# Multi-Mixed Fractional Brownian Motions and Orstein–Uhlenbeck Processes

Hamidreza Maleki Almani  School of Technology and Innovations, University of
Vaasa, P.O. Box 700, FIN-65101 Vaasa, FINLAND
<EMAIL_ADDRESS>and  Tommi Sottinen  School of Technology and
Innovations, University of Vaasa, P.O. Box 700, FIN-65101 Vaasa, FINLAND
<EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

We study the so-called multi-mixed fractional Brownian motions (mmfBm) and
multi-mixed fractional Ornstein–Ulhenbeck (mmfOU) processes. These processes
are constructed by mixing by superimposing (infinitely many) independent
fractional Brownian motions (fBm) and fractional Ornstein–Uhlenbeck processes
(fOU), respectively. We prove their existence as $L^{2}$ processes and study
their path properties, viz. long-range and short-range dependence, Hölder
continuity, $p$-variation, and conditional full support.

###### Key words and phrases:

fractional Brownian motion, Gaussian processes, long-range dependence, multi-
mixed fractional Brownian motion, multi-mixed fractional Ornstein–Uhlenbeck
process, short-range dependence, stationary-increment processes, stationary
processes

###### 2020 Mathematics Subject Classification:

60G10, 60G15, 60G22

##  1\. Introduction and Preliminaries

The fractional Brownian motion (fBm) $B^{H}$, with parameter $H\in(0,1)$
called the Hurst index, is the unique (up to a multiplicative constant)
centered $H$-self-similar stationary-increment Gaussian process. The fBm was
first studied in [15]. The name fractional Brownian motion comes from the
influential article [18]. For further information of the fBm, see the
monographs [5, 19]. The covariance of the fBm with Hurst index $H$ is given by

$r_{H}(t,s)=\frac{1}{2}\left[t^{2H}+s^{2H}-|t-s|^{2H}\right].$

For $H=1/2$ this process is well-known as the Brownian motion (Bm) or the
Wiener process. As a stationary-increment process, the fBm admits the spectral
density

(1.1) $f_{H}(x)=\frac{\sin(\pi H)\Gamma(1+2H)}{2\pi}\,|x|^{1-2H},$

where $\Gamma$ is the complete gamma function

$\Gamma(\alpha)=\int_{0}^{\infty}t^{\alpha-1}\mathrm{e}^{-t}\,\mathrm{d}t,$

see [22].

Let

$\varrho_{H}(\delta;t)=\mathbb{E}\left[(B^{H}_{\delta}-B^{H}_{0})(B^{H}_{t+\delta}-B^{H}_{t})\right]$

be the incremental autocovariance (with lag $\delta$) of the fBm. For
$t\to\infty$ we have the power decay

$\varrho_{H}(\delta;t)\sim H(2H-1)\delta^{2}t^{2H-2}.$

This means that the increments of fBm, called the fractional Gaussian noise
(fGn), are positively correlated and long-range dependent of $H>\frac{1}{2}$.
For $H<\frac{1}{2}$ they are negatively correlated and short-range dependent.

In the Bm case $H=\frac{1}{2}$ we have independent increments, i.e., no
dependence:

$\varrho_{\frac{1}{2}}(\delta;t)=0.$

A process $X$ is (locally) Hölder continuous with exponent $H$ if

$\sup_{t,s\in[0,T]}\frac{|X_{t}-X_{s}|}{|t-s|^{H}}<\infty$

The Hölder index of a process $X$ is

$\mathrm{Hol}_{T}(X)=\sup\left\\{H>0\,;\,\sup_{t,s\in[0,T]}\frac{|X_{t}-X_{s}|}{|t-s|^{H}}<\infty\right\\}$

The fBm has almost surely locally Hölder continuous paths with any order
$H-\varepsilon$ for any $\varepsilon>0$, but not with order $H$. This follows,
e.g., from Theorem 1 of [1]. Consequenlty, $\mathrm{Hol}(B^{H})=H$.

