factors. The same statement
holds if $t^{\alpha}\mathbb{P}(X\geq t)$ is $\log$-concave and bounded from
below by some $\delta>0$.

In particular, the conditions of Proposition 3 hold for
$X\sim\mathrm{Par}(\alpha,b)$ (which even happens to be the only non-trivial
example for the log-convex case as the proof demonstrates), so that we get
back the correct tail behaviour this way. Assuming that
$t^{\alpha}\mathbb{P}(X\geq t)\geq\delta$ in the case of log-concavity has
technical reasons, but one can easily see that it does not exclude any
situations of interest. Indeed, otherwise we must have
$t^{\alpha}\mathbb{P}(X\geq t)\to 0$ as $t\to\infty$, and by log-concavity the
rate of decay must be so fast that $X$ cannot be a heavy-tailed random
variable.

###### Proof.

First recall that as already mentioned in the introduction, $\mathbb{P}(X\geq
t)\leq C_{5}/t^{\alpha}$ always implies part (a) of Proposition 1 with
$C_{1}=\alpha C_{5}$ without any further assumptions. (To be precise, here we
assume that the tail bound holds for any $t\geq 1$, but even if we can only
access it for $t\geq\mathrm{e}$ replacing $C_{5}$ by
$\max(C_{5},\mathrm{e}^{\alpha})$ will lead to the same result.)

Conversely, set $h(t)=t^{\alpha}\mathbb{P}(X\geq t)$ and assume part (a) of
Proposition 1. As in (2), we have $h(t)\leq C_{1}\mathrm{e}\log(t)$ by
Chebyshev’s inequality and optimization. In particular, $\log(h(t))$ is upper-
bounded by the concave function $1+\log(C_{1})+\log(\log(t))$. Since
$\log(h(t))$ is convex by assumption, this implies that $h(t)$ must be non-
increasing, so that for any $t\geq\mathrm{e}$, we have $h(t)\leq
C_{1}\mathrm{e}$ (in fact, $h$ must even be constant).

If $h(t)$ is log-concave, assume part (c) of Proposition 1. Fixing any
$r\geq\mathrm{e}$, we first consider the case where $h(t_{n})$ is unbounded,
i. e. there exists a monotone sequence $(t_{n})_{n\in\mathbb{N}}$ with
$t_{n}\rightarrow\infty$ and $h(t_{n})\rightarrow\infty$. Additionally, we can
assume that $h(r)\leq h(t_{n})$. By log-concavity of $h$, we have

$h(u)=h(r(1-s)+st_{n})\geq
h(r)^{1-s}h(t_{n})^{s}\geq\min\\{h(r),h(t_{n})\\}=h(r)$

for all $u=r(1-s)+st_{n}$ with some $s\in[0,1]$, i. e. $h$ is quasiconcave.
Thus, it follows that

$C_{3}\log(t_{n})\geq\int_{r}^{t_{n}}\frac{h(u)}{u}\,\mathrm{d}u\geq
h(r)\log(t_{n}/r).$ (4)

Dividing by $\log(t_{n})$ and letting $n\rightarrow\infty$ yields $h(r)\leq
C_{3}$.

Now assume $h$ to be bounded by some constant $K>1$. Using the log-concavity
again gives

$\displaystyle C_{3}\log(t)\geq\int_{r}^{t}\frac{h(u)}{u}\,\mathrm{d}u\geq
h(r)\Big{(}\frac{h(r)}{h(t)}\Big{)}^{\frac{r}{t-r}}\int_{r}^{t}\Big{(}\frac{h(t)}{h(r)}\Big{)}^{\frac{u}{t-r}}\,\frac{\mathrm{d}u}{u}.$

Since $\delta\leq h(t)\leq K$, we find that

$\lim_{t\rightarrow\infty}\Big{(}\frac{h(r)}{h(t)}\Big{)}^{\frac{r}{t-r}}=1.$
(5)

Dividing by $\log(t)$ and writing $\beta(t)=\log(h(t)/h(r))$, we have

$\displaystyle\frac{1}{\log(t)}\int_{r}^{t}\Big{(}\frac{h(t)}{h(r)}\Big{)}^{\frac{u}{t-r}}\,\frac{\mathrm{d}u}{u}=\frac{1}{\log(t)}\int_{\frac{r}{t-r}}^{\frac{t}{t-r}}\exp(\beta(t)u)\,\frac{\mathrm{d}u}{u},$

so that using the boundedness of $\beta(t)$ again, we find by partial
integration that

$\displaystyle\lim_{t\rightarrow\infty}\frac{1}{\log(t)}\int_{\frac{r}{t-r}}^{\frac{t}{t-r}}\exp(\beta(t)u)\,\frac{\mathrm{d}u}{u}$
$\displaystyle=\lim_{t\to\infty}\Big{(}\frac{\log(u)\exp(\beta(t)u)}{\log(t)}\Big{|}_{\frac{r}{t-r}}^{\frac{t}{t-r}}$
$\displaystyle-\frac{\beta(t)}{\log(t)}\int_{\frac{r}{t-r}}^{\frac{t}{t-r}}\log(u)\exp(\beta(t)u)\Big{)}\,\mathrm{d}u=1,$

where we have employed (5). Altogether, it follows that $h(r)\leq C_{3}$. ∎

###  2.2. Examples

As we have seen in Proposition 3, in certain situations one may avoid the
logarithmic error (2) which appears when combing Pareto-type moment bounds
with the Chebyshev inequality. Next, we will present a number of distributions
on the real line or, equivalently, random variables, which demonstrate that in
general, this is not possible. Here we also consider situations where the
tails satisfy certain regularity properties.

We first recall a number of basic definitions and results. A function
$L\colon(0,\infty)\to(0,\infty)$ is called slowly varying if for any $a>0$,

$\lim_{t\to\infty}\frac{L(at)}{L(t)}=1.$

It is called regularly varying if for any $a>0$,

$h_{L}(a):=\lim_{t\to\infty}\frac{L(at)}{L(t)}\in(0,\infty).$

By Karamata’s characterization theorem, if $g\colon(0,\infty)\to(0,\infty)$ is
regularly varying, it has to be of the form $g(t)=t^{\beta}L(t)$ for some
$\beta\in\mathbb{R}$ and some slowly varying function $L$. In particular,
$h_{g}(a)=a^{\beta}$, and $\beta$ is called the index of $g$. For a reference,
see [BGT87, Ch. 1.3 & 1.4].

######  Proposition 4.

Fix any $\alpha>0$, and let $h\colon[0,\infty)\to[0,\infty)$ be any
monotonously increasing function such that $h(t)=\mathcal{O}(\log(t))$. Then,
there exists a random variable $X$ such that

$t^{\alpha}\mathbb{P}(X\geq t)=\Theta(h(t)).$

If $h(t)=o(\log(t))$, then $X$ can be chosen such that its tails are a
regularly varying function of $t$ with index $-\alpha$.

