# Non-local in time telegraph equations and very slowly growing variances

Francisco Alegría  Instituto de Ciencias Físicas y Matemáticas. Facultad de
Ciencias. Universidad Austral de Chile, Valdivia, Chile. Departamento de
Matemática y Estadística. Facultad de Ingeniería y Ciencias. Universidad de La
Frontera, Temuco, Chile<EMAIL_ADDRESS>and  Juan C. Pozo
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Las
Palmeras 3425, Ñuñoa, Santiago, Chile. Partially supported by Fondecyt grant
1181084<EMAIL_ADDRESS>
###### Abstract.

In this paper we consider a class of non-local in time telegraph equations.
Recently, in [16] it has been proved that the fundamental solutions of such
equations can be interpreted as the probability density function of a
stochastic process. We study the asymptotic behavior of the variance of this
process at large and short times. In this context, we develop a method to
construct new examples such the variance has a slowly growth behavior,
extending the earlier results. Finally, we show that our approach can be
adapted to define new integro-differential operators which are interesting in
sub-diffusion processes.

###### 2010 Mathematics Subject Classification:

Primary 45K05, 34K25, 35R10

##  1\. Introduction

The study of non-local in time differential equations have received a lot of
attention in last years due to its deep connection with non-local transport
phenomena, control of stochastic jump processes, description of anomalous
diffusion in physics and memory effects in parabolic equations, see [8, 9, 20,
16, 15, 10] and references therein.

Let $k\in L_{1,loc}(\mathbb{R}_{+})$ and $\eta,\nu$ be positive constants. In
this paper we consider the non-local in time telegraph equation

$\partial_{t}^{2}\bigl{(}k\ast k\ast
u(\cdot,z)\bigr{)}(t)+\eta\,\partial_{t}\bigl{(}k\ast
u(\cdot,z)\bigr{)}(t)-\nu\,\partial_{z}^{2}u(t,z)=0,\ t>0,\ z\in\mathbb{R},$
(1.1)

which has been recently proposed in [16]. We are interested into study some
properties of the fundamental solution of (1.1). For this reason we consider
the initial conditions

$u(0,z)=\delta_{0}(z)\quad\text{and}\quad\partial_{t}u(0,z)=0,\
z\in\mathbb{R},$ (1.2)

where the $\partial_{t}u(0,z)=0$ must be considered whenever it exists. We
point out that the fundamental solution of (1.1) has been already studied in
[16, Section 4] assuming that $k$ is a kernel of type $(\mathcal{PC})$, which
means that the following condition is satisfied.

  1. $(\mathcal{PC})$

$k\in L_{1,loc}(\mathbb{R}_{+})$ is nonnegative, nonincreasing, and there is
$\ell\in L_{1,loc}(\mathbb{R}_{+})$ such that $(k*\ell)=1$ in $(0,\infty)$. In
this case we also write $(k,\ell)\in(\mathcal{PC})$.

In this paper we also assume that $k$ is a kernel of type $(\mathcal{PC})$ and
we extend some of the results obtained in [16].

It is worth mentioning that the most classical example of a pair
$(k,\ell)\in(\mathcal{PC})$ is given by $(g_{1-\alpha},g_{\alpha})$ with
$\alpha\in(0,1)$, where $g_{\beta}$ with $\beta>0$ is the standard notation
for the function

$g_{\beta}(t)=\dfrac{t^{\beta-1}}{\Gamma(\beta)},\quad t>0.$

In this case (1.1)-(1.2) takes the form of the time fractional telegraph
equation

$\displaystyle\partial_{t}^{2\alpha}u(t,z)+\eta\,\partial_{t}^{\alpha}u(t,z)-\nu\,\partial_{z}^{2}u(t,z)$
$\displaystyle=0,\ t>0,\ z\in\mathbb{R},$ (1.3) $\displaystyle
u(0,z)=\delta_{0}(z)\quad\text{and}\quad\partial_{t}u(0,z)$ $\displaystyle=0,\
z\in\mathbb{R},$ (1.4)

which have been extensively studied, for instance see [14, 19] and references
therein. In this case the initial condition on the derivative of $u$ only must
be considered if $\alpha\in(\frac{1}{2},1]$. The solution $U_{\alpha}$ of
(1.3)-(1.4) exhibits several interesting properties, one of them being that
$U_{\alpha}$ can be viewed as the probability density function of a stochastic
process denoted by $X_{\alpha}(t)$, (cf. [14]). Moreover, the variance of the
process $X_{\alpha}(t)$ is given by

$\textrm{Var}[X_{\alpha}(t)]=2\nu t^{2\alpha}E_{\alpha,2\alpha+1}(-\eta
t^{\alpha}),\ t\geq 0,$

where $E_{\alpha,2\alpha+1}$ is an example of the so-called Mittag-Leffler
function of two parameters, see [5, Chapter 4] or [12, Appendix E] for several
properties and results of this function.

Motivated by this result, it has been established in [16, Theorem 1.1] that
for every $(k,\ell)\in(\mathcal{PC})$ the fundamental solution $U(t,z)$ of
(1.1) can be interpreted as a probability density function on $\mathbb{R}$ and
there exists a process, denoted by $X(t)$, whose distribution coincides with
$U(t,\cdot)$ for all time $t>0$. Further, the corresponding variance of this
process is positive, increasing on $(0,\infty)$ and it is given by

$\mathrm{Var}[X(t)]=2\nu(1\ast r_{\eta}\ast\ell)(t),\quad t\geq 0,$ (1.5)

where $r_{\eta}$ is the integrated resolvent associated to $\ell$, (see
Definition 2.2 below).

The importance of knowing the variance of a stochastic process relies in the
fact that allows measuring how far the set of random values are spread out
from their average. Although (1.5) provides an exact representation of
$\textrm{Var}[X(t)]$, in general the function $r_{\eta}$ cannot be computed
explicitly. In consequence, get more analytical properties of the variance
could be a hard task. For example, we are particularly interested in knowing
how slow its growth rate can be. In this context, in [16, Section 4] it has
been proved that the asymptotic behavior of $\textrm{Var}[X(t)]$ could be of
very different kinds, which are e.g., exponential, algebraic and logarithmic.
To the best of our knowledge, the slowest growth rate of $\textrm{Var}[X(t)]$
known in the literature is logarithmic, and this rate is obtained considering

$k(t)=\int_{0}^{1}\dfrac{t^{\alpha-1}}{\Gamma(\alpha)}d\alpha,\quad\text{and}\quad\ell(t)=\int_{0}^{\infty}\dfrac{e^{-st}}{1+s}ds,\quad
t>0.$ (1.6)

So a basic question arises: Is there a pair $(k,\ell)\in(\mathcal{PC})$ such
that the variance grows slower than a logarithmic function at infinity? In
this paper we show that the answer to this question is affirmative. Indeed, we
develop a method to construct infinitely many examples of pairs
$(k,\ell)\in(\mathcal{PC})$ answering affirmatively this question. These
examples cover a broad range of slowly growing functions.

The paper is organized as follows. In Section 2, we give some preliminaries
concept that we need to our work. Section 3 is the central part of this work.
In this section we prove the main results of our work. In Section 4, we
explain why this method could be interesting in another contexts such as sub-
diffusion processes.

##  2\. Preliminaries

The Laplace transform of a subexponential function
$f\colon[0,\infty)\to\mathbb{R}$ defined on the half-line will be denoted by

$\widehat{f}(\lambda)=\int_{0}^{\infty}e^{-\lambda t}f(t)dt,\quad\lambda>0.$

######  Definition 2.1.

An infinitely differentiable function $f\colon(0,\infty)\to\mathbb{R}$ is
called completely monotone if $(-1)^{n}f^{(n)}(\lambda)\geq 0$ for all
$n\in\mathbb{N}_{0}$ and $\lambda>0$. An infinitely differentiable function
$f\colon(0,\infty)\to\mathbb{R}$ is called Bernstein function if
$f(\lambda)\geq 0$ for $\lambda>0$ and $f^{\prime}$ is a completely monotonic
function. We will denote the class of completely monotonic functions by
$(\mathcal{CM})$, and the class of Bernstein functions will be denoted by
$(\mathcal{BF})$.

A detailed collection of the most important properties and results about the
classes $(\mathcal{CM})$ and $(\mathcal{BF})$ can be found in [7, Chapter 3]
and [18].

######  Definition 2.2.

Let $\eta\in\mathbb{C}$ and $\ell\in L_{1,loc}(\mathbb{R}_{+})$. The solution
$r_{\eta}\colon\mathbb{R}_{+}\to\mathbb{C}$ of the scalar Volterra equation

$r_{\eta}(t)+\eta(r_{\eta}\ast\ell)(t)=\ell(t),\quad t>0,$ (2.1)

is called integral scalar resolvent associated to $\ell$.

The integral scalar resolvent $r_{\eta}$ has proved to be essential for the
treatment of non-homogeneous Volterra equations, see [2, 3, 6, 15] and
references therein.

We remark that if $\eta\in\mathbb{R}_{+}$ then the condition
$(k,\ell)\in(\mathcal{PC})$ implies that $r_{\eta}$ is positive, see [17,
Proposition 4.5]. Further, if additionally $\ell$ is completely monotonic then
$r_{\eta}$ is completely monotonic as well, (see [17, Lemma 4.1]). More
properties and results about $r_{\eta}$ can be found in [6, Chapter 5].

Now we introduce a version of Karamata-Feller Tauberian theorem. The proof can
be found in [1, Section 1.7, Chapter I] or [4, Chapter XIII].

######  Definition 2.3.

Let $\varrho\in\mathbb{R}$. We say that a function $L:(0,\infty)\to(0,\infty)$
is a regularly varying function of index $\varrho$ if for every fixed $x>0$ we
have that

$\lim_{t\to\infty}\dfrac{L(tx)}{L(t)}=x^{\varrho}.$

The regularly varying functions at infinity of index $\varrho=0$ are called
slowly varying functions.