In addition to Hölder continuity, we have the equidistant $p$-variation as a
measure of the path regularity. For a process $X$ and $p\in[1,\infty)$ for the
equidistant partitions $\pi_{n}:=\\{t_{k}=\frac{k}{n}T:k=0,1,\ldots,n\\},$
consider the limit in probability

$V^{p}_{T}(X):=\lim_{n\to\infty}\sum_{k=1}^{n}|X_{t_{k}}-X_{t_{k-1}}|^{p}.$

If this limit is finite, it is called the equidistant $p$-variation on $[0,T]$
of $X$. The equidistant $p$-variation index of a process $X$ is

$\mathrm{var}_{T}(X)=\sup\left\\{p\,;\,V_{T}^{p}(X)<\infty\right\\}.$

For the fBm we have

$V^{p}_{T}(B^{H})=\left\\{\begin{array}[]{lll}\infty&;&pH<1\\\
T\mu_{p}&;&pH=1\\\ 0&;&pH>1\end{array}\right.$

where $\mu_{p}$ is the $p$th moment of a standard Gaussian random variable,
see [8, 9]. Consequently, $\mathrm{var}_{T}(B^{H})=1/H$.

While the fBm has been proposed as a model for financial time series, modeling
with it makes arbitrage possible, see [3]. To eliminate this problem, a
generalization called mixed fractional Brownian motion (mfBm) was introduced
in [6]. This is the mixture model

$M^{a,b}=aB+bB^{H},$

where $a,b\in\mathbb{R}$ and $B$ is a standard Brownian motion (Bm)
independent of the fBm $B^{H}$. If $H>1/2$, the mfBm has the path roughness
governed by the Bm part and the long-range dependence governed by the fBm
part. Hence, e.g., in pricing of financial derivatives the corresponding mixed
Black–Scholes model yields the same option prices as the standard Brownian
model, see [4].

A natural generalization of the mfBm is to consider two (or $n$) independent
fBm mixtures, see [17]. In this paper, we study an independent infinite-
mixture generalization that we call the multi-mixed fractional Brownian motion
(mmfBm) with parameters $\sigma_{k}$, $H_{k}$, $k\in\mathbb{N}$:

$M=\sum_{k=1}^{\infty}\sigma_{k}B^{H_{k}},$

where $B^{H_{k}}$’s are independent fBm’s with Hurst indices $H_{k}\in(0,1)$,
and $\sigma_{k}$’s are positive volatility constants satisfying
$\sum_{k=1}^{\infty}\sigma_{k}^{2}<\infty$.

For other kinds of generalizations of the fBm, see e.g. [12, 16, 20, 21].

The fractional Ornstein–Uhlenbeck process (fOU) $U^{\lambda,H}$, with
parameters $\lambda>0$ and $H\in(0,1)$ is the stationary solution of the
Langevin equation

$\mathrm{d}U^{\lambda,H}_{t}=-\lambda
U^{\lambda,H}_{t}\mathrm{d}t+\mathrm{d}B^{H}_{t},$

which is given by

$U^{\lambda,H}_{t}=\int_{-\infty}^{t}\mathrm{e}^{-\lambda(t-s)}\,\mathrm{d}B^{H}_{s},$

where $(B^{H}_{s})_{s\leq 0}$ is an independent copy of the fBm
$(B^{H}_{s})_{s\geq 0}$, see [7]. Note that the Langevin equation and its
solution can be understood via integration-by-parts. As a stationary process,
the fOU admits the spectral density

(1.2) $f_{\lambda,H}(x)=\frac{f_{H}(x)}{x^{2}+\lambda^{2}},$

where $f_{H}$ is the spectral density of the driving fBm (1.1), see [2].
Denote, for $\alpha\in(-1,0)\cup(0,1)$