###### Proof.

Consider functions of the form

$g(t):=t^{-\alpha}L(t)$ (6)

with $L(t):=\exp(\int_{a}^{t}\varepsilon(s)s^{-1}\,\mathrm{d}s)$ for some
$a>0$ (for simplicity, we will set $a=1$ in the sequel) and a bounded
measurable function $\varepsilon$. If $g^{\prime}(t)\leq 0$ and $g(t)\to 0$ as
$t\to\infty$, the function $g$ gives rise to the tails of a random variable
on, say, $[1,\infty)$, by setting $\mathbb{P}(X\geq t):=g(t)$, noting that
$g(1)=1$. Assuming $\varepsilon$ to be piecewise continuous, the condition
$g^{\prime}(t)\leq 0$ holds if and only if $\varepsilon(t)\leq\alpha$, since
we have

$g^{\prime}(t)=(-\alpha+\varepsilon(t))t^{-1}g(t).$

In particular, if $\varepsilon$ is bounded away from $\alpha$, say,
$\varepsilon(t)\leq\alpha/2$, we also have $g(t)\to 0$ (indeed, note that in
this case, $0\leq g(t)\leq t^{-\alpha/2}$).

To specify a choice of $\varepsilon$, now define sequences
$a_{n}=\exp(n\exp(n))$ and $b_{n}:=\exp(n\exp(n)+n)$. Moreover, let
$\gamma(n)$ be any function on the natural numbers such that
$\mathrm{e}^{-n}\leq\gamma(n)\leq 1$ which is bounded away from $\alpha$ and
which satisfies $\exp(n\gamma(n))=\Theta(h(b_{n}))$. Here, the technical
condition $\gamma(n)\geq\mathrm{e}^{-n}$ can be assumed without loss of
generality since already for $\gamma(n)=\mathrm{e}^{-n}$, $\exp(n\gamma(n))$
is decreasing, while $h$ was assumed to be increasing. For instance, one may
take
$\gamma(n)=\min\\{1,\alpha/2,\max\\{\log(h(b_{n}))/n,\mathrm{e}^{-n}\\}\\}$.
Set

$\varepsilon(t)=\begin{cases}\gamma(n)&\text{ if }t\in[a_{n},b_{n}]\\\
-\gamma(n)&\text{ if }t\in[b_{n},b_{n}\exp(n)]\\\ 0&\text{ if
}t\in(b_{n}\exp(n),a_{n+1}]\end{cases}.$ (7)

Clearly, $\varepsilon(t)\rightarrow 0$ if $h(t)=o(\log(t))$, so that by
Karamata’s representation theorem, $L$ is slowly varying and hence, $g$ is
regularly varying with index $-\alpha$ in this case.

It is now easy to see that

$\int_{1}^{t}\frac{\varepsilon(s)}{s}\,\mathrm{d}s=\begin{cases}\gamma(n)(\log(t)-\log(a_{n}))&\text{
if }t\in[a_{n},b_{n}]\\\ \gamma(n)(n+\log(b_{n})-\log(t))&\text{ if
}t\in(b_{n},b_{n}\exp(n))\\\ 0&\text{ if
}t\in(b_{n}\exp(n),a_{n+1}]\end{cases},$

and hence,

$L(t)=\begin{cases}\frac{t^{\gamma(n)}}{a_{n}^{\gamma(n)}}&\text{ if
}t\in[a_{n},b_{n}]\\\
\exp(n\gamma(n))\frac{b_{n}^{\gamma(n)}}{t^{\gamma(n)}}&\text{ if
}t\in(b_{n},b_{n}\exp(n))\\\ 1&\text{ if
}t\in(b_{n}\exp(n),a_{n+1}]\end{cases}.$

In particular,
$b_{n}^{\alpha}g(b_{n})=L(b_{n})=\exp(n\gamma(n))=\Theta(h(b_{n}))$ by
definition.

It remains to verify that the random variable $X$ induced by the function $g$
satisfies condition (c) of Proposition 1. To this end, we first consider
$r\in[a_{n},b_{n}]$ and note that
$\log(r)\geq\log(a_{n})=n\exp(n)\geq\log(b_{n})/2$. Thus, we may assume that
$r=b_{n}$. For any $k\leq n$, we get

$\displaystyle\int_{a_{k}}^{b_{k}}t^{\alpha-1}\mathbb{P}(X\geq
t)\,\mathrm{d}t=\int_{a_{k}}^{b_{k}}\frac{L(t)}{t}\,\mathrm{d}t$
$\displaystyle=a_{k}^{-\gamma(k)}\int_{a_{k}}^{b_{k}}t^{-1+\gamma(k)}\,\mathrm{d}t$
$\displaystyle=\gamma(k)^{-1}\Big{(}\Big{(}\frac{b_{k}}{a_{k}}\Big{)}^{\gamma(k)}-1\Big{)}\leq\gamma(k)^{-1}\exp(k\gamma(k)).$

Summing this up to $n$ gives a contribution of at most
$n\gamma(n)^{-1}\exp(n\gamma(n))\leq C\log(b_{n})$ for a suitable constant
$C>0$ (to see this, note that the slightly stronger inequality
$n\gamma(n)^{-1}\exp(n\gamma(n))\leq\log(a_{n})$ can be rewritten as
$n\gamma(n)-\log(\gamma(n))\leq n$ and use that
$\mathrm{e}^{-n}\leq\gamma(n)\leq 1$). If $r\in[b_{k},\exp(k)b_{k}]$, then
again $\log(r)\geq\log(b_{k})\geq\log(\exp(k)b_{k})/2$ and thus we may
consider $r=\exp(k)b_{k}$. Here we have

$\int_{b_{k}}^{b_{k}\exp(k)}\frac{L(t)}{t}\,\mathrm{d}t=\exp(k\gamma(k))b_{k}^{\gamma(k)}\int_{b_{k}}^{b_{k}\exp(k)}\frac{1}{t^{1+\gamma(k)}}\,\mathrm{d}t\leq\gamma(k)^{-1}\exp(k\gamma(k)),$

so that we may argue similarly as above. Finally, if
$r\in[b_{k}\exp(k),a_{k+1}]$,

$\int_{b_{k}\exp(k)}^{a_{k+1}}\frac{L(t)}{t}\,\mathrm{d}t=\int_{b_{k}\exp(k)}^{a_{k+1}}\frac{1}{t}\,\mathrm{d}t\leq\log(a_{k+1})-\log(b_{k}).$

Because of $a_{k}\leq b_{k}$, summing up and telescoping yields a bound of
$\log(r)$ again. ∎

As for the auxiliary function $\gamma(n)$ from the proof of Proposition 4,
simple examples like $\gamma(n):=n^{\delta-1}$ for any $\delta<1$ yield
$b_{n}^{\alpha}\mathbb{P}(X\geq
b_{n})=\exp(n^{\delta}))\approx\exp((\log\log(b_{n}))^{\delta})$.

In view of Proposition 3, one may wonder whether in order to avoid the
logarithmic error, it might already suffice to require $X$ to be log-convex.
It turns out that this is not true as we demonstrate in the following
proposition, whose proof is based on the proof of Proposition 4 together with
a smoothening argument.

######  Proposition 5.

In the situation of Proposition 4, the random variable $X$ can be chosen to be
log-convex.