######  Remark 2.4.

Let $F\colon(0,\infty)\to(0,\infty)$ be a regularly varying function at
infinity of index $\varrho$. It follows from [1, Theorem 1.4.1] that there is
a slowly varying function $L\colon(0,\infty)\to(0,\infty)$ such that
$F(x)=x^{\varrho}L(x)$ for $x>0$.

######  Theorem 2.5 (Karamata-Feller Theorem).

Let $L_{1},L_{2}\colon(0,\infty)\to(0,\infty)$ slowly varying functions. Let
$\beta>0$ and $w:(0,\infty)\rightarrow\mathbb{R}$ be a monotone function whose
Laplace transform $\widehat{w}(\lambda)$ exists for all
$\lambda\in\mathbb{C}_{+}$. Then

$\widehat{w}(\lambda)\sim\,\frac{1}{\lambda^{\beta}}\,L_{1}(\lambda),\
\text{as}\;\lambda\to\infty,\ \;\mbox{if and only if}\
\;w(t)\sim\frac{t^{\beta-1}}{\Gamma(\beta)}L_{1}\left(\dfrac{1}{t}\right),\
\mbox{as}\;t\to 0^{+},$

and

$\widehat{w}(\lambda)\sim\,\frac{1}{\lambda^{\beta}}\,L_{2}\left(\frac{1}{\lambda}\right),\
\mbox{as}\ \lambda\to 0^{+},\ \;\mbox{if and only if}\ \
w(t)\sim\frac{t^{\beta-1}}{\Gamma(\beta)}L_{2}(t),\ \mbox{as}\ t\to\infty.$

Here the approaches are on the positive real axis and the notation $f(t)\sim
g(t)$ as $t\to t_{*}$ means that $\lim_{t\to t_{*}}f(t)/g(t)=1$.

##  3\. Very slowly growing variances

We begin this section recalling the result established in [16, Theorem 1.1]
which is the base of our research.

######  Theorem 3.1.

Let $\eta,\nu$ positive constants and $(k,\ell)\in(\mathcal{PC})$. The
variance of the process $X(t)$, whose density function coincides with the
fundamental solution of (1.1), satisfies the following Volterra equation

$\textrm{Var}[X(t)]+\eta(\ell\ast\textrm{Var}[X(\cdot)])(t)=2\nu(1\ast\ell\ast\ell)(t),\quad
t\geq 0.$ (3.1)

Further, $\textrm{Var}[X(t)]$ is positive and increasing on $(0,\infty)$ and
it satisfies the formula

$\textrm{Var}[X(t)]=2\nu(1\ast\ell\ast r_{\eta})(t),\quad t\geq 0.$ (3.2)

Since (3.2) is given by means of convolutions, we can use Theorem 2.5 to study
the behavior of $\mathrm{Var}[X(t)]$ at large times and short times. The
following theorem is the main results of our work.

######  Theorem 3.2.

Let $(k,\ell)\in(\mathcal{PC})$. If the Laplace transform $\widehat{\ell}$ is
a regularly varying function of index $\varrho_{1}<\frac{1}{2}$, then

$\mathrm{Var}[X(t)]\sim\dfrac{2\nu}{\Gamma(1-2\varrho_{1})}\left(\widehat{\ell}\bigl{(}t^{-1}\bigr{)}\right)^{2},\
\text{as}\ t\to 0^{+}.$ (3.3)

Further, if the function $t\mapsto\widehat{\ell}(t^{-1})$ is a regularly
varying function of index $\varrho_{2}>-1$, then

$\mathrm{Var}[X(t)]\sim\dfrac{2\nu}{\eta\,\Gamma(1+\varrho_{2})}\,\widehat{\ell}\Bigl{(}\frac{1}{t}\Bigr{)},\quad\text{as}\quad
t\to\infty.$ (3.4)

###### Proof.

Set $V(t)=\mathrm{Var}[X(t)]$ for $t\geq 0$. It follows from (3.2) that the
Laplace transform of $V$ is given by

$\widehat{V}(\lambda)=\dfrac{2\nu}{\lambda}\cdot\dfrac{\widehat{\ell}(\lambda)}{1+\eta\widehat{\ell}(\lambda)}\cdot\widehat{\ell}(\lambda),\quad\lambda>0.$

Since $\widehat{\ell}(\lambda)\to 0$ as $\lambda\to\infty$, we have that

$\widehat{V}(\lambda)\sim\dfrac{2\nu}{\lambda^{1-2\varrho_{1}}}L_{1}(\lambda),\quad\text{as}\quad\lambda\to\infty,$

where $L_{1}(t)=t^{-2\varrho_{1}}\bigl{(}\widehat{\ell}(t)\bigr{)}^{2}$. Since
$\widehat{\ell}$ is a regular variation function of index $\varrho_{1}$, by
Remark 2.4, we have that $L_{1}$ is a slowly varying function. Since
$\frac{1}{2}>\varrho_{1}$, it follows from Theorem 2.5 that

$\mathrm{Var}[X(t)]\sim\dfrac{2\nu}{\Gamma(1-2\varrho_{1})}\left(\widehat{\ell}\bigl{(}t^{-1}\bigr{)}\right)^{2},\
\text{as}\ t\to 0^{+}.$

On the other hand, we note that

$\widehat{V}(\lambda)=\dfrac{2\nu}{\lambda^{1+\varrho_{2}}}L_{2}\left(\dfrac{1}{\lambda}\right),\quad\lambda>0.$

where $L_{2}\colon(0,\infty)\to(0,\infty)$ is defined by

$L_{2}(t)=\frac{\widehat{\ell}(t^{-1})}{1+\eta\widehat{\ell}(t^{-1})}\cdot\frac{\widehat{\ell}(t^{-1})}{t^{\varrho_{2}}},\quad\text{for}\quad
t>0.$

Since $t\mapsto\widehat{\ell}(t^{-1})$ is a regularly varying function of
index $\varrho_{2}$, by Remark 2.4 we have that $L_{2}$ is a slowly varying
function. Furthermore,

$L_{2}(\lambda^{-1})\sim\frac{1}{\eta\lambda^{\varrho_{2}}}\widehat{\ell}(\lambda^{-1})\quad\text{as}\quad\lambda\to
0^{+}.$

Since $\varrho_{2}>-1$, it follows from Theorem 2.5 that

$\mathrm{Var}[X(t)]\sim\dfrac{2\nu}{\eta}\cdot\dfrac{t^{\varrho_{2}}}{\Gamma(1+\varrho_{2})}L_{2}(t)\sim\dfrac{2\nu}{\eta\,\Gamma(1+\varrho_{2})}\,\widehat{\ell}\Bigl{(}\frac{1}{t}\Bigr{)},\quad\text{as}\quad
t\to\infty.$

∎

We remark that the conditions of Theorem 3.2 are satisfied for many pairs of
functions $(k,\ell)\in(\mathcal{PC})$. For instance, all the examples
presented in [16, Section 5] satisfy these conditions. For the sake of brevity
of the text we omit their proof.

######  Example 3.3.

(Time fractional telegraph equation). If $k=g_{1-\alpha}$ with
$\alpha\in(0,1)$, then

$\textrm{Var}[X(t)]\sim\dfrac{2\nu}{\eta}\dfrac{t^{\alpha}}{\Gamma(1+\alpha)},\quad\text{as}\quad
t\to\infty,$

and

$\textrm{Var}[X(t)]\sim
2\nu\,\dfrac{t^{2\alpha}}{\Gamma(1+2\alpha)},\quad\text{as}\quad t\to 0^{+}.$

######  Example 3.4.

(Sum of two time fractional derivatives). If $k=g_{1-\alpha}+g_{1-\beta}$ with
$0<\alpha<\beta<1$, then

$\textrm{Var}[X(t)]\sim\dfrac{2\nu}{\eta}\dfrac{t^{\alpha}}{\Gamma(1+\alpha)},\quad\text{as}\quad
t\to\infty,$

and

$\textrm{Var}[X(t)]\sim
2\nu\,\dfrac{t^{2\beta}}{\Gamma(1+2\beta)},\quad\text{as}\quad t\to 0^{+}.$

######  Example 3.5.

(Time-fractional telegraph equation with Mittag-Leffler weight). If
$k(t)=t^{\beta-1}E_{\alpha,\beta}(-\omega t^{\alpha})$ with $0<\alpha,\beta<1$
and $\omega>0$, then

$\textrm{Var}[X(t)]\sim\dfrac{2\nu}{\eta}\dfrac{\omega\,t^{\alpha+1-\beta}}{\Gamma(2+\alpha-\beta)},\quad\text{as}\quad
t\to\infty,$

and

$\textrm{Var}[X(t)]\sim
2\nu\,\dfrac{t^{2-2\beta}}{\Gamma(2+\alpha-\beta)},\quad\text{as}\quad t\to
0^{+}.$

######  Example 3.6.

(Time distributed order telegraph equation). If $\displaystyle
k(t)=\int_{a}^{b}g_{\alpha}(t)d\alpha$, with $0\leq a<b\leq 1$, then

$\textrm{Var}[X(t)]\sim\dfrac{2\nu}{\eta}\dfrac{t^{1-b}\log(t)}{\Gamma(2-b)},\quad\text{as}\quad
t\to\infty,$

and

$\textrm{Var}[X(t)]\sim
2\nu\,\dfrac{t^{1+b-2a}\bigl{(}\log(t)\bigr{)}^{2}}{\Gamma(2-b)},\quad\text{as}\quad
t\to 0^{+}.$

######  Remark 3.7.

We note that if $b=1$ and $a\in(0,1)$ in Example 3.6 then we have an infinity
family of processes whose variance behaves like a logarithmic function at
infinity.