(1.3) $\displaystyle{\gamma}_{\alpha}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\alpha)}\int_{0}^{x}s^{\alpha-1}\mathrm{e}^{s}\,\mathrm{d}s,$
(1.4) $\displaystyle{\Gamma}_{\alpha}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\alpha)}\int_{x}^{\infty}s^{\alpha-1}\mathrm{e}^{-s}\,\mathrm{d}s,$

and ${\gamma}_{0}(x)=1$, ${\Gamma}_{0}(x)=0$. The autocovariance function of
the fOU process can be written as

(1.5) $\rho_{\lambda,H}(t)=\frac{\Gamma(1+2H)}{4}\frac{\mathrm{e}^{-\lambda
t}}{\lambda^{2H}}\bigg{\\{}1+{\gamma}_{2H-1}(\lambda t)+\mathrm{e}^{2\lambda
t}{\Gamma}_{2H-1}(\lambda t)\bigg{\\}},$

see Proposition 2.4 below. For $H=\frac{1}{2}$ we recover the well-known Bm
case

$\rho_{\lambda,\frac{1}{2}}(t)=\frac{\mathrm{e}^{-\lambda t}}{2\lambda}.$

For $t\to\infty$ we have the power decay

$\rho_{\lambda,H}(t)=\frac{1}{2}\sum_{n=1}^{N}\lambda^{-2n}\left(\prod_{j=0}^{2n-1}(2H-j)\right)t^{2H-2n}+O(t^{2H-2N-2}),$

for $N=1,2,\ldots$, i.e., the fOU process with $H>\frac{1}{2}$ is long-range
dependent, and for $H\leq\frac{1}{2}$ it is short-range dependent, see [7].

The Hölder index and the $p$-variation $p$-variation index of fOU is the same
as for the fBm: $\mathrm{Hol}_{T}(U^{\lambda,H})=\mathrm{Hol}_{T}(B^{H})=H$
and $\mathrm{var}_{T}(U^{\lambda,H})=\mathrm{var}_{T}(B^{H})=1/H$. These
results follow e.g. from our Theorem 4.1 and Theorem 5.1.

In this paper we study the multi-mixed fractional Ornstein–Uhlenbeck process
(mmfOU) with parameters $\lambda>0$ and $\sigma_{k}$, $H_{k}$,
$k\in\mathbb{N}$, that is defined naturally as the stationary solution of
Langevin equation with mmfBm as the driving noise:

$\mathrm{d}U_{t}=-\lambda U_{t}\,\mathrm{d}t+\mathrm{d}M_{t},$

with

$U_{0}=\int_{-\infty}^{0}\mathrm{e}^{\lambda s}\,\mathrm{d}M_{s},$

where $(M_{s})_{s\leq 0}$ is an independent copy of the mmfBm.

The rest of the paper is organized as follows. In Section 2 we define the
multi-mixed fractional Brownian motions (mmfBm) and the associated multi-mixed
fractional Ornstein–Uhlenbeck (mmfOU) processes, prove their existence in
$L^{2}(\Omega\times[0,T])$, and provide their basic properties. The long-range
dependence of these processes are what we studied in Section 3 In Section 4 we
analyze the Hölder continuity and $p$-variation of mmfBm’s and mmfOU
processes. The $p$-variation of these processes are calculated in Section 5 In
Section 6 we show that the mmfBm’s and mmfOU processes have the conditional
full support property. Finally, In Section 7 some simulated path of these
processes are given.

##  2\. Definitions and Basic Properties

######  Definition 2.1.

Let $\sigma_{k}$, $k\in\mathbb{N}$, satisfy

(2.1) $\sum_{k=1}^{\infty}\sigma_{k}^{2}<\infty,$

and let $H_{k}$, $k\in\mathbb{N}$, satisfy

$\displaystyle H_{k}\neq H_{l}\mbox{ for }k\neq l,$ (2.2) $\displaystyle
H_{\inf}=\inf_{k\in\mathbb{N}}H_{k}>0$ $\displaystyle
H_{\sup}=\sup_{k\in\mathbb{N}}H_{k}<1.$

The multi-mixed fractional Brownian motion (mmfBm) is

$M=\sum_{k=1}^{\infty}\sigma_{k}B^{H_{k}},$

where $B^{H_{k}}$, $k\in\mathbb{N}$, are independent fBm’s.