###### Proof.

Consider a function $g$ as in (6) and set $h(t):=\log(g(t))$. Assuming
$\varepsilon(s)$ to be continuously differentiable, one can check in the same
way as in the proof of Proposition 4 that $h^{\prime\prime}(t)\geq 0$ is
equivalent to $g^{\prime\prime}(t)g(t)-(g^{\prime}(t))^{2}\geq 0$ and thus to

$\varepsilon^{\prime}(t)+\frac{\alpha}{t}\geq\frac{\varepsilon(t)}{t}.$ (8)

Therefore, once (8) holds, the corresponding random variable has log-convex
tails. By continuity of $h^{\prime}$, the same arguments remain valid if
$\varepsilon(t)$ is only piecewise continuously differentiable (with (8) for
all continuity points of $\varepsilon^{\prime}$).

To construct such a function, we write $\Lambda(s):=\min\\{2s,2-2s\\}$ for
$s\in[0,1]$ and replace (7) by the smoothed version

$\varepsilon(t)=\begin{cases}\gamma(n)c_{n}^{-1}\Lambda(\tfrac{t-a_{n}}{b_{n}-a_{n}})&\text{
if }t\in[a_{n},b_{n}]\\\
-\gamma(n)c_{n}^{-1}\Lambda(\tfrac{t-b_{n}}{b_{n}(\mathrm{e}^{n}-1)})&\text{
if }t\in[b_{n},b_{n}\exp(n)]\\\ 0&\text{ if
}t\in(b_{n}\exp(n),a_{n+1}]\end{cases}$

with correction factor

$c_{n}:=\int_{0}^{1}\frac{\Lambda(t)}{t+(\mathrm{e}^{n}-1)^{-1}}\,\mathrm{d}t=\int_{a_{n}}^{b_{n}}\Lambda(\tfrac{t-a_{n}}{b_{n}-a_{n}})t^{-1}\,\mathrm{d}t=\int_{b_{n}}^{b_{n}\exp(n)}\Lambda(\tfrac{t-b_{n}}{b_{n}(\mathrm{e}^{n}-1)})t^{-1}\,\mathrm{d}t.$

Note that $c_{n}$ is monotoneously increasing with

$c_{1}\approx
0.480156,\qquad\lim_{n\rightarrow\infty}c_{n}=\int_{0}^{1}\frac{\Lambda(t)}{t}\,\mathrm{d}t=\log(4).$

In particular, assuming without loss of generality that $\gamma(n)\leq\alpha
c_{1}/2$, as in the proof of Proposition 4 we have that $g$ is monotoneously
decreasing with $g(t)\to 0$, and $\varepsilon(t)\to 0$ if $h(t)=o(\log(t))$.
Moreover, it also follows that

$\int_{1}^{b_{n}}\varepsilon(s)s^{-1}\,\mathrm{d}s=\gamma(n)n,\qquad\int_{1}^{b_{n}\exp(n)}\varepsilon(s)s^{-1}\,\mathrm{d}s=0,$

so that the relation
$b_{n}^{\alpha}g(b_{n})=\exp(n\gamma(n))=\Theta(h(b_{n}))$ continues to hold.
Furthermore, part (c) of Proposition 1 can be verified as in the proof of
Proposition 4 by invoking the trivial bound $\Lambda(t)\leq 1$.

It remains to check (8), which is trivial if $\varepsilon^{\prime}(t)\geq 0$.
For $t\in((b_{n}-a_{n})/2,b_{n})$, (8) reads

$\varepsilon(t)=\frac{\gamma(n)}{c_{n}}\Big{(}2-2\frac{t-a_{n}}{b_{n}-a_{n}}\Big{)}\leq\alpha-2\frac{\gamma(n)}{c_{n}}\frac{t}{b_{n}-a_{n}}=\alpha-t\varepsilon^{\prime}(t),$

and the inequality in the middle is equivalent to

$2\frac{\gamma(n)}{c_{n}}\frac{1}{\mathrm{e}^{n}-1}\leq\alpha-2\frac{\gamma(n)}{c_{n}}.$

Recalling that $\gamma(n)\leq\alpha c_{1}/2$ and that $c_{n}$ is strictly
increasing and bounded, this inequality is true for all large enough $n$
(formally, one may replace $\varepsilon(t)$ by $0$ for small values of $n$).
Similar arguments also hold in the remaining case of
$t\in(b_{n},b_{n}(1+\mathrm{e}^{n})/2)$, which is even simpler since
$\varepsilon(t)\leq 0$ in this range. ∎

##  3\. Tail inequalities for polynomial chaos

###  3.1. Results

Typical applications of $L^{p}$ norm inequalities include concentration
results of higher order, in particular bounds for polynomials (or polynomial
chaos) of some order $d$ in independent random variables $X_{1},\ldots,X_{n}$.
Concentration results of this type have been shown in [Lat06] (polynomials in
Gaussian random variables), [AW15] (functions of sub-Gaussian random
variables) and [GSS21] (polynomials in $\alpha$-sub-exponential random
variables), for example. All these papers in fact prove bounds on the
respective $L^{p}$ norms, from which concentration bounds are subsequently
derived.

We may proceed similarly, using the following generalization of (2): given a
random variable $Z$ which satisfies the moment condition

$\lVert
Z\rVert_{L^{p}}\leq\gamma\Big{(}\frac{\alpha}{\alpha-p}\Big{)}^{\frac{\beta}{p}}$

for any $p\in(0,\alpha)$ and some $\gamma,\beta>0$, a standard Chebyshev-type
argument shows that

$\mathbb{P}(|Z|\geq
t)\leq(\alpha/\beta)^{\beta}\mathrm{e}^{\beta}\log(t/\gamma)^{\beta}(\gamma/t)^{\alpha}$
(9)

for any $t>\gamma\mathrm{e}^{\beta/\alpha}$ (here one argues similarly as in
(2), choosing $p=\alpha-\beta/\log(t/\gamma)$).

In the sequel, for a random vector $X=(X_{1},\ldots,X_{n})$ with independent
centered components (i. e., $\mathbb{E}[X_{i}]=0$ for all $i$), we study
functions of type

$f_{d}(X):=f_{d,A}(X):=\sum_{i_{1},\ldots,i_{d}=1}^{n}a_{i_{1}\ldots
i_{d}}X_{i_{1}}\cdots X_{i_{d}},$ (10)

where $A=(a_{i_{1}\ldots i_{d}})\in\mathbb{R}^{n^{d}}$ is a tensor of real
coefficients such that $a_{i_{1}\ldots i_{d}}=0$ whenever
$i_{j}=i_{j^{\prime}}$ for some $j\neq j^{\prime}$ (we also say that $A$ has
_generalized diagonal_ $0$). These polynomials are linear in every component,
hence they are also referred to as multilinear (or tetrahedral) polynomials.

For multilinear polynomials in heavy-tailed random variables, we may show the
following elementary $L^{p}$ (and concentration) bound, which shares some
similarities with [GSS21, Theorem 1.5] (whose proof it partially mimics). Here
and all over the rest of this section, absolute constants with values in
$(0,\infty)$ are typically denoted by $C$, constants which depend on some
quantities $Q_{1},\ldots,Q_{m}$ only are denoted by $C_{Q_{1},\ldots,Q_{m}}$,
and the precise values of the constants may vary from line to line (especially
in the course of the proofs).