The following example has not been established in the literature before. In
order to present it, we define recursively the functions $\Theta_{n}$ as
follows

$\Theta_{1}(t,x)=\int_{0}^{x}g_{\alpha}(t)d\alpha\quad\text{and}\quad\Theta_{n+1}(t,x)=\int_{0}^{x}\Theta_{n}(t,y)dy,\quad
t>0,\ x>0.$ (3.5)

Further, for $n\in\mathbb{N}$ we define the functions
$\theta_{n}\colon(0,\infty)\to(0,\infty)$ by

$\theta_{n}(t)=\Theta_{n}(t,1),\ \text{for}\ t>0.$ (3.6)

######  Example 3.8.

Let $n\in\mathbb{N}$. Consider $\displaystyle k=\theta_{n}$ where $\theta_{n}$
has been defined in (3.6), then

$\textrm{Var}[X(t)]\sim\dfrac{2\nu}{\eta}\bigl{(}\log(t)\bigr{)}^{n},\quad\text{as}\quad
t\to\infty,$

and

$\textrm{Var}[X(t)]\sim\nu\big{(}(n-1)!\bigr{)}^{2}\bigl{(}t\cdot\log(t)\bigr{)}^{2},\quad\text{as}\quad
t\to 0^{+}.$

###### Proof.

Let $n\in\mathbb{N}$. We note that $\theta_{n}$ is positive and locally
integrable on $\mathbb{R}_{+}$. Since $\alpha\in(0,1)$ we have that
$g_{\alpha}\in(\mathcal{CM})$. We recall that the class of completely
monotonic functions is closed under addition and pointwise limits, see [18,
Corollary 1.6 and Corollary 1.7]. Therefore, we have that
$\theta_{n}\in(\mathcal{CM})$ as well. It follows from [6, Theorem 5.4 and
Theorem 5.5] that there exists $\zeta_{n}\in(\mathcal{CM})$ such that

$\theta_{n}\ast\zeta_{n}=1.$ (3.7)

Consequently, we have that $(\theta_{n},\zeta_{n})\in(\mathcal{PC})$. Further,
for $\lambda>0$ and $x>0$ we note that
$\widehat{\Theta}_{1}(\lambda,x)=\dfrac{\lambda^{x}-1}{\lambda^{x}\log(\lambda)}$.
Hence, it follows from (3.5) and an inductive procedure that

$\widehat{\Theta}_{n}(\lambda,x)=\dfrac{1}{\lambda^{x}\bigl{(}\log(\lambda)\bigr{)}^{n}}\left((-1)^{n}+\lambda^{x}\sum_{k=1}^{n}\dfrac{(-1)^{k-1}x^{n-k}(\log(\lambda))^{n-k}}{(n-k)!}\right),\
\lambda>0,\ x>0,$

for $n\geq 2$. This in turn implies that

$\widehat{\theta}_{n}(\lambda)=\dfrac{1}{\lambda\bigl{(}\log(\lambda)\bigr{)}^{n}}\left((-1)^{n}+\lambda\sum_{k=1}^{n}\dfrac{(-1)^{k-1}(\log(\lambda))^{n-k}}{(n-k)!}\right),\quad\lambda>0,$

and

$\widehat{\zeta}_{n}(\lambda)=\dfrac{\bigl{(}\log(\lambda)\bigr{)}^{n}}{\displaystyle(-1)^{n}+\lambda\sum_{k=1}^{n}\dfrac{(-1)^{k-1}(\log(\lambda))^{n-k}}{(n-k)!}},\quad\lambda>0,\
$

for $n\geq 2$. We note that $\widehat{\zeta}_{n}(\lambda)$ can be rewritten as
follows

$\widehat{\zeta}_{n}(\lambda)=\dfrac{\log(\lambda)}{\displaystyle\frac{(-1)^{n}}{\big{(}\log(\lambda)\big{)}^{n-1}}+\frac{\lambda}{(n-1)!}+\lambda\sum_{k=2}^{n}\dfrac{(-1)^{k-1}}{(n-k)!(\log(\lambda))^{k-1}}},\quad\lambda>0$

Hence, $\widehat{\zeta}_{n}(\lambda)\sim\frac{(n-1)!\log(\lambda)}{\lambda}$
as $\lambda\to\infty$. This implies that $\widehat{\zeta}_{n}$ is a regularly
varying function of index $\varrho=-1$. On the other hand, we have that

$\widehat{\zeta}_{n}(t^{-1})=\dfrac{(-1)^{n}\big{(}\log(t)\big{)}^{n}}{\displaystyle(-1)^{n}+\frac{(-1)^{n-1}}{t}\sum_{k=1}^{n}\frac{\big{(}\log(t)\big{)}^{n-k}}{(n-k)!}},\quad
t>0,$

which implies that $\widehat{\zeta}_{n}(t^{-1})\sim\big{(}\log(t)\big{)}^{n}$
as $t\to\infty$. Since the logarithmic function is regularly varying, and the
multiplication of regularly varying functions is regularly varying as well,
this in turn implies that the function $t\mapsto\zeta_{n}(t^{-1})$ is a
regularly varying function. Moreover,

$\widehat{\zeta}_{n}(t^{-1})=\dfrac{(-1)^{n}\,t\log(t)}{\displaystyle\frac{(-1)^{n}\,t}{(\log(t))^{n-1}}+\frac{(-1)^{n-1}}{(n-1)!}+\sum_{k=2}^{n}\frac{1}{(n-k)!\,\big{(}\log(t)\big{)}^{k-1}}},\quad
t>0,$

which implies that $\widehat{\zeta}_{n}(t^{-1})\sim(-1)(n-1)!\,t\log(t)$ as
$t\to 0^{+}$. Consequently, as direct application of Theorem 3.2 we obtain
that

$\textrm{Var}[X(t)]\sim\dfrac{2\nu}{\eta}\bigl{(}\log(t)\bigr{)}^{n},\quad\text{as}\quad
t\to\infty,$

and

$\textrm{Var}[X(t)]\sim\nu\big{(}(n-1)!\bigr{)}^{2}(\bigl{(}t\cdot\log(t)\bigr{)}^{2},\quad\text{as}\quad
t\to 0^{+}.$

∎

We remark that all the examples above are very interesting, However, as we
have mentioned in the Introduction, we are interested into find examples of
$(k,\ell)\in(\mathcal{PC})$ such that $\textrm{Var}[X(t)]$ grows slower than a
logarithmic function at infinity. To this end we prove the following result.

######  Lemma 3.9.

Let $f,g\in L_{1,loc}(\mathbb{R}_{+})$. Assume that $f,g\in(\mathcal{CM})$,
then there exists $h\in(\mathcal{CM})$ such that

$\widehat{h}(\lambda)=\widehat{f}(\lambda)\
\widehat{g}\bigl{(}\widehat{f}(\lambda)\bigr{)},\quad\lambda>0.$

###### Proof.

Consider $a=1\ast f$, $b\equiv 1$, $c=1\ast g$. It is clear that $a,b$ and $c$
are Bernstein functions. According to [17, Lemma 4.3], there exists a
Bernstein function $e\colon(0,\infty)\to(0,\infty)$ such that
$e(0^{+})=a(0^{+})=0$ and

$\widehat{e}(\lambda)=\widehat{a}(\lambda)\widehat{dc}\left(\dfrac{\widehat{a}(\lambda)}{\widehat{b}(\lambda)}\right),\
\lambda>0,$

where $\widehat{dc}$ stands for the Laplace transform of the measure $dc$
which is defined by

$\widehat{dc}(\lambda)=\int_{0}^{\infty}e^{-\lambda t}dc(t),\quad\lambda>0.$

Since $c=1\ast g$, we have that $dc(t)=g(t)dt$, and consequently
$\widehat{dc}=\widehat{g}$. Let us now define $h(t)=\frac{d}{dt}e(t)$ for
$t>0$. Since $e\in(\mathcal{BF})$ we have that $h\in(\mathcal{CM})$ and

$\widehat{h}(\lambda)=\lambda\widehat{e}(\lambda)-e(0^{+})=\widehat{f}(\lambda)\
\widehat{g}\bigl{(}\widehat{f}(\lambda)\bigr{)},\quad\lambda>0.$

∎

######  Corollary 3.10.

For all $\delta\in(0,1]$ there exists a pair
$(\phi^{\delta}_{1},\psi^{\delta}_{1})\in(\mathcal{PC})$ such that

$\widehat{\psi^{\delta}_{1}}\bigl{(}t^{-1}\bigr{)}\sim\bigl{(}\log(t)\bigr{)}^{\delta},\quad\text{as}\quad
t\to\infty,$

and

$\widehat{\psi^{\delta}_{1}}\bigl{(}t^{-1}\bigr{)}\sim(t\cdot\log(t^{-1})\bigr{)}^{\delta}\quad\text{as}\quad
t\to 0^{+}.$

If $\delta=1$, we will simply write $(\phi_{1},\psi_{1})\in(\mathcal{PC})$.