The following proposition shows the existence of the mmfBm.

######  Proposition 2.1.

The mmfBm $M$ exist as a random function taking values in
$L^{2}(\Omega\times[0,T])$ for all $T>0$.

###### Proof.

Let $M^{n}=\sum_{k=1}^{n}\sigma_{k}B^{H_{k}}.$ Clearly $M^{n}$ takes values in
$L^{2}(\Omega\times[0,T])$. Let $n,m\in\mathbb{N}$ with $n>m$. Then

$\displaystyle{\|M^{n}-M^{m}\|}^{2}_{L^{2}(\Omega\times[0,T])}$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\mathbb{E}\left[(M_{t}^{n}-M_{t}^{m})^{2}\right]\,\mathrm{d}t$
$\displaystyle=$
$\displaystyle\int_{0}^{T}\mathbb{E}\left[\left(\sum_{k=m+1}^{n}\sigma_{k}B^{H_{k}}_{t}\right)^{2}\right]\,\mathrm{d}t$
$\displaystyle=$
$\displaystyle\sum_{k=m+1}^{n}\int_{0}^{T}\sigma_{k}^{2}\mathbb{E}\left[(B_{t}^{H_{k}})^{2}\right]\,\mathrm{d}t$
$\displaystyle=$
$\displaystyle\sum_{k=m+1}^{n}\int_{0}^{T}\sigma_{k}^{2}t^{2H_{k}}\,\mathrm{d}t$
$\displaystyle=$
$\displaystyle\sum_{k=m+1}^{n}\sigma_{k}^{2}\frac{T^{1+2H_{k}}}{1+2H_{k}}$
$\displaystyle\leq$
$\displaystyle\sum_{k=m+1}^{n}\sigma_{k}^{2}\,\max\left\\{1,T^{3}\right\\},$

which shows that the sequence $(M^{n})_{n\in\mathbb{N}}$ is Cauchy. Thus
$M^{n}\to M$ in $L^{2}(\Omega\times[0,T])$ showing the existence. ∎

In the same way we see that the mmfBm $(M_{t})_{t\geq 0}$ exist in the sense
that $M^{n}_{t}\to M_{t}$ in $L^{2}(\Omega)$ for all $t\geq 0$.

The following is now obvious:

######  Proposition 2.2.

The mmfBm has stationary increments, its covariance function is

(2.3)
$r(t,s)=\sum_{k=1}^{\infty}\sigma_{k}^{2}r_{H_{k}}(s,t)=\frac{1}{2}\sum_{k=1}^{\infty}\sigma_{k}^{2}\left[|t|^{2H_{k}}+|s|^{2H_{k}}-|t-s|^{2H_{k}}\right],$

and it admits the spectral density

(2.4)
$f(x)=\sum_{k=1}^{\infty}\sigma_{k}^{2}f_{H_{k}}(x)=\sum_{k=1}^{\infty}\frac{\sin(\pi
H_{k})\Gamma(1+2H_{k})}{2\pi}\sigma_{k}^{2}|x|^{1-2H_{k}}.$

######  Definition 2.2.

The multi-mixed fractional Ornstein–Uhlenbeck process (mmfOU) $U$ with
parameter $\lambda>0$, is the stationary solution of the Langevin equation

(2.5) $\mathrm{d}U_{t}=-\lambda U_{t}\mathrm{d}t+\mathrm{d}M_{t},$

where the equation is understood in the integration-by-parts sense.

######  Proposition 2.3.

On $L^{2}(\Omega\times[0,T])$, the mmfOU can be represented as the integral

$U_{t}=\mathrm{e}^{-\lambda
t}\xi+\int_{0}^{t}\mathrm{e}^{-\lambda(t-s)}\,\mathrm{d}M_{s},$

where the integral is understood in the integration-by-parts sense, and

$\xi=\int_{-\infty}^{0}\mathrm{e}^{\lambda s}\,\mathrm{d}M_{s},$

where $(M_{s})_{s\leq 0}$ is an independent copy of the mmfBm $(M_{s})_{s\geq
0}$.