######  Proposition 6.

Let $X=(X_{1},\ldots,X_{n})$ be a random vector with independent centered
components such that there exist $\alpha>2$ and $b>0$ with
$\mathbb{P}(|X_{i}|\geq t)\leq(b/t)^{\alpha}$ for any $t\geq b$ and any
$i=1,\ldots,n$. Let $A=(a_{i_{1}\ldots i_{d}})$ be a tensor with generalized
diagonal $0$. Then,

$\lVert f_{d,A}\rVert_{p}\leq C_{d,\alpha}\lVert
A\rVert_{\mathrm{HS}}b^{d}\Big{(}\frac{\alpha}{\alpha-p}\Big{)}^{d/p}$

for any $p\in[2,\alpha)$. In particular,

$\mathbb{P}(|f_{d,A}(X)|\geq t)\leq C_{d,\alpha}\log^{d}\Big{(}\frac{t}{\lVert
A\rVert_{\mathrm{HS}}b^{d}}\Big{)}\Big{(}\frac{\lVert
A\rVert_{\mathrm{HS}}b^{d}}{t}\Big{)}^{\alpha}$

for all $t\geq C_{d,\alpha}\lVert A\rVert_{\mathrm{HS}}b^{d}$.

###### Proof.

In view of (9), it suffices to prove the $L^{p}$ bound, and moreover, we may
clearly assume $b=1$. Let $X^{(1)},\ldots,X^{(d)}$ be independent copies of
the vector $X$ and $(\varepsilon_{i}^{(j)})$, $i\leq n$, $j\leq d$, a set of
i.i.d. Rademacher variables which are independent of the $(X^{(j)})_{j}$.
First, by standard decoupling (cf. [PG99, Theorem 3.1.1]) and symmetrization
inequalities ([PG99, Lemma 1.2.6] applied iteratively), we have

$\displaystyle\Big{\lVert}\sum_{i_{1},\ldots,i_{d}}a_{i_{1}\ldots
i_{d}}X_{i_{1}}\cdots X_{i_{d}}\Big{\rVert}_{p}$ $\displaystyle\leq
C_{d}\Big{\lVert}\sum_{i_{1},\ldots,i_{d}}a_{i_{1}\ldots
i_{d}}X_{i_{1}}^{(1)}\cdots X_{i_{d}}^{(d)}\Big{\rVert}_{p}$
$\displaystyle\leq C_{d}\Big{\lVert}\sum_{i_{1},\ldots,i_{d}}a_{i_{1}\ldots
i_{d}}\varepsilon_{i_{1}}^{(1)}X_{i_{1}}^{(1)}\cdots\varepsilon_{i_{d}}^{(d)}X_{i_{d}}^{(d)}\Big{\rVert}_{p}.$

By Kwapień’s contraction principle [Kwa87, Theorem 1] (more precisely, its
decoupled analogue, which follows easily by iteration of Kwapień’s result for
linear forms conditionally on the other random variables), we furthermore
obtain that

$\Big{\lVert}\sum_{i_{1},\ldots,i_{d}}a_{i_{1}\ldots
i_{d}}\varepsilon_{i_{1}}^{(1)}X_{i_{1}}^{(1)}\cdots\varepsilon_{i_{d}}^{(d)}X_{i_{d}}^{(d)}\Big{\rVert}_{p}\leq\Big{\lVert}\sum_{i_{1},\ldots,i_{d}}a_{i_{1}\ldots
i_{d}}Y_{i_{1}}^{(1)}\cdots Y_{i_{d}}^{(d)}\Big{\rVert}_{p}$

for any $p\geq 1$, where $Y_{1},\ldots,Y_{n}$ is a set of i.i.d. random
variables with symmetrized Pareto distribution
$X_{i}\sim\mathrm{Par}_{\mathrm{s}}(\alpha,1)$.

Let us now first assume $d=1$. By [HMO97, Theorem 1.1] applied to the
symmetric log-convex random variables $a_{1}Y_{1},\ldots,a_{n}Y_{n}$, we
obtain

$\displaystyle\Big{\lVert}\sum_{i=1}^{n}a_{i}Y_{i}\Big{\rVert}_{p}$
$\displaystyle\leq C\Big{(}\Big{(}\sum_{i=1}^{n}|a_{i}|^{p}\lVert
Y_{i}\rVert_{p}^{p}\Big{)}^{1/p}+\sqrt{p}\Big{(}\sum_{i=1}^{n}a_{i}^{2}\lVert
Y_{i}\rVert_{2}^{2}\Big{)}^{1/2}\Big{)}$
$\displaystyle=C\Big{(}\Big{(}\sum_{i=1}^{n}|a_{i}|^{p}\Big{)}^{1/p}\Big{(}\frac{\alpha}{\alpha-p}\Big{)}^{1/p}+\sqrt{p}\Big{(}\sum_{i=1}^{n}a_{i}^{2}\Big{)}^{1/2}\Big{(}\frac{\alpha}{\alpha-2}\Big{)}^{1/2}\Big{)}$
(11) $\displaystyle\leq
C_{\alpha}\Big{(}\sum_{i=1}^{n}a_{i}^{2}\Big{)}^{1/2}\Big{(}\frac{\alpha}{\alpha-p}\Big{)}^{1/p}=C_{\alpha}\lVert
A\rVert_{\mathrm{HS}}\Big{(}\frac{\alpha}{\alpha-p}\Big{)}^{1/p}$

for any $p\in[2,\alpha)$. This may be iterated to arrive at

$\Big{\lVert}\sum_{i_{1},\ldots,i_{d}}a_{i_{1}\ldots
i_{d}}Y_{i_{1}}^{(1)}\cdots Y_{i_{d}}^{(d)}\Big{\rVert}_{p}\leq
C_{\alpha}^{d}\lVert
A\rVert_{\mathrm{HS}}\Big{(}\frac{\alpha}{\alpha-p}\Big{)}^{d/p}$ (12)