###### Proof.

Let $\delta\in(0,1]$. Consider the pair $(\theta,\zeta)\in(\mathcal{PC})$
given by

$\theta(t)=\int_{0}^{1}\dfrac{t^{\alpha-1}}{\Gamma(\alpha)}d\alpha,\quad\text{and}\quad\zeta(t)=\int_{0}^{\infty}\dfrac{e^{-st}}{1+s}ds,\quad
t>0.$ (3.8)

It is a well known fact that both $\theta$ and $\zeta$ are completely
monotonic functions. So, applying Lemma 3.9 with $f=\theta$ and
$g=g_{1-\delta}$, we conclude that there exists $h_{1\delta}\in(\mathcal{CM})$
such that

$\widehat{h_{1\delta}}(\lambda)=\left(\dfrac{\lambda-1}{\lambda\,\log(\lambda)}\right)^{\delta},\quad\lambda>0.$

Applying again Lemma 3.9 with $f=\zeta$ and $g=g_{1-\delta}$, we note that
there exists $h_{2\delta}\in(\mathcal{CM})$ such that

$\widehat{h_{2\delta}}(\lambda)=\left(\dfrac{\log(\lambda)}{\lambda-1}\right)^{\delta},\quad\lambda>0.$

Now define the kernels

$\phi^{\delta}_{1}=g_{1-\delta}\ast
h_{2\delta},\quad\text{and}\quad\psi^{\delta}_{1}=h_{1\delta}.$

Applying directly the Laplace transform, we have

$\widehat{\phi^{\delta}_{1}}(\lambda)=\dfrac{1}{\lambda}\left(\dfrac{\lambda-1}{\log(\lambda)}\right)^{\delta},\quad\text{and
}\quad\widehat{\psi^{\delta}_{1}}(\lambda)=\left(\dfrac{\log(\lambda)}{\lambda-1}\right)^{\delta},\quad\lambda>0.$
(3.9)

We note that by construction $\psi^{\delta}_{1}\in(\mathcal{CM})$. Hence, it
follows from [6, Theorem 5.4 and Theorem 5.5] that
$\phi^{\delta}_{1}\in(\mathcal{CM})$. In consequence
$(\phi^{\delta}_{1},\psi^{\delta}_{1})\in(\mathcal{PC})$. Furthermore, it
clear that

$\widehat{\psi^{\delta}_{1}}(t^{-1})=\left(\dfrac{\log(t)}{1-t^{-1}}\right)^{\delta},\quad
t>0,$

which in turn implies that

$\widehat{\psi^{\delta}_{1}}(t^{-1})\sim\bigl{(}\log(t)\bigr{)}^{\delta},\quad\text{as}\quad
t\to\infty.$

To compute the behavior $\widehat{\psi^{\delta}_{1}}(t^{-1})$ as $t\to 0^{+}$,
we rewrite this function as follows

$\widehat{\psi}_{\delta}\bigl{(}t^{-1}\bigr{)}=\left(\dfrac{t\,\log(t)}{t-1}\right)^{\delta},\quad
t>0,$

which implies that

$\widehat{\psi}_{\delta}\bigl{(}t^{-1}\bigr{)}\sim(-t\cdot\log(t)\bigr{)}^{\delta}\quad\text{as}\quad
t\to 0^{+}.$

∎

######  Remark 3.11.

Let $\delta\in(0,1]$ and $(\phi_{\delta},\psi_{\delta})\in(\mathcal{PC})$
given in Corollary 3.10. If $\delta=1$, then $(\phi_{1},\psi_{1})$ is the same
pair of functions defined by Kochubei in [11]. On the other hand, since
$\widehat{\psi^{\delta}_{1}}(t^{-1})\sim(\log(t))^{\delta}$ as $t\to\infty$ we
have that $\psi^{\delta}_{1}\notin L_{1}(\mathbb{R}_{+})$.

######  Example 3.12.

Let $\delta\in(0,1]$. Consider the pair
$\displaystyle(k,\ell)=(\phi^{\delta}_{1},\psi^{\delta}_{1})$ given in
Corollary 3.10, then

$\textrm{Var}[X(t)]\sim\dfrac{2\nu}{\eta}\bigl{(}\log(t)\bigr{)}^{\delta},\quad\text{as}\quad
t\to\infty,$

and

$\textrm{Var}[X(t)]\sim
2\nu\dfrac{\bigl{(}t\cdot\log(t^{-1})\bigr{)}^{2\delta}}{\Gamma(1+2\delta)},\quad\text{as}\quad
t\to 0^{+}.$

###### Proof.

It follows from (3.9) that $\widehat{\ell}$ is a regularly varying function of
index $\varrho=-\delta$. Further, since

$\widehat{\ell}(t^{-1})\sim(\log(t))^{\delta},\ \text{as}\ t\to\infty,$

it follows that $\widehat{\ell}(t^{-1})$ is a slowly varying function.
Therefore, Theorem 3.2 implies that

$\textrm{Var}[X(t)]\sim\dfrac{2\nu}{\eta}\bigl{(}\log(t)\bigr{)}^{\delta},\quad\text{as}\quad
t\to\infty,$

and

$\textrm{Var}[X(t)]\sim
2\nu\dfrac{\bigl{(}t\cdot\log(t^{-1})\bigr{)}^{2\delta}}{\Gamma(1+2\delta)},\quad\text{as}\quad
t\to 0^{+}.$

∎

######  Corollary 3.13.

For all $n\in\mathbb{N}$ there exists a pair
$(\phi_{n},\psi_{n})\in(\mathcal{PC})$ such that $\phi_{n}\in(\mathcal{CM})$
and

$\widehat{\psi_{n}}(t^{-1})\sim\log^{[n]}(t),\ \text{as}\ t\to\infty,$

and

$\widehat{\psi_{n}}(t^{-1})\sim t\,\bigl{(}\log(t^{-1})\bigr{)}^{n}\
\text{as}\ t\to 0^{+},$

where
$\log^{[n]}=\underbrace{\log\circ\log\circ\dots\circ\log}_{n-\rm{times}}$.

###### Proof.

The proof will be done by induction. For $n=1$, we consider the pair
$(\phi_{1},\psi_{1})\in(\mathcal{PC})$ given by Corollary 3.10 with
$\delta=1$.

Let $n>1$. Assume that there exists a pair
$(\phi_{n},\psi_{n})\in(\mathcal{PC})$ such that $\phi_{n}\in(\mathcal{CM})$
satisfying

$\widehat{\psi}_{n}(t^{-1})\sim\log^{[n]}(t),\ \text{as}\ t\to\infty.$

and

$\widehat{\psi_{n}}(t^{-1})\sim t\,\bigl{(}\log(t^{-1})\bigr{)}^{n},\
\text{as}\ t\to 0^{+}.$

Since $\phi_{n}\in(\mathcal{CM})$ and $\phi_{n}\ast\psi_{n}=1$, it follows
from [6, Theorem 5.4 and Theorem 5.5] that $\psi_{n}\in(\mathcal{CM})$. Hence,
applying Lemma 3.9 with $f=\psi_{n}$ and $g=g_{1-\delta}$ with
$\delta\in(0,1)$, we conclude that for all $\delta\in(0,1)$ there exists
$\omega_{n}^{\delta}\in(\mathcal{CM})$ such that

$\widehat{\omega_{n}^{\delta}}(\lambda)=\big{(}\widehat{\psi}_{n}(\lambda)\big{)}^{\delta},\quad\lambda>0.$

Using again [6, Theorem 5.4 and Theorem 5.5] we can establish the existence of
$\varphi^{\delta}_{n}\in(\mathcal{CM})$ such that
$\omega_{n}^{\delta}\ast\varphi_{n}^{\delta}=1$, which in turn implies that

$\widehat{\varphi_{n}^{\delta}}(\lambda)=\frac{1}{\lambda}\bigl{(}\widehat{\psi}_{n}(\lambda)\bigr{)}^{-\delta},\quad\lambda>0.$

Let us now define

$\phi_{n+1}(t)=\int_{0}^{1}\varphi_{n}^{\delta}(t)d\delta,\quad t>0.$ (3.10)

Since the class of completely monotonic functions is closed under addition and
pointwise limits, we conclude that $\phi_{n+1}\in(\mathcal{CM})$. In
consequence, it follows from [6, Theorem 5.4 and Theorem 5.5] that there
exists $\psi_{n+1}\in(\mathcal{CM})$ such that

$\psi_{n+1}\ast\varphi_{n+1}=1.$ (3.11)

Furthermore, we have that

$\widehat{\phi_{n+1}}(\lambda)=\int_{0}^{1}\widehat{\varphi^{\delta}_{n}}(\lambda)d\delta=\int_{0}^{1}\dfrac{1}{\lambda}\Bigl{(}\widehat{\psi}_{n}(\lambda)\Bigr{)}^{-\delta}d\delta=\dfrac{\widehat{\psi}_{n}(\lambda)-1}{\lambda\,\widehat{\psi}_{n}(\lambda)\,\log\bigl{(}\widehat{\psi}_{n}(\lambda)\bigr{)}},\quad\lambda>0.$

Since $(\phi_{n+1},\psi_{n+1})\in(\mathcal{PC})$, this in turn implies that

$\widehat{\psi_{n+1}}(\lambda)=\dfrac{\widehat{\psi}_{n}(\lambda)\log\bigl{(}\widehat{\psi}_{n}(\lambda)\bigr{)}}{\widehat{\psi}_{n}(\lambda)-1},\quad\lambda>0.$
(3.12)

Therefore, we have that

$\widehat{\psi_{n+1}}(t^{-1})=\dfrac{\widehat{\psi_{n}}(t^{-1})\log\bigl{(}\widehat{\psi_{n}}(t^{-1})\bigr{)}}{\widehat{\psi_{n}}(t^{-1})-1},\quad
t>0.$

We note that the inductive hypothesis implies that

$\dfrac{\widehat{\psi}_{n}(t^{-1})}{\widehat{\psi}_{n}(t^{-1})-1}\sim\dfrac{\log^{[n]}(t)}{\log^{[n]}(t)-1}\sim
1,\ \text{as}\ t\to\infty.$

In consequence we have

$\widehat{\psi_{n+1}}\bigl{(}t^{-1}\bigr{)}\sim\log\bigl{(}\widehat{\psi}_{n}(t^{-1})\bigr{)}\sim\log^{[n+1]}(t),\
\text{as}\ t\to\infty.$