###### Proof.

Let $M^{n}=\sum_{k=1}^{n}\sigma_{k}B^{H_{k}}$. Then, the stationary solution
of the Langevin equation

$\mathrm{d}U^{n}_{t}=-\lambda U^{n}_{t}\mathrm{d}t+\mathrm{d}M^{n}_{t},$

is given by

$U^{n}_{t}=\mathrm{e}^{-\lambda
t}\xi_{n}+\int_{0}^{t}\mathrm{e}^{-\lambda(t-s)}\,\mathrm{d}M^{n}_{s},$

where

$\xi_{n}=\int_{-\infty}^{0}\mathrm{e}^{\lambda s}\,\mathrm{d}M^{n}_{s}.$

Then, with integration-by-parts

$\int_{0}^{t}\mathrm{e}^{\lambda s}\,\mathrm{d}M^{n}_{s}=\mathrm{e}^{\lambda
t}M^{n}_{t}-\lambda\int_{0}^{t}\mathrm{e}^{\lambda s}M^{n}_{s}\,\mathrm{d}s\
\to\ \mathrm{e}^{\lambda t}M_{t}-\lambda\int_{0}^{t}\mathrm{e}^{\lambda
s}M_{s}\,\mathrm{d}s=\int_{0}^{t}\mathrm{e}^{\lambda s}\,\mathrm{d}M_{s},$

because $M^{n}\to M$ in $L^{2}(\Omega\times[0,T])$. With the same arguments
$\xi_{n}\to\xi$ in $L^{2}(\Omega)$. This yields $U^{n}\to U$ in
$L^{2}(\Omega\times[0,T])$. ∎

The following technical lemma is used to calculate spectral densities.

######  Lemma 2.1.

For $0\neq p\in(-1,1),\,\lambda>0,\,t>0$

(2.6)
$\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i}tx}\frac{|x|^{p}}{\lambda^{2}+x^{2}}\,\mathrm{d}x=\frac{\pi\mathrm{e}^{-\lambda
t}}{2\cos(\frac{p\pi}{2})\lambda^{1-p}}\left\\{1+\gamma_{-p}(\lambda
t)+\mathrm{e}^{2\lambda t}\Gamma_{-p}(\lambda t)\right\\},$

where $\gamma_{-p}$ and $\Gamma_{-p}$ are given by (1.3) and (1.4).

###### Proof.

Recall that for the Fourier transform

$\mathscr{F}(f)(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\mathrm{e}^{-\mathrm{i}tx}f(t)\,\mathrm{d}t,$

we have the convolution theorem

(2.7)
$\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i}tx}\mathscr{F}(f)(x)\mathscr{F}(g)(x)\,\mathrm{d}x=\int_{-\infty}^{\infty}f(t-\xi)g(\xi)\,\mathrm{d}\xi.$

Moreover, we have

(2.8)
$\displaystyle\mathscr{F}\left(e^{-\lambda|t|}\right)=\sqrt{\frac{2}{\pi}}\cdot\frac{\lambda}{\lambda^{2}+x^{2}},$
(2.9)
$\displaystyle\mathscr{F}\left(|t|^{\alpha}\right)=\sqrt{\frac{2}{\pi}}\cdot\Gamma(\alpha+1)\cos\left(\frac{(\alpha+1)\pi}{2}\right)|x|^{-(\alpha+1)}.$

The first formula (2.8) is valid for $\lambda>0$. The second formula (2.9) is
valid for $-1<\alpha<0$. For $-2<\alpha<-1$, because of the function
$|t|^{\alpha}$, some singular terms arise at the origin. Nevertheless, it
admits a unique meromorphic extension as a tempered distribution, also denoted
$|t|^{\alpha}$ as a homogeneous distribution on all real line $\mathbb{R}$
including the origin (see [11]). So, we use that extension and formula (2.9)
will be valid for all $-1\neq\alpha\in(-2,0)$. So, using
$f(t)=e^{-\lambda|t|}$ and $g(t)=|t|^{\alpha}$ in (2.7) we obtain