To see this, assume that (12) holds up to order $d-1$. It follows that

$\displaystyle\qquad\Big{\|}\sum_{i_{1},\ldots,i_{d}}a_{i_{1}\ldots
i_{d}}Y_{i_{1}}^{(1)}\cdots Y_{i_{d}}^{(d)}\Big{\|}_{p}^{2}$
$\displaystyle\leq
C_{\alpha}^{2(d-1)}\Big{(}\frac{\alpha}{\alpha-p}\Big{)}^{2(d-1)/p}\Big{\|}(\sum_{i_{1},\ldots,i_{d-1}}(\sum_{i_{d}=1}^{n}a_{i_{1}\ldots
i_{d}}Y_{i_{d}}^{(d)})^{2})^{1/2}\Big{\|}_{p}^{2}$
$\displaystyle=C_{\alpha}^{2(d-1)}\Big{(}\frac{\alpha}{\alpha-p}\Big{)}^{2(d-1)/p}\Big{\|}\sum_{i_{1},\ldots,i_{d-1}}(\sum_{i_{d}=1}^{n}a_{i_{1}\ldots
i_{d}}Y_{i_{d}}^{(d)})^{2}\Big{\|}_{p/2}\allowdisplaybreaks$
$\displaystyle\leq
C_{\alpha}^{2(d-1)}\Big{(}\frac{\alpha}{\alpha-p}\Big{)}^{2(d-1)/p}\sum_{i_{1},\ldots,i_{d-1}}\Big{\|}(\sum_{i_{d}=1}^{n}a_{i_{1}\ldots
i_{d}}Y_{i_{d}}^{(d)})^{2}\Big{\|}_{p/2}$
$\displaystyle=C_{\alpha}^{2(d-1)}\Big{(}\frac{\alpha}{\alpha-p}\Big{)}^{2(d-1)/p}\sum_{i_{1},\ldots,i_{d-1}}\Big{\|}\sum_{i_{d}=1}^{n}a_{i_{1}\ldots
i_{d}}Y_{i_{d}}^{(d)}\Big{\|}_{p}^{2}$ $\displaystyle\leq
C_{\alpha}^{2d}\Big{(}\frac{\alpha}{\alpha-p}\Big{)}^{2d/p}\sum_{i_{1},\ldots,i_{d-1}}\lVert
a_{i_{1}\ldots
i_{d-1}\cdot}\rVert_{2}^{2}=C_{\alpha}^{2d}\Big{(}\frac{\alpha}{\alpha-p}\Big{)}^{2d/p}\lVert
A\rVert_{\mathrm{HS}}^{2},$

using the induction in the first and the case of $d=1$ in the last inequality
and denoting by $a_{i_{1}\ldots i_{d-1}\cdot}$ the vector in $\mathbb{R}^{n}$
defined by fixing all the coordinates of the tensor $A$ but the last one. ∎

Note that the $L^{p}$ estimates from [HMO97] we used in the proof hold for
$p\geq 2$ only, so that we must assume $\alpha>2$ in Proposition 6. Similarly,
one may think of replacing the assumption $\mathbb{P}(|X_{i}|\geq
t)\leq(b/t)^{\alpha}$ by a condition on the $L^{p}$ norms. However, in the
proof we need stochastic dominance by a log-concave distribution. In
particular, it is possible to reformulate Proposition 6 with the Pareto-type
tails replaced by the tails of the random variables introduced in Proposition
5.

Proposition 6 yields concentration bounds which, up to logarithmic error and
rescaling, do not essentially change as $d$ increases. While this is not too
surprising in view of (3), it remarkably differs from the behaviour of
polynomial chaos in sub-Gaussian (and subexponential) random variables, where
the tails get significantly heavier as $d$ increases. For instance, the tails
of a $d$-th order chaos in sub-Gaussian random variables will be
$2/d$-subexponential (i. e. of order $\exp(-t^{2/d})$) for large $t$.

Next, we show a version of Proposition 6 which also includes diagonal terms.
Note that if $X\sim\mathrm{Par}(\alpha,b)$ and $k\in\mathbb{N}$, then
$X^{k}\sim\mathrm{Par}(\alpha/k,b^{k})$. As a consequence, unlike in the sub-
Gaussian or sub-exponential case again, the tail behaviour depends on the
order of the largest power to which a single random variable $X_{i}$ is
raised. More precisely, we may write $A=A_{1}+\ldots+A_{d}$ with $d$-tensors
$A_{1},\ldots,A_{d}$ such that

$(A_{k})_{i_{1}\ldots i_{d}}=\begin{cases}a_{i_{1}\ldots i_{d}}&\text{if
}k=\max\\{\nu\colon\exists\,i_{j_{1}}=\ldots=i_{j_{\nu}},j_{1},\ldots,j_{\nu}\text{
pw.\ different}\\}\\\ 0&\text{else.}\end{cases}$

For instance, if $A$ has generalized diagonal $0$ as in Proposition 6, then
$A=A_{1}$.

Furthermore, if diagonal terms appear in a $d$-th order chaos, it is necessary
to “recenter” it. This is already obvious for $d=2$, where the functional
under consideration may be rewritten as

$\sum_{i\neq
j}a_{ij}(X_{i}-\mathbb{E}[X_{i}])(X_{j}-\mathbb{E}[X_{j}])+\sum_{i=1}^{n}a_{ii}(X_{i}^{2}-\mathbb{E}[X_{i}^{2}]).$

Similarly, in higher orders, we need to replace each of the monomials
$X_{i_{1}}^{k_{1}}\cdots X_{i_{m}}^{k_{m}}$ by
$(X_{i_{1}}^{k_{1}}-\mathbb{E}[X_{i_{1}}^{k_{1}}])\cdots(X_{i_{m}}^{k_{m}}-\mathbb{E}[X_{i_{m}}^{k_{m}}])$.
As a consequence, given any $d$-tensor $A\in\mathbb{R}^{n^{d}}$, the natural
generalization of (10) are functionals of the form

$f_{d,A}(X):=\sum_{\nu=1}^{d}\sum_{\begin{subarray}{c}k_{1}\geq\ldots\geq
k_{\nu}\geq 1\\\
k_{1}+\ldots+k_{\nu}=d\end{subarray}}\sum_{i_{1}\neq\ldots\neq
i_{\nu}}\tilde{a}_{i_{1}\ldots i_{\nu}}^{k_{1}\ldots
k_{\nu}}(X_{i_{1}}^{k_{1}}-\mathbb{E}[X_{i_{1}}^{k_{1}}])\cdots(X_{i_{m}}^{k_{m}}-\mathbb{E}[X_{i_{m}}^{k_{m}}]).$
(13)

Here, in terms of the tensor $A$, we have

$\tilde{a}_{i_{1}\ldots i_{\nu}}^{k_{1}\ldots k_{\nu}}=\sum a_{j_{1}\ldots
j_{d}},$

where the sum extends over all $d$-tuples $j_{1},\ldots,j_{d}$ in which every
index $i_{1},\ldots,i_{\nu}$ appears exactly $k_{1},\ldots,k_{\nu}$ times. In
particular, if $a_{j_{1}\ldots j_{d}}\neq 0$, then it is a non-zero entry of
$A_{k_{1}}$ (recall that $k_{1}\geq\ldots\geq k_{\nu}$). In this sense, (13)
centers and “regroups” the indexes according to how many _different_ indexes
appear in $i_{1},\ldots,i_{d}$ and to which powers they are raised.