On the other hand, since $\widehat{\psi_{n}}(t^{-1})\to 0$ as $t\to 0^{+}$, we
have that

$\displaystyle\widehat{\psi_{n+1}}\bigl{(}t^{-1}\bigr{)}$
$\displaystyle\sim\widehat{\psi}_{n}(t^{-1})\log\bigl{(}\widehat{\psi}_{n}(t^{-1})\bigr{)},\
\text{as}\quad t\to 0^{+},$

which by the inductive hypothesis is equivalent to

$\displaystyle\widehat{\psi_{n+1}}\bigl{(}t^{-1}\bigr{)}$
$\displaystyle\sim(t\,\bigl{(}\log(t^{-1})\bigr{)}^{n})\Bigl{(}\log(t)+\log(\log(t^{-1})^{n})\Bigr{)},\quad\text{as}\
t\to 0^{+}.$

We recall that

$\lim_{t\to 0^{+}}\dfrac{\log(\log(t^{-1})^{n})}{\log(t)}=0,$

for all $n\in\mathbb{N}$. Hence

$\widehat{\psi_{n+1}}(t^{-1})\sim t\,\bigl{(}\log(t^{-1})\bigr{)}^{n+1},\
\text{as}\ t\to 0^{+},$

and the proof is complete. ∎

######  Remark 3.14.

Let $n\in\mathbb{N}$ and $(\phi_{n},\psi_{n})\in(\mathcal{PC})$ given in
Corollary 3.13. Since $\widehat{\psi_{n}}(t^{-1})\sim\log^{[n]}(t)$ as
$t\to\infty$, we have that for all $n\in\mathbb{N}$ the functions
$\psi_{n}\notin L_{1}(\mathbb{R}_{+})$.

######  Example 3.15.

Let $n\in\\{2,3,\cdots\\}$. Consider pair $(k,\ell)=(\phi_{n},\psi_{n})$ given
in Corollary 3.13, then

$\textrm{Var}[X(t)]\sim\dfrac{2\nu}{\eta}\log^{[n]}(t),\quad\text{as}\quad
t\to\infty,$

and

$\textrm{Var}[X(t)]\sim\nu
t^{2}\,\bigl{(}\log(t^{-1})\bigr{)}^{2n},\quad\text{as}\quad t\to 0^{+},$

where
$\log^{[n]}=\underbrace{\log\circ\log\circ\dots\circ\log}_{n-\rm{times}}$.

###### Proof.

We note that for every $n\geq 2$ the function $\widehat{\psi_{n}}$ is a
regularly varying function of index $\varrho=-1$. Indeed, recall that
$\widehat{\psi}_{1}$ is defined by

$\widehat{\psi}_{1}(t)=\dfrac{\log(t)}{t-1},\quad t>0.$

Hence, it clear that $\widehat{\psi}_{1}$ is a regularly varying function of
index $\varrho=-1$. Assume now that $\widehat{\psi}_{n}$ is a regularly
varying function of index $\varrho=-1$. Since $\widehat{\psi_{n}}(t)\to 0$ as
$t\to\infty$, we have that

$t\mapsto\dfrac{\widehat{\psi}_{n}(t)}{\widehat{\psi}_{n}(t)-1},\ t>0,$

is a regularly varying function of index $\varrho=-1$. Moreover, by properties
of the logarithmic function we have that

$t\mapsto\log(\widehat{\psi}_{n}(t)),\ t>0,$

is a slowly varying function. Therefore, it follows from (3.12) that
$\widehat{\psi}_{n+1}$ is a regularly varying function of index $\varrho=-1$.
On the other hand, it follows from Corollary 3.13 that

$\widehat{\psi}_{n}(t^{-1})\sim\log^{[n]}(t),\ \text{as}\ t\to\infty.$ (3.13)

Therefore, $\widehat{\ell}(t^{-1})$ is a slowly varying function and Theorem
3.2 implies that

$\textrm{Var}[X(t)]\sim\dfrac{2\nu}{\eta}\log^{[n]}(t),\quad\text{as}\quad
t\to\infty,$

and

$\textrm{Var}[X(t)]\sim\nu
t^{2}\,\bigl{(}\log(t^{-1})\bigr{)}^{2n},\quad\text{as}\quad t\to 0^{+}.$

∎

######  Corollary 3.16.

For all $n\in\mathbb{N}$ and $\delta\in(0,1)$ there exists a pair
$(\phi_{n}^{\delta},\psi_{n}^{\delta})\in(\mathcal{PC})$ such that
$\phi^{\delta}_{n}\in(\mathcal{CM})$ and

$\widehat{\psi^{\delta}_{n}}(t^{-1})\sim\bigl{(}\log^{[n]}(t)\bigr{)}^{\delta},\
\text{as}\ t\to\infty,$

and

$\widehat{\psi_{n}}(t^{-1})\sim
t^{\delta}\,\bigl{(}\log(t^{-1})\bigr{)}^{\delta n}\ \text{as}\ t\to 0^{+}.$

###### Proof.

Let $n\in\mathbb{N}$ and $\delta\in(0,1)$. Consider the pair
$(\phi_{n},\psi_{n})\in(\mathcal{PC})$ given in Corollary 3.13. According to
Lemma 3.9 there are completely monotonic functions $h_{1n}^{\delta}$ and
$h_{2n}^{\delta}$ such that

$\widehat{h_{1n}^{\delta}}(\lambda)=\bigl{(}\widehat{\phi_{n}}(\lambda)\bigr{)}^{\delta},\quad\lambda>0,$

and

$\widehat{h_{2n}^{\delta}}(\lambda)=\bigl{(}\widehat{\psi_{n}}(\lambda)\bigr{)}^{\delta},\quad\lambda>0.$

Now define the kernels

$\phi_{n}^{\delta}=g_{1-\delta}\ast
h_{2n}^{\delta},\quad\text{and}\quad\psi_{n}^{\delta}=h_{1n}^{\delta}.$ (3.14)

Applying directly the Laplace transform, we have

$\widehat{\phi_{n}^{\delta}}(\lambda)=\dfrac{1}{\lambda}\left(\widehat{\phi_{n}}(\lambda)\right)^{\delta}\quad\text{and
}\quad\widehat{\psi_{n}^{\delta}}(\lambda)=\left(\widehat{\psi_{n}}(\lambda)\right)^{\delta},\quad\lambda>0.$

We note that by construction $\psi_{n}^{\delta}\in(\mathcal{CM})$. Hence, it
follows from [6, Theorem 5.4 and Theorem 5.5] that
$\phi^{\delta}_{n}\in(\mathcal{CM})$. In consequence
$(\phi^{\delta}_{n},\psi^{\delta}_{n})\in(\mathcal{PC})$. The rest of the
proof follows the same ideas of Corollary 3.13. ∎

######  Example 3.17.

Let $n\in\\{2,3,\cdots\\}$ and $\delta\in(0,1)$. Consider pair
$(k,\ell)=(\phi_{n},\psi_{n})$ given in Corollary 3.16, then

$\textrm{Var}[X(t)]\sim\dfrac{2\nu}{\eta}\bigl{(}\log^{[n]}(t)\bigr{)}^{\delta},\quad\text{as}\quad
t\to\infty,$

and

$\textrm{Var}[X(t)]\sim
2\nu\dfrac{t^{2\delta}\,\bigl{(}\log(t^{-1})\bigr{)}^{2n\delta}}{\Gamma(1+2\delta)},\quad\text{as}\quad
t\to 0^{+},$

where
$\log^{[n]}=\underbrace{\log\circ\log\circ\dots\circ\log}_{n-\rm{times}}$.

###### Proof.

Following the same ideas of Corollary 3.13 we note that for all
$n\in\mathbb{N}$ the functions $\widehat{\psi_{n}^{\delta}}$ are regularly
varying functions of index $\varrho=-\delta$ and
$\widehat{\psi_{n}^{\delta}}(t^{-1})$ is a slowly varying function. The rest
of the proof is similar to the proof of Example 3.15. ∎

##  4\. Application to ultra slow diffusion equations

There are another contexts where the pairs $(k,\ell)\in(\mathcal{PC})$ play a
fundamental role. For example, this type of functions has been successfully
exploited to study the so-called sub-diffusion processes, see [8, 15, 20] and
references therein. In order to fix some ideas and explain why the results
developed in this work could be interesting in the theory of subdiffusion
processes, we consider the following equation

$\displaystyle\partial_{t}(k\ast(u(\cdot,x)-u_{0}(x)))(t)-\Delta u(t,x)$
$\displaystyle=0,\quad t>0,x\in\mathbb{R}^{d},$ (4.1) $\displaystyle u(0,x)$
$\displaystyle=u_{0}(x),\quad x\in\mathbb{R}^{d},$ (4.2)

where $k$ is a kernel of type $(\mathcal{PC})$ and $u_{0}$ is a given
function. It has been proved in [8, Section 2] that the fundamental solution
of (4.1) coincides with the probability density function of a stochastic
process $X(t)$. Moreover, in [8, Lemma 2.1] it has been proved that the mean
square displacement $M(t)$ of such process is given by

$M(t)=2d(1\ast\ell)(t),\quad t\geq 0.$ (4.3)

The function $M(t)$ allows to measure how fast or slow is the diffusion of the
equation (4.1). It is worthwhile to mention that the slowest known rate of
growth of $M(t)$ follows a logarithmic law, for instance see [11] and
references therein. In such work Kochubei considered functions of the form

$k(t)=\int_{0}^{1}g_{\alpha}(t)\sigma(\alpha)d\alpha,\quad t>0,$

where $\sigma(t)$ with $t\in[0,1]$ is a continuous, non-negative function
different from zero on a set of positive measure.