$\displaystyle\frac{2}{\pi}\cdot\Gamma(\alpha+1)\cos\left(\frac{(\alpha+1)\pi}{2}\right)\lambda\int_{-\infty}^{\infty}\mathrm{e}^{itx}\frac{|x|^{-(\alpha+1)}}{\lambda^{2}+x^{2}}\,\mathrm{d}x$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}|\xi|^{\alpha}\mathrm{e}^{-\lambda|t-\xi|}\,\mathrm{d}\xi$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{0}(-\xi)^{\alpha}\mathrm{e}^{-\lambda(t-\xi)}\,\mathrm{d}\xi+\int_{0}^{t}\xi^{\alpha}\mathrm{e}^{-\lambda(t-\xi)}\,\mathrm{d}\xi+\int_{t}^{\infty}\xi^{\alpha}\mathrm{e}^{-\lambda(\xi-t)}\,\mathrm{d}\xi$
$\displaystyle=$ $\displaystyle\frac{\mathrm{e}^{-\lambda
t}}{\lambda^{(\alpha+1)}}\int_{0}^{\infty}u^{\alpha}\mathrm{e}^{-u}\,\mathrm{d}u+\frac{\mathrm{e}^{-\lambda
t}}{\lambda^{(\alpha+1)}}\int_{0}^{\lambda
t}u^{\alpha}\mathrm{e}^{u}\,\mathrm{d}u+\frac{\mathrm{e}^{\lambda
t}}{\lambda^{(\alpha+1)}}\int_{\lambda
t}^{\infty}u^{\alpha}\mathrm{e}^{-u}\,\mathrm{d}u$ $\displaystyle=$
$\displaystyle\frac{\mathrm{e}^{-\lambda
t}\Gamma(\alpha+1)}{\lambda^{(\alpha+1)}}\left\\{1+\gamma_{(\alpha+1)}(\lambda
t)+\mathrm{e}^{2\lambda t}\Gamma_{(\alpha+1)}(\lambda t)\right\\}.$

Now, choosing $p=-(\alpha+1)$ proves (2.6). ∎

It follows from Lemma 2.1 that:

######  Proposition 2.4.

The covariance function of the fOU is

(2.10)
$\rho_{\lambda,H}(t)=\mathbb{E}[U^{\lambda,H}_{s}U^{\lambda,H}_{s+t}]=\frac{\Gamma(1+2H)}{4}\frac{\mathrm{e}^{-\lambda
t}}{\lambda^{2H}}\bigg{\\{}1+{\gamma}_{2H-1}(\lambda t)+\mathrm{e}^{2\lambda
t}{\Gamma}_{2H-1}(\lambda t)\bigg{\\}}.$

######  Proposition 2.5.

The covariance function of the mmfOU is

(2.11)
$\rho_{\lambda}(t)=\mathbb{E}[U_{s}U_{s+t}]=\sum_{k=1}^{\infty}\sigma_{k}^{2}\frac{\Gamma(1+2H_{k})\mathrm{e}^{-\lambda
t}}{4\lambda^{2H_{k}}}\bigg{\\{}1+\gamma_{2H_{k}-1}(\lambda
t)+\mathrm{e}^{2\lambda t}\Gamma_{2H_{k}-1}(\lambda t)\bigg{\\}},$

and it admits the spectral density

(2.12) $f_{\lambda}(x)=\sum_{k=1}^{\infty}\sigma_{k}^{2}\frac{\sin(\pi
H_{k})\Gamma(1+2H_{k})}{2\pi}\frac{|x|^{1-2H_{k}}}{x^{2}+\lambda^{2}}.$