######  Proposition 7.

Let $X=(X_{1},\ldots,X_{n})$ be a random vector with independent centered
components such that there exist $\alpha>2$ and $b>0$ with
$\mathbb{P}(|X_{i}|\geq t)\leq(b/t)^{\alpha}$ for any $t\geq b$ and any
$i=1,\ldots,n$. Let $A=(a_{i_{1}\ldots i_{d}})\in\mathbb{R}^{n^{d}}$ be a
$d$-tensor and $f_{d,A}$ the functional defined in (13). Then,

$\displaystyle\lVert f_{d,A}(X)\rVert_{p}$ $\displaystyle\leq
C_{d,\alpha}\sum_{k=1}^{k^{\ast}}\lVert
A_{k}\rVert_{\mathrm{HS}}b^{d}\Big{(}\frac{\alpha/k}{\alpha/k-p}\Big{)}^{(d-k+1)/p}$
(14) $\displaystyle\leq C_{d,\alpha}\lVert
A\rVert_{\mathrm{HS}}b^{d}\Big{(}\frac{\alpha/k^{\ast}}{\alpha/k^{\ast}-p}\Big{)}^{(d-k^{\ast}+1)/p}$

for any $p\in[2,\alpha/k^{\ast})$, where $k^{\ast}:=\max\\{k\colon A_{k}\neq
0\\}$. In particular,

$\mathbb{P}(|f_{d,A}(X)|\geq t)\leq
C_{d,\alpha}\max_{k=1,\ldots,k^{\ast}}\Big{(}\log^{d-k+1}\Big{(}\frac{t}{\lVert
A_{k}\rVert_{\mathrm{HS}}b^{d}}\Big{)}\Big{(}\frac{\lVert
A_{k}\rVert_{\mathrm{HS}}b^{d}}{t}\Big{)}^{\alpha/k}\Big{)}$

for all $t\geq C_{d,\alpha}b^{d}\max_{k=1,\ldots,k^{\ast}}\lVert
A_{k}\rVert_{\mathrm{HS}}$.

Note that Proposition 7 implicitly requires that $\alpha/k^{\ast}>2$. To
illustrate the result, let us consider the case of $d=2$. Here,
$A=A_{1}+A_{2}=A^{\mathrm{od}}+A^{\mathrm{d}}$ for the off-diagonal and
diagonal parts of $A$, so that the tail bound given in Proposition 7 reads

$\displaystyle\begin{split}\mathbb{P}(|f_{2,A}(X)|\geq t)\leq
C_{\alpha}\max\Big{(}&\log^{2}\Big{(}\frac{t}{\lVert
A^{\mathrm{od}}\rVert_{\mathrm{HS}}b^{2}}\Big{)}\Big{(}\frac{\lVert
A^{\mathrm{od}}\rVert_{\mathrm{HS}}b^{2}}{t}\Big{)}^{\alpha},\\\
&\log\Big{(}\frac{t}{\lVert
A^{\mathrm{d}}\rVert_{\mathrm{HS}}b^{2}}\Big{)}\Big{(}\frac{\lVert
A^{\mathrm{d}}\rVert_{\mathrm{HS}}b^{2}}{t}\Big{)}^{\alpha/2}\Big{)}.\end{split}$
(15)

As we will discuss in more detail in the following subsection, this can be
regarded as a crude or preliminary Hanson–Wright type inequality for heavy-
tailed random variables.

###### Proof.

We again assume $b=1$. Note that $\mathbb{P}(|X_{i_{j}}^{k_{j}}|\geq t)\leq
t^{-\alpha/k_{j}}$ and therefore also
$\mathbb{P}(|X_{i_{j}}^{k_{j}}-\mathbb{E}[X_{i_{j}}^{k_{j}}]|\geq t)\leq
C_{\alpha,k_{j}}t^{-\alpha/k_{j}}$ for $t$ sufficiently large (depending on
$\alpha$ and $k_{j}$). Therefore, we may apply Proposition 6 with $\nu$
instead of $d$ and $\alpha/k_{1}$ instead of $\alpha$ to each of the summands

$f^{k_{1},\ldots,k_{\nu}}_{\nu,\tilde{A}}(X):=\sum_{i_{1}\neq\ldots\neq
i_{\nu}}\tilde{a}_{i_{1}\ldots i_{\nu}}^{k_{1}\ldots
k_{\nu}}(X_{i_{1}}^{k_{1}}-\mathbb{E}[X_{i_{1}}^{k_{1}}])\cdots(X_{i_{\nu}}^{k_{\nu}}-\mathbb{E}[X_{i_{\nu}}^{k_{\nu}}])$

from the representation (13), where we suppress the dependency of $\tilde{A}$
on $k_{1},\ldots,k_{\nu}$ in our notation. Clearly,
$\lVert\tilde{A}\rVert_{\mathrm{HS}}\leq C_{d}\lVert
A_{k_{1}}\rVert_{\mathrm{HS}}$ (recall that $k_{1}=\max_{j}k_{j}$), and hence,

$\displaystyle\lVert f^{k_{1},\ldots,k_{\nu}}_{\nu,\tilde{A}}(X)\rVert_{p}$
$\displaystyle\leq
C_{d,\alpha}\lVert\tilde{A}\rVert_{\mathrm{HS}}\Big{(}\frac{\alpha/k_{1}}{\alpha/k_{1}-p}\Big{)}^{\nu/p}$
$\displaystyle\leq C_{d,\alpha}\lVert
A_{k_{1}}\rVert_{\mathrm{HS}}\Big{(}\frac{\alpha/k_{1}}{\alpha/k_{1}-p}\Big{)}^{(d-k_{1}+1)/p}.$

Summing up according to $k_{1}=k\in\\{1,\ldots,k^{\ast}\\}$ we arrive at (14).
To derive the tail bound, use that $f_{d,A}=\sum_{k=1}^{k^{\ast}}f_{k,A_{k}}$,
so that by union bound,

$\mathbb{P}(|f_{d,A}(X)|\geq
t)\leq\sum_{k=1}^{k^{\ast}}\mathbb{P}(|f_{k,A_{k}}(X)|\geq t/k^{\ast}).$

As in Proposition 6, the claim now follows from the $L^{p}$ estimates we
derived above (for each $k$) in combination with (9). ∎

###  3.2. Discussion and open questions

For general $d\geq 2$, the results from Propositions 6 and 7 seem new.
However, alternate bounds for linear forms, i. e., the case of $d=1$, can be
deduced from existing literature. In particular, we shall consider the
Fuk–Nagaev inequality which is due to [FN71], cf. also [Fuk73] and [Nag79]
(for a more recent partial refinement, see [Che09]). Given a sequence
$X_{1},\ldots,X_{n}$ of independent random variables with
$\mathbb{E}[X_{i}]=0$ and finite second moments, let
$S_{n}:=X_{1}+\ldots+X_{n}$,
$\sigma^{2}:=\mathbb{E}[X_{1}^{2}]+\ldots+\mathbb{E}[X_{n}^{2}]$, and

$C_{p}(X):=\Big{(}\sum_{i=1}^{n}\mathbb{E}[\max(0,X_{i})^{p}]\Big{)}^{1/p}$

for any $p\geq 1$. Then, by [Nag79, Corollary 1.8],

$\mathbb{P}(S_{n}>t)\leq\Big{(}\frac{(p+2)C_{p}(X)}{pt}\Big{)}^{p}+\exp\Big{(}-\frac{2t^{2}}{(p+2)^{2}\mathrm{e}^{p}\sigma^{2}}\Big{)}$

for any $t>0$.