Those equations of the form (4.1) whose mean square displacement follows a
logarithmic rate (or even slower) are known in the specialized literature as
ultra slow diffusion equations and they are strongly related with ultraslow
inverse subordinators, see [13].

Our work allows to study some ultra-slow diffusion equations which are not
considered before. For instance, the equation (4.1) with
$(k,\ell)=(\phi_{n},\psi_{n})$ for some $n\in\mathbb{N}$, where
$(\phi_{n},\psi_{n})$ has been defined in Corollary 3.13. According to (4.3)
we have that the Laplace transform of $M$ is given by

$\widehat{M}(\lambda)=\frac{2d}{\lambda}\,\widehat{\psi_{n}}(\lambda),\quad\lambda>0,$

which by Karamata-Feller’s Theorem 2.5 and the asymptotic behavior of
$\widehat{\psi_{n}}$ given in (3.13) imply that

$M(t)\sim 2d\,\log^{[n]}(t),\quad\text{as}\quad t\to\infty.$

In consequence, the mean square displacement $M(t)$ grows (at infinity) slower
than a logarithmic function.

This procedure can be applied to all the pairs of functions in
$(\mathcal{PC})$ defined in the Corollary 3.10 and Corollary 3.16. As far we
know, this implies that there are an infinite number of ultra-slow diffusion
equations which have not been analyzed before. All these new interesting
examples will be studied in a forthcoming work.

## References

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2D

    Two Dimensions
2G

    Second Generation
3D

    Three Dimensions
3G

    Third Generation
3GPP

    Third Generation Partnership Project
3GPP2

    Third Generation Partnership Project 2
4G

    Fourth Generation
5G

    Fifth Generation
AI

    Artificial Intelligence
AoA

    Angle of Arrival
AoD

    Angle of Departure
AR

    Augmented Reality
AP

    Access Point
AE

    Antenna Element
AC

    Anechoic Chamber
AUT

    Antenna Under Test
BER

    Bit Error Rate
BPSK

    Binary Phase-Shift Keying
BRDF

     Bidirectional Reflectance Distribution Function
BS

    Base Station
CA

    Carrier Aggregation
CDF

    Cumulative Distribution Function
CDM

    Code Division Multiplexing
CDMA

    Code Division Multiple Access
CPU

    Central Processing Unit
CUDA

    Compute Unified Device Architecture
CDF

    Cumulative Distribution Function
CI

    Confidence Interval
CVRP

    Constrained-View Radiated Power
CATR

    Compact Antenna Test Range
D2D

    Device-to-Device
DL

    Down Link
DS

    Delay Spread
DAS

    Distributed Antenna System
DKED

    double knife-edge diffraction
DUT

    Device Under Test
EDGE

    Enhanced Data rates for GSM Evolution
EIRP

    Equivalent Isotropic Radiated Power
eMBB

    Enhanced Mobile Broadband
eNodeB

    evolved Node B
ETSI

    European Telecommunications Standards Institute
ER

    Effective Roughness
E-UTRA

    Evolved UMTS Terrestrial Radio Access
E-UTRAN

    Evolved UMTS Terrestrial Radio Access Network
EF

    Electric Field
EMC

    Electromagnetic Compatibility
FDD

    Frequency Division Duplexing
FDM

    Frequency Division Multiplexing
FDMA

    Frequency Division Multiple Access
FoM

    Figure of Merit
FoV

    Field of View
FSA

    Frequency Selective Absorber
GI

    Global Illumination
GIS

    Geographic Information System
GO

    Geometrical Optics
GPU

    Graphics Processing Unit
GPGPU

    General Purpose Graphics Processing Unit
GPRS

    General Packet Radio Service
GSM

    Global System for Mobile Communication
GNSS

    Global Navigation Satellite System
GoF

    Goodness of Fit
H2D

    Human-to-Device
H2H

    Human-to-Human
HDRP

    High Definition Render Pipeline
HSDPA

    High Speed Downlink Packet Access
HSPA

    High Speed Packet Access
HSPA+

    High Speed Packet Access Evolution
HSUPA

    High Speed Uplink Packet Access
HPBW

    Half-Power Beamwidth
HA

    Horn Antenna
IEEE

    Institute of Electrical and Electronic Engineers
InH

    Indoor Hotspot
IMT

    International Mobile Telecommunications
IMT-2000

     International Mobile Telecommunications (IMT) 2000
IMT-2020

     IMT 2020
IMT-Advanced

     IMT Advanced
IoT

    Internet of Things
IP

    Internet Protocol
ITU

    International Telecommunications Union
ITU-R

     International Telecommunications Union (ITU) Radiocommunications Sector
IS-95

    Interim Standard 95
IES

    Inter-Element Spacing
IF

    Intermediate Frequency
KPI

    Key Performance Indicator
K-S

    Kolmogorov-Smirnov
LB

    Light Bounce
LIM

    Light Intensity Model
LOS

    Line-Of-Sight
LTE

    Long Term Evolution
LTE-Advanced

     Long Term Evolution (LTE) Advanced
LSCP

    Lean System Control Plane
LSI

    Light Source Intensity
M2M

    Machine-to-Machine
MatSIM

    Multi Agent Transport Simulation
METIS

    Mobile and wireless communications Enablers for Twenty-twenty Information Society
METIS-II

    Mobile and wireless communications Enablers for Twenty-twenty Information Society II
MIMO

    Multiple-Input Multiple-Output
mMIMO

    massive MIMO
mMTC

    massive Machine Type Communications
mmW

    millimeter-wave
MU-MIMO

    Multi-User MIMO
MMF

    Max-Min Fairness
MKED

    Multiple Knife-Edge Diffraction
MF

    Matched Filter
mmWave

    Millimeter Wave
NFV

    Network Functions Virtualization
NLOS

    Non-Line-Of-Sight
NR

    New Radio
NRT

    Non Real Time
NYU

    New York University
N75PRP

    Near-75-degrees Partial Radiated Power
NHPRP

    Near-Horizon Partial Radiated Power
O2I

    Outdoor to Indoor
O2O

    Outdoor to Outdoor
OFDM

    Orthogonal Frequency Division Multiplexing
OFDMA

    Orthogonal Frequency Division Multiple Access
OtoI

    Outdoor to Indoor
OTA

    Over-The-Air
PDF

    Probability Distribution Function
PDP

    Power Delay Profile
PHY

    Physical
PLE

    Path Loss Exponent
PRP

    Partial Radiated Power
QAM

    Quadrature Amplitude Modulation
QoS

    Quality of Service
RCSP

    Receive Signal Code Power
RAN

    Radio Access Network
RAT

    Radio Access Technology
RAN

    Radio Access Network
RMa

    Rural Macro-cell
RMSE

    Root Mean Square Error
RSCP

    Receive Signal Code Power
RT

    Ray Tracing
RX

    receiver
RMS

    Root Mean Square
Random-LOS

    Random Line-Of-Sight
RF

    Radio Frequency
RC

    Reverberation Chamber
RIMP

    Rich Isotropic Multipath
SB

    Shadow Bias
SC

    small cell
SDN

    Software-Defined Networking
SGE

    Serious Game Engineering
SF

    Shadow Fading
SIMO

    Single Input Multiple Output
SINR

    Signal to Interference plus Noise Ratio
SISO

    Single Input Single Output
SMa

    Suburban Macro-cell
SNR

    Signal to Noise Ratio
SU

    Single User
SUMO

    Simulation of Urban Mobility
SS

    Shadow Strength
STD

    Standard Deviation
SW

    Sliding Window
TDD

    Time Division Duplexing
TDM

    Time Division Multiplexing
TD-CDMA

    Time Division Code Division Multiple Access
TDMA

    Time Division Multiple Access
TX

    transmitter
TZ

    Test Zone
TRP

    Total Radiated Power
UAV

    Unmanned Aerial Vehicle
UE

    User Equipment
UI

    User Interface
UHD

    Ultra High Definition
UL

    Uplink
UMa

    Urban Macro-cell
UMi

    Urban Micro-cell
uMTC

    ultra-reliable Machine Type Communications
UMTS

    Universal Mobile Telecommunications System
UPM

    Unity Package Manager
UTD

    Uniform Theory of Diffraction
UTRA

    UMTS Terrestrial Radio Access
UTRAN

    UMTS Terrestrial Radio Access Network
URLLC

    Ultra-Reliable and Low Latency Communications
UHRP

    Upper Hemisphere Radiated Power
V2V

    Vehicle-to-Vehicle
V2X

    Vehicle-to-Everything
VP

    Visualization Platform
VR

    Virtual Reality
VNA

    Vector Network Analyzer
VIL

    Vehicle-in-the-loop
WCDMA

    Wideband Code Division Multiple Access
WINNER

    Wireless World Initiative New Radio
WINNER+

    Wireless World Initiative New Radio +
WiMAX

    Worldwide Interoperability for Microwave Access
WRC

    World Radiocommunication Conference
xMBB

    extreme Mobile Broadband
ZF

    Zero Forcing

# K-factor Evaluation in a Hybrid Reverberation Chamber plus CATR OTA Testing
Setup

Alejandro Antón Ruiz1, Samar Hosseinzadegan2, John Kvarnstrand2, Klas
Arvidsson2, Andrés Alayón Glazunov13  1University of Twente, Enschede, The
Netherlands, {a.antonruiz<EMAIL_ADDRESS>2Bluetest AB,
Gothenburg, Sweden<EMAIL_ADDRESS>3Linköping University,
Norrköping Campus, Sweden<EMAIL_ADDRESS>
###### Abstract

This paper investigates achieving diverse K-factors using a Reverberation
Chamber (RC) with a Compact Antenna Test Range (CATR) system. It explores six
hybrid "RC plus CATR" configurations involving different excitations of the
Rich Isotropic Multipath (RIMP) field and CATR-generated plane waves, with
some setups including absorbers. A fixed horn antenna points towards the CATR
in all configurations. The study found that the null hypothesis of Rayleigh or
Rician probability distributions for the received signal envelope could not be
rejected, with RIMP setups primarily conforming to Rayleigh distribution and
all setups showing Rician distribution. Various K-factors were obtained, but
no generalizable method for achieving the desired K-factor was identified. The
paper also estimates the K-factor as a function of frequency in the 24.25-29.5
GHz band. Smaller K-factors exhibit larger fluctuations, while larger
K-factors remain relatively stable, with consistent fluctuations across the
frequency range.

###### Index Terms:

OTA, K-factor, Reverberation Chamber, CATR, mmWave.

##  I Introduction

Over-The-Air (OTA) testing is key for developing and final product testing of
wireless devices. It provides an assessment of real-life performance in
controlled and repeatable conditions. OTA testing is currently the only option
for fully integrated devices without any connectors available and for smart
antenna systems [1]. It is impractical to characterize the device in infinite
scenarios to ensure it always performs well. In practice, it is necessary to
constrain the testing to a subset of meaningful scenarios. A relevant
parameter for a testing scenario is the distribution of the received signal at
the Device Under Test (DUT). Two limiting propagation scenarios can be
identified: Rich Isotropic Multipath (RIMP) [2], and pure Line-Of-Sight (LOS),
having an application in the Random Line-Of-Sight (Random-LOS) technique for
OTA testing [3]. RIMP can be achieved in a Reverberation Chamber (RC), where a
distribution of the received signal close to theoretical Rayleigh is generated
if the chamber is properly designed [4]. Pure LOS can be achieved in an
Anechoic Chamber (AC) or also in a semi-anechoic as used in the
Electromagnetic Compatibility (EMC) tests.

The above two OTA environments offer excellent methods to recreate the ideal
limiting propagation scenarios. However, real propagation environments are
much more complex and may require combining the RIMP and the Compact Antenna
Test Range (CATR), depending on the application, as provided by the Bluetest
solution. Thus, a LOS component and other Non-Line-Of-Sight (NLOS) components
can be generated following a desired mixture. In this case, the distribution
of the envelope of the received signal at the DUT should be a Rician. This
distribution is characterized by two parameters: the $K-$factor defined as the
ratio of the LOS over the NLOS powers, or the ratio between the unstirred
power over the stirred power and the average received power [5]. An ideal RIMP
scenario has a $K-$factor of 0 (or -$\infty$ dB), while pure LOS would have a
$K-$factor of $\infty$.

On another note, Millimeter Wave (mmWave) has been deemed as one of the
enablers for Fifth Generation (5G) communications [6] due to the large
available bandwidth at those frequencies. Due to the smaller wavelengths,
diffraction is less relevant in the propagation. Also, channels at mmWave tend
to be sparser, more directional, and have higher pathloss than sub-10 GHz
frequencies. Therefore, it is expected that, in a general sense, $K-$factors
of mmWave channels are higher than those of lower frequencies. Highly
directive antennas such as phased arrays are used to overcome pathloss.

In this work, an investigation of the use of different configurations of a RC
to achieve different $K-$factors is performed. The frequency range of interest
is the lower FR2 bands for 5G ($24.25$ GHz to $29.5$ GHz). The RC used for
this experiment is a Bluetest RTS65, with the CATR option installed [7]. This
allows to generate, in addition to the RIMP and the CATR environments
separately, also other channels where both are mixed. In addition, using an
absorber designed to attenuate the reflections coming from the CATR has also
been considered in this work. The emulation of Rician channels in RCs is not
new in the literature [8, 9, 10, 11]. However, to the author’s best knowledge,
this investigation has only been carried out at sub-6 GHz frequencies. This
work considers using a CATR system at mmWave frequencies. Existing commercial
products are used novelly since the Bluetest RTS65 was not originally designed
to emulate Rician channels.

##  II Signal Probability Distribution

###  II-A Rayleigh Probability Distribution

In a RIMP environment, the distribution of the receive signal envelope by the
DUT positioned in the test zone is Rayleigh and results from multipath
interference. The scale parameter $\sigma$ fully determines the distribution
[4]. In this case, the power follows an exponential distribution, with the
parameter $\sigma^{2}$ as its only parameter, the average receive power.
Estimating this parameter is straightforward by performing the mean of the
obtained data.

###  II-B Rician Probability Distribution

The Rician distribution model signals have two components: deterministic or
direct LOS and NLOS random or scattered field components. The latter is
Rayleigh distributed [8]. The $K-$factor is the power ratio of the LOS
component over the NLOS component. In an ideal RC, which produced a perfect
Rayleigh distribution, the $K-$factor would be 0 (or $-\infty$ dB). However,
that is not achievable in practice. In the RC context, the $K-$factor can be
better understood as

$K=P_{d}/P_{s},$ (1)

where $P_{s}$ is the stirred power, and $P_{d}$ is the power of the unstirred
paths, which includes a direct LOS when the CATR is used.

In this work, we have decided to use the estimator found in [9] since it is
unbiased. A major note is that this estimator requires acquiring the complex
value of the transmission parameter $S_{21}$. While we do not reproduce the
estimator equation, we say that it depends on the number of samples, which are
assumed to be independent. This we take for granted in this work. However,
future work will fully consider their independence and impact on the
estimation.

##  III Experimental setup

###  III-A Chamber

Fig. 1 shows the experimental setup using a Bluetest RTS65 RC chamber with the
CATR option installed [7]. The chamber is designed to be used as a 2-in-1 test
system, providing a RIMP field in the test zone when it is excited through the
RIMP port or a LOS only when it is excited through the CATR port. The latter
mode uses absorbers on the walls, enabling radiation pattern measurements
within the same space.

Figure 1: Bluetest RTS65 with the CATR option and the back absorber installed.
The dimensions of the chamber are $1945$x$2000$x$1440$ mm (WxHxD). Figure 2:
CAD render showing geometry and signal flow (in green). The test zone is
located within the lightest green volume. Source: [7]. Figure 3: Experiment
setup for calibrating the VNA.

The CATR mode of the chamber is realized by the use of a parabolic reflector
with a dual-polarized antenna feed shown in yellow in Fig. 1. The generated
plane wave has a $0.6$ dB amplitude and a $4^{\circ}$ phase ripples. The
Standard Deviation (STD) ripples are at frequencies from $24.25-42$ GHz within
a $30$ cm diameter cylindrical test zone [12] (see Fig. 2). The polarization
of the antenna is selected through a passive switch. To handle the reflections
coming from the reflector or the DUT, a set of carbon-loaded foam absorbers is
placed around the chamber, as shown in Fig. 2. Except for the absorber on the
wall towards which the plane wave is directed ("back absorber"), all the other
absorbers are Frequency Selective Absorber (FSA). They are covered by a
metallic honeycomb pattern designed with spacing such that the sub-6 GHz
frequency signals for which the chamber is also designed to operate in RIMP
mode are not significantly attenuated by the FSA. However, they provide
significant attenuation at mmWave. The back absorber has the same material and
size as the FSA, except that the metallic honeycomb part is removed, thus
providing better absorption. This is because most of the energy coming from
the reflector is directed to this back absorber. In this work, only the back
absorber scenario was considered (see Fig. 1).

###  III-B Antenna

The experiment uses a linearly-polarized double ridged Horn Antenna (HA),
designed to operate from $4.5-50$ GHz [13]. The choice of antenna is because
of its directional pattern and an accessible port so that the complex-valued
$S_{21}$ can be measured to estimate the $K-$factor. The use of a directive
antenna for mmWave experiments is highly relevant because the obtained
$K-$factor can serve as a reference value for other directive antennas. Even
though results are valid for this specific antenna because of the spatial
filtering that the directive antenna performs, they might be similar to those
obtained for a different antenna with similar directivity. For example,
results obtained with a phased array can be compared to the reference value.
However, a characterization of the $K-$factor using an omnidirectional antenna
is acknowledged to be relevant but is left for future work.

###  III-C Considered test cases

A total of six cases are considered:

  * •

NoAs_R: no absorbers, only RIMP excitation.

  * •

NoAs_C_PS1: no absorbers, only CATR excitation with the matching polarization
of the HA.

  * •

NoAs_RC_PS1: no absorbers, RIMP and CATR excitation with the CATR in the
matching polarization of the HA.

  * •

BAs_R: back absorber, only RIMP excitation.

  * •

BAs_C_PS1: back absorber, only CATR excitation with the matching polarization
of the HA.

  * •

BAs_RC_PS1: back absorber, RIMP and CATR excitation with the CATR in the
matching polarization of the HA.

Note that PS1 stands for "Passive Switch set to 1", which means that the CATR
port is routed to the port of the reflector feeder antenna designed as "1",
which is the one that produces the polarization that matches that of the HA,
as it was installed in the experiment.

###  III-D Instrument and measurement

####  III-D1 Cabling

To perform this experiment, a 1:4 splitter was used to connect one of the VNA
ports to the RIMP and/or CATR ports, while the other port of the VNA was
connected to the HA via a port in the front panel of the chamber. For RIMP or
CATR only measurements, only one of the four splitter ports was used, while
the other three were terminated. Two of the four ports were used in the RIMP
plus CATR measurements, while the other two were terminated.

####  III-D2 VNA and measurement configuration

The VNA was configured to perform a $S_{21}$ sweep in the frequency from
$24-29.5$ GHz with a $10$ MHz step, although only the $24.25$ to $29.5$ GHz
range is presented here to adequate to the FR2 bands. An Intermediate
Frequency (IF) bandwidth of $1$ KHz was used.