Now assume that $X_{1}^{\prime},\ldots,X_{n}^{\prime}$ have $L^{p}$ norms at
most of Pareto type (1), say, with $b=1$, and let
$X_{1}:=a_{1}X_{1}^{\prime},\ldots,a_{n}X_{n}^{\prime}$ for real-valued
coefficients $a_{1},\ldots,a_{n}$. Writing $a=(a_{1},\ldots,a_{n})$ and
$\lVert a\rVert_{\ell^{p}}$ for its $\ell^{p}$ norm, we clearly have

$C_{p}(X)^{p}\leq\lVert
a\rVert_{\ell^{p}}^{p}\frac{\alpha}{\alpha-p},\qquad\sigma^{2}\leq\lVert
a\rVert_{\ell^{2}}^{2}\frac{\alpha}{\alpha-2}.$

It follows that in this case,

$\mathbb{P}(S_{n}>t)\leq\frac{(p+2)^{p}}{p^{p}}\frac{\alpha}{\alpha-p}\frac{\lVert
a\rVert_{\ell^{p}}^{p}}{t^{p}}+\exp\Big{(}-\frac{2(\alpha-2)t^{2}}{(p+2)^{2}\mathrm{e}^{p}\lVert
a\rVert_{\ell^{2}}^{2}\alpha}\Big{)}$

for any $p\in(2,\alpha)$ and any $t>0$. Plugging in $p=\alpha-1/\log(t/\lVert
a\rVert_{\ell^{\alpha}})$ together with some elementary estimates thus leads
to the bound

$\mathbb{P}(S_{n}\geq
t)\leq\frac{(\alpha+2)^{\alpha}}{\alpha^{\alpha}}\mathrm{e}\alpha\log\Big{(}\frac{t}{\lVert
a\rVert_{\ell^{\alpha}}}\Big{)}\frac{\lVert
a\rVert_{\ell^{\alpha}}^{\alpha}}{t^{\alpha}}+\exp\Big{(}-\frac{2(\alpha-2)}{(\alpha+2)^{2}\mathrm{e}^{\alpha}\alpha}\cdot\frac{t^{2}}{\lVert
a\rVert_{\ell^{2}}^{2}}\Big{)}.$

The same holds for the lower tails $\mathbb{P}(S_{n}\leq-t)$. The resulting
(two-sided) inequality is obviously sharper than the bound given by
Proposition 6 for $d=1$ and $b=1$, which reads

$\mathbb{P}\Big{(}\Big{|}\sum_{i=1}^{n}a_{i}X_{i}\Big{|}\geq t\Big{)}\leq
C_{\alpha}\log(t/\lVert a\rVert_{\ell^{2}})\Big{(}\frac{\lVert
a\rVert_{\ell^{2}}}{t}\Big{)}^{\alpha}$

for all $t\geq C_{\alpha}\lVert a\rVert_{\ell^{2}}$.

In fact, this result corresponds to what we heuristically get from (11) if we
naively optimize each of the two summands of (11) on its own, recalling (1)
and standard results about sub-Gaussian tail behavior like [Ver18, Proposition
2.5.2] (in particularly pretending we may consider the second summand for any
$p\in[2,\infty)$). However, this is an informal argument, and in practice we
can only address the “sub-Gaussian” part for $p\in[2,\alpha)$, where it is
almost meaningless (one needs to access $p\to\infty$ to obtain sub-Gaussian
tail bounds). Yet, on a heuristical level the $L^{p}$ bounds do seem to imply
the Fuk–Nagaev results, and it is an interesting question whether this
observation can be made rigorous.

Similar remarks hold for $d\geq 2$. Here, the aforementioned results for
Gaussian chaos or polynomials in sub-Gaussian or sub-exponential random
variables identify a wealth of different tail levels which scale with
different tensor-type norms of the coefficient tensor $A$ (or more generally
of the derivatives of the polynomial $f$ under consideration). By contrast,
Propositions 6 and 7 only involve much fewer levels of tail decay and always
the Hilbert–Schmidt norm, which is the largest norm among the family of norms
which appears in those results.

For instance, in the case of $d=2$ the by now classical Hanson–Wright
inequality (cf. e. g. [HW71, Wri73, RV13]) states that if the
$X_{1},\ldots,X_{n}$ are independent sub-Gaussian random variables with mean
$0$ and variance $1$,

$\mathbb{P}(|f_{2,A}(X)-\mathbb{E}[f_{2,A}(X)]|\geq t)\leq
2\exp\Big{(}-\frac{1}{C}\min\Big{(}\Big{(}\frac{t}{\lVert
A\rVert_{\mathrm{HS}}}\Big{)}^{2},\frac{t}{\lVert
A\rVert_{\mathrm{op}}}\Big{)}\Big{)},$

where $\lVert A\rVert_{\mathrm{op}}$ is the operator norm of $A$ and $C>0$ is
some constant which only depends on the sub-Gaussian norms of the $X_{i}$.
Analogues of the Hanson-Wright inequality for sub-exponential random vectors
can be found in [GSS21, Sam23]. While identifying two regimes of tail decay,
the “heavy-tailed” Hanson–Wright type inequality (15) does not yet seem to be
optimal in the sense of involving different norms (of the matrix $A$) and
scaling regimes. One may invoke a sub-Gaussian part by treating the diagonal
part with the Fuk–Nagaev result instead of Proposition 6, which yields the
bound

$\displaystyle\begin{split}&\mathbb{P}(|f_{2,A}(X)|\geq t)\leq
C_{\alpha}\max\Big{(}\exp\Big{(}-C_{\alpha}^{\prime}\Big{(}\frac{t}{\lVert
a^{\mathrm{d}}\rVert_{\ell^{2}}b^{2}}\Big{)}^{2}\Big{)},\\\
&\qquad\log\Big{(}\frac{t}{{\lVert
a^{\mathrm{d}}\rVert_{\ell^{\alpha/2}}}b^{2}}\Big{)}\Big{(}\frac{{\lVert
a^{\mathrm{d}}\rVert_{\ell^{\alpha/2}}}b^{2}}{t}\Big{)}^{\alpha/2},\log^{2}\Big{(}\frac{t}{\lVert
A^{\mathrm{od}}\rVert_{\mathrm{HS}}b^{2}}\Big{)}\Big{(}\frac{\lVert
A^{\mathrm{od}}\rVert_{\mathrm{HS}}b^{2}}{t}\Big{)}^{\alpha}\Big{)},\end{split}$
(16)

where $a^{\mathrm{d}}$ is the vector consisting of the diagonal entries of
$A$. However, even if (16) improves upon (15) it should still be non-optimal.

In principle, an obvious strategy would be to sharpen the proof of Proposition
[HMO97] by the results derived in [KL15] (similar to the methods used in
[GSS21]). However, even if we possibly arrive at $L^{p}$ bounds of the form

$\lVert
f_{d,A}(X)\rVert_{L^{p}}\leq\sum_{i}C(i,\alpha,A)\varphi_{i,\alpha}(p)$

for suitable coefficients $C(i,\alpha,A)$ and $p$-dependent functions
$\varphi_{i,\alpha}(p)$, it is again hard to derive multilevel tail
inequalities from them for similar reasons as sketched in the case of $d=1$.
Indeed, in the case of exponential-type tails, $\varphi(p)\approx p^{\kappa}$
for suitable exponents $\kappa>0$, while in the heavy-tailed situation
$\varphi(p)\approx(\kappa/(\kappa-p))^{1/p}$ for $\kappa\approx\alpha/i$. In
particular, these inequalities will only hold for $p\in[p,\alpha/d)$, so that
we can only access $p$ in a region where most of the $L^{p}$ bounds are
basically meaningless.