For each frequency point, a total of 600 samples were collected. Those 600
samples were taken at 600 unique positions of the two mode-stirrers of the
chamber located on the ceiling and right wall while keeping the turntable
fixed where the HA was pointed at the reflector. It is worth noting in Fig. 1
that the "right wall" is located on the right, behind the large reflector and
the metallic plate behind it. Hence, keeping the antenna in a fixed position
relative to the reflector was done to have a more controlled interaction
between the HA and the reflector. However, it is left for future work to use
the turntable as a stirring mechanism, making the results less dependent on
the direction towards which the main beam of the antenna is pointing.

Regarding the VNA calibration, the calibration plane was, on one side, at the
port of the HA and, on the other side, at the single port of the 1:4 splitter.
This is depicted in more detail in Fig 3.

(a) NoAs_R

(b) NoAs_C_PS1

(c) NoAs_RC_PS1

(d) BAs_R

(e) BAs_C_PS1

(f) BAs_RC_PS1

Figure 4: $K-$factor for different SWs and $|S_{21}|^{2}$, as a function of
the frequency; where only $|S_{21}|^{2}$ belongs to the right y-axis. TABLE I:
Relevant statistics of $K-$factor and $|S_{21}|^{2}$.

##  IV Results

Fig. 4 shows the results of both $K-$factor (left y-axis) and $|S_{21}|^{2}$
(right y-axis) as a function of the considered frequency. The estimated
$K-$factor is computed for each frequency point of the considered band and
shown as “No SW,” where SW stands for Sliding Window. Then, an averaging of
the $K-$factor over SWs of 100, 200, and 400 MHz is presented. This averaging
is common in the literature to reduce statistical variations and is known as
frequency stirring [5] and, in this case, it can approximate what the average
$K-$factor value of a 5G FR2 channel of 100, 200, or 400 MHz centered at the
x-axis frequency would be in this setup. The $|S_{21}|^{2}$ is plotted to
assess what occurs in each studied case regarding the system losses. On the
other hand, Table. I shows some relevant statistics of both the $K-$factor and
$|S_{21}|^{2}$, for the whole frequency range, including the results of the
one-sample Kolmogorov-Smirnov tests that were performed to assess if the
distributions of the 600 samples at each frequency could come from Rayleigh or
Rician distributions. The presented result in Table I is the pass rate of
these Goodness of Fit (GoF) tests, which indicates the percentage of
frequencies for which it could not be rejected that the samples $|S_{21}|$
came from a fitted Rayleigh or Rician distribution, at a $5\%$ significance
level. Regarding this, the results align with the expectations since for the
RIMP only excitations, the pass rate for Rayleigh is high, while, for the
cases with CATR excitation, it is 0. The pass rate for Rician distribution is
high in all cases, aligned with the expectations, which enables the use of the
$K-$factor as a relevant parameter for all cases. Note that the Rayleigh
distribution is a particular case of the Rician distribution. For the CATR
only, even if we had a pure LOS contribution, we would still have a particular
case of Rician distribution (with $\infty$ $K-$factor).

In Fig. 4, it can be observed that the $K-$factor heavily fluctuates, in
relative terms, i.e., in dB, whenever the chamber is not excited through the
CATR port (a, d), remaining much more stable when the CATR port is excited (b,
c, e, f). This can also be observed in Table I, in both the dynamic range (or
peak-to-peak variation) and the normalized STD of the $K-$factor. The
stability when the CATR is excited is caused by the fact that the LOS
component, which is relatively stable with frequency, dominates over the
stirred components. The large variations of $K-$factor when RIMP is only
excited are caused by several factors. On the one hand, there is the effect of
limited sampling in conjunction with a low $K-$factor, as observed in Fig. 8
of [9]. In short, the Confidence Intervals (in dB scale) for the estimates of
the $K-$factor, which can be found in (43) and (44) from [9], when $K-$factor
is low, are broadened significantly (in dB scale). For reference, the $95\%$
CI for 600 samples and a $K-$factor of $-9.28$ dB is $[-11.09,-8.01]$ dB,
while, for a $K-$factor of $35.01$ dB, it is $[34.65,35.35]$ dB. However, the
$K-$factor varies much further than those confidence intervals, so it is
likely that the actual $K-$factor experiences large frequency variations when
it is low. On the other hand, these variations could also be partially due to
the absence of turntable stirring. This leaves the antenna in a fixed
position, resulting in frequency-dependant unstirred paths at the antenna
position. Therefore, it would be interesting to use turntable stirring and
observe if variations are smoothed.

When the back absorber is placed, there is an increase in the system’s losses,
i.e., average $|S_{21}|^{2}$. In particular, the total averaged $|S_{21}|^{2}$
decreases in $2.82$ dB for the RIMP excitation only, $0.09$ dB for the CATR
excitation only, and $0.15$ dB for the RIMP and CATR excitation. As expected,
the increase in the system’s losses in relative terms, i.e., in dB, is lower
for the cases where the CATR is on and, therefore, the dominant contribution
is that of the LOS, which does not interact with the absorber. It can be
observed that there is a direct relation between higher average $K-$factor and
smaller increases in the system’s losses, being the CATR only case the one
with the smallest increase, of just $0.09$ dB.

As for the use of SWs, they only make a difference in the RIMP only cases
since there is not such a large variation on the $K-$factor with the frequency
for the rest of the considered cases. For the RIMP only cases, the $100$ MHz
SW already provides a relevant smoothing, which improves with the larger SWs.
However, even with the $400$ MHz SW, there are some variations of the
$K-$factor with the frequency. This is not observed when the CATR is excited.
This result is relevant because it implies that within the presented cases,
the proposed setup can only provide a quite frequency-stable $K-$factor when
the CATR is excited, resulting in high $K-$factors. Conversely, for low
$K-$factors, it cannot provide, in relative terms, a frequency-stable
$K-$factor, which occurs when only the RIMP branch is excited. As a point for
future work, it is left to assess if a frequency-stable, low (at least lower
than the ones obtained when the CATR is excited) $K-$factor can be achieved
by, e.g., exciting the CATR and RIMP, but attenuating the CATR branch, or
using turntable stirring. On another note, it is also worth noting that the
$K-$factor variations do not follow any clear trend with respect to the
frequency, i.e., there is not an appreciable linear dependency of neither the
SW averaged $K-$factor nor the $K-$factor variation.

Finally, when observing the full frequency band averaged $K-$factor from Table
I, it is worth noting that the fact of adding the back absorber increases the
$K-$factor in around $14.31$ dB when only the CATR is excited, but only in
around $4.12$ dB when both the RIMP and CATR are excited. Therefore, the
estimated stirred power decreases in $14.36$ dB for the CATR and $4.18$ dB for
the RIMP and CATR case, when adding the back absorber. As for the estimated
unstirred power, it decreases in $0.06$ dB for both the CATR only and the RIMP
and CATR cases, when adding the back absorber. Therefore, it is clear that the
addition of the back absorber has a greater effect on reducing the stirred
power coming from the CATR by capturing a significant amount of the energy
that is not directly coupled to the horn antenna. This leads to a greater
increase of $K-$factor for the CATR only case than for the RIMP and CATR one.
As for the RIMP only case, adding the back absorber decreases the $K-$factor
of $1.62$ dB. This comes from a decrease of the estimated stirred power of
$3.03$ dB and a decrease of the estimated unstirred power of $1.41$ dB. All in
all, adding the back absorber for RIMP excitation only decreases the
resemblance of the environment to a pure RIMP one.

##  V Conclusion

This paper presents six RC setups equipped with a CATR system operating at the
$24.25-29.5$ GHz band, analyzing field distribution using one-sample
Kolmogorov-Smirnov GoF tests and focusing on $K-$factor. The GoF tests align
with expectations, confirming a Rician distribution for all cases and a
Rayleigh distribution for RIMP-only excitation. Different $K-$factors are
achievable with varying setups. Low estimated $K-$factors are less stable with
frequency relative to high $K-$factors. $K-$factor variations and sliding
window averages show no frequency dependence. The directional horn antenna
used in this study aims to provide a reference for directive antennas in
spatially selective channels. However, further work can improve predictability
by incorporating turntable stirring and omnidirectional antennas. This will
also allow us to characterize the $K-$factor independently from the
orientation of the antenna, at least on the 2D horizontal plane. Other setups
should be explored, such as attenuators in RIMP and/or CATR configurations.
This work marks an initial step toward utilizing a commercialized mmWave
product for novel OTA testing with a controllable $K-$factor.

## Acknowledgment

The work of Alejandro Antón is supported by the European Union’s Horizon 2020
Marie Skłodowska-Curie grant agreement No. 955629. Andrés Alayón Glazunov also
kindly acknowledges funding from the ELLIIT strategic research environment
(https://elliit.se/).

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  *[IMT]: International Mobile Telecommunications
  *[ITU]: International Telecommunications Union
  *[LTE]: Long Term Evolution
  *[OTA]: Over-The-Air
  *[DUT]: Device Under Test
  *[RIMP]: Rich Isotropic Multipath
  *[LOS]: Line-Of-Sight
  *[Random-LOS]: Random Line-Of-Sight
  *[RC]: Reverberation Chamber
  *[AC]: Anechoic Chamber
  *[EMC]: Electromagnetic Compatibility
  *[CATR]: Compact Antenna Test Range
  *[NLOS]: Non-Line-Of-Sight
  *[mmWave]: Millimeter Wave
  *[5G]: Fifth Generation
  *[RCs]: Reverberation Chamber
  *[VNA]: Vector Network Analyzer
  *[STD]: Standard Deviation
  *[FSA]: Frequency Selective Absorber
  *[HA]: Horn Antenna
  *[IF]: Intermediate Frequency
  *[SWs]: Sliding Window
  *[GoF]: Goodness of Fit
  *[CI]: Confidence Interval
  *[SW]: Sliding Window