Note in passing that for this reason, the proof of the tail bound in
Proposition 7 involves a union bound (thus splitting the functional into
several parts), which is not necessary in the case of sub-exponential random
variables. On the other hand, slightly generalizing the case of $d=2$ in
Proposition 7, observe that if $X$ is a positive random variable such that

$P(X\geq
x)=\max\Big{(}\frac{b^{2\alpha}}{x^{2\alpha}},\frac{a^{\alpha}}{x^{\alpha}}\Big{)}$
(17)

for all $x\geq b$, where $0<a<b<\infty$ and $\alpha>0$, we have

$\mathbb{E}[X^{p}]\leq
b^{p}\frac{2\alpha}{2\alpha-p}+a^{p}\frac{\alpha}{\alpha-p}.$ (18)

Indeed, an easy calculation yields

$\displaystyle\mathbb{E}[X^{p}]$
$\displaystyle=p\int_{0}^{b}x^{p-1}dx+p\int_{b}^{b^{2}/a}x^{p-1}\frac{b^{2\alpha}}{x^{2\alpha}}dx+p\int_{b^{2}/a}^{\infty}x^{p-1}\frac{a^{\alpha}}{x^{\alpha}}dx$
$\displaystyle=b^{p}+\frac{pb^{2\alpha}}{p-2\alpha}\Big{(}\frac{b^{2p-4\alpha}}{a^{p-2\alpha}}-b^{p-2\alpha}\Big{)}-\frac{pa^{\alpha}}{p-\alpha}\frac{b^{2p-2\alpha}}{a^{p-\alpha}}$
$\displaystyle=b^{p}\frac{2\alpha}{2\alpha-p}+a^{p}\frac{b^{2p-2\alpha}}{a^{2p-2\alpha}}\frac{p}{2\alpha-p}\frac{\alpha}{\alpha-p}\leq
b^{p}\frac{2\alpha}{2\alpha-p}+a^{p}\frac{\alpha}{\alpha-p}.$

However, it is not clear whether (18) also implies (17) (maybe up to a
logarithmic error), which could serve as a starting point for more precise
tail bounds for chaos in heavy-tailed random variables.

## References

  * [Akh65] N.. Akhiezer  “The classical moment problem and some related questions in analysis.” Translated by N. Kemmer  Hafner Publishing Co., New York,, 1965, pp. x+253
  * [AW15] Radosław Adamczak and Paweł Wolff  “Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order”  In _Probab. Theory Related Fields_ 162.3-4, 2015, pp. 531–586  DOI: 10.1007/s00440-014-0579-3
  * [BGT87] N.. Bingham, C.. Goldie and J.. Teugels  “Regular variation” 27, Encyclopedia of Mathematics and its Applications  Cambridge University Press, Cambridge, 1987, pp. xx+491  DOI: 10.1017/CBO9780511721434
  * [Che09] Christophe Chesneau  “A tail bound for sums of independent random variables and application to the Pareto distribution”  In _Appl. Math. E-Notes_ 9, 2009, pp. 300–306
  * [FN71] D.. Fuk and S.. Nagaev  “Probabilistic inequalities for sums of independent random variables”  In _Teor. Verojatnost. i Primenen._ 16, 1971, pp. 660–675
  * [Fuk73] D.. Fuk  “Certain probabilistic inequalities for martingales”  In _Sibirsk. Mat. Ž._ 14, 1973, pp. 185–193239
  * [GSS21] Friedrich Götze, Holger Sambale and Arthur Sinulis  “Concentration inequalities for polynomials in $\alpha$-sub-exponential random variables”  In _Electron. J. Probab._ 26, 2021, pp. Paper No. 4822  DOI: 10.1214/21-ejp606
  * [HMO97] Paweł Hitczenko, Stephen J. Montgomery-Smith and Krzysztof Oleszkiewicz  “Moment inequalities for sums of certain independent symmetric random variables”  In _Studia Math._ 123.1, 1997, pp. 15–42  URL: http://eudml.org/doc/216377
  * [HW71] David L. Hanson and Farroll T. Wright  “A bound on tail probabilities for quadratic forms in independent random variables”  In _Ann. Math. Statist._ 42, 1971, pp. 1079–1083  DOI: 10.1214/aoms/1177693335
  * [Kaw72] T. Kawata  “Fourier analysis in probability theory” Probability and Mathematical Statistics, No. 15  Academic Press, New York-London, 1972, pp. xii+668
  * [KL15] K. Kolesko and R. Latała  “Moment estimates for chaoses generated by symmetric random variables with logarithmically convex tails”  In _Statist. Probab. Lett._ 107, 2015, pp. 210–214  DOI: 10.1016/j.spl.2015.08.019
  * [Kwa87] Stanisław Kwapień  “Decoupling inequalities for polynomial chaos”  In _Ann. Probab._ 15.3, 1987, pp. 1062–1071  DOI: 10.1214/aop/1176992081
  * [Lat06] Rafał Latała  “Estimates of moments and tails of Gaussian chaoses”  In _Ann. Probab._ 34.6, 2006, pp. 2315–2331  DOI: 10.1214/009117906000000421
  * [MP16] M. Matsui and Z. Pawlas  “Fractional absolute moments of heavy tailed distributions”  In _Braz. J. Probab. Stat._ 30.2, 2016, pp. 272–298  DOI: 10.1214/15-BJPS280
  * [Nag79] S.. Nagaev  “Large deviations of sums of independent random variables”  In _Ann. Probab._ 7.5, 1979, pp. 745–789  URL: http://links.jstor.org/sici?sici=0091-1798(197910)7:5%3C745:LDOSOI%3E2.0.CO;2-S&origin=MSN
  * [PG99] Víctor H. Peña and Evarist Giné  “Decoupling”, Probability and its Applications (New York)  Springer-Verlag, New York, 1999, pp. xvi+392
  * [RV13] Mark Rudelson and Roman Vershynin  “Hanson-Wright inequality and sub-Gaussian concentration”  In _Electron. Commun. Probab._ 18, 2013, pp. no. 829  DOI: 10.1214/ECP.v18-2865
  * [Sam23] Holger Sambale  “Some notes on Concentration for $\alpha$-subexponential random variables”  In _High dimensional probability IX—the Ethereal volume_ 80, Progr. Probab.  Birkhäuser/Springer, Cham, 2023, pp. 167–192  URL: https://doi.org/10.1007/978-3-031-26979-0_7
  * [Ver18] Roman Vershynin  “High-dimensional probability” An introduction with applications in data science, With a foreword by Sara van de Geer 47, Cambridge Series in Statistical and Probabilistic Mathematics  Cambridge University Press, Cambridge, 2018, pp. xiv+284  DOI: 10.1017/9781108231596
  * [Wri73] Farroll T. Wright  “A bound on tail probabilities for quadratic forms in independent random variables whose distributions are not necessarily symmetric”  In _Ann. Probability_ 1.6, 1973, pp. 1068–1070  URL: https://projecteuclid.org/euclid.aop/1176996815