###### Proof.

Let $U^{n}$ be like in the proof of Proposition 2.3, then

$f_{\lambda,n}(x)=\sum_{k=1}^{n}\sigma_{k}^{2}\frac{\sin(\pi
H_{k})\Gamma(1+2H_{k})}{2\pi}\frac{|x|^{1-2H_{k}}}{x^{2}+\lambda^{2}},$

and $f_{\lambda,n}(x)\to f_{\lambda}(x)$ because $U^{n}\to U$ in
$L^{2}(\Omega\times[0,T])$. This proves (2.12). Similarly, (2.11) follows by
Proposition 2.4. ∎

######  Remark 2.1.

Proposition 2.4 represents the covariance function $\rho_{\lambda,H}(t)$ in a
form involving special functions. However, these special complex functions are
usually not suitable for numerical computations. For example,in [2], Lemma
B.1, the following representation was used for $H>\frac{1}{2}$

$\displaystyle\rho_{\lambda,H}(t)$ $\displaystyle=$ $\displaystyle
H\Gamma(2H)\frac{e^{-\lambda t}}{\lambda^{2H}}\left\\{\frac{1+e^{2\lambda
t}}{2}-\frac{\lambda}{\Gamma(2H-1)}I_{\lambda,H}(t)\right\\},$ $\displaystyle
I_{\lambda,H}(t)$ $\displaystyle=$ $\displaystyle\int_{0}^{t}\int_{0}^{\lambda
v}e^{2\lambda v}e^{-s}s^{2H-2}\,\mathrm{d}s\mathrm{d}v.$

the double integral above seems reasonable enough, but yields slow numerical
calculation in practice. This can be remedied by calculating the inner
integral as follows:

$\displaystyle I_{\lambda,H}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{\lambda t}\int_{s/{\lambda}}^{t}e^{2\lambda
v}e^{-s}s^{2H-2}\,\mathrm{d}s\mathrm{d}v$ $\displaystyle=$
$\displaystyle\frac{1}{2\lambda}\int_{0}^{\lambda t}s^{2H-2}(e^{2\lambda
t-s}-e^{s})\,\mathrm{d}s$ $\displaystyle=$ $\displaystyle\frac{e^{\lambda
t}}{\lambda}\int_{0}^{\lambda t}s^{2H-2}\sinh(\lambda t-s)\,\mathrm{d}s.$

Consequently,

$\rho_{\lambda,H}(t)=\frac{\Gamma(2H+1)}{2\lambda^{2H}}\bigg{\\{}\cosh(\lambda
t)-\frac{1}{\Gamma(2H-1)}\int_{0}^{\lambda t}s^{2H-2}\sinh(\lambda
t-s)\,ds\bigg{\\}},$

For the case $H<1/2$ we use the following developed version of Lemma 5.1 in
[13] for $\alpha>-1$. The proof is similar.

######  Lemma 2.2.

For $\alpha>-1$

$\int_{0}^{\infty}\int_{0}^{\infty}\mathrm{e}^{-(x+y)}|x-y|^{\alpha}\,\mathrm{d}x\mathrm{d}y=\Gamma(\alpha+1).$

######  Theorem 2.1.

For the fOU process $U^{\lambda,H}$. We have

$\displaystyle\rho_{\lambda,H}(t)=\frac{\Gamma(2H+1)}{2\lambda^{2H}}\,\bigg{\\{}\cosh(\lambda
t)-\frac{1}{\Gamma(2H)}\int_{0}^{\lambda t}s^{2H-1}\cosh(\lambda
t-s)\,ds\bigg{\\}},$

and so for mmfOU process we have

$\displaystyle\rho_{\lambda}(t)=\sum_{k=0}^{\infty}\sigma_{k}^{2}\frac{\Gamma(2H_{k}+1)}{2\lambda^{2H_{k}}}\,\bigg{\\{}\cosh(\lambda
t)-\frac{1}{\Gamma(2H_{k})}\int_{0}^{\lambda t}s^{2H_{k}-1}\cosh(\lambda
t-s)\,ds\bigg{\\}}.$

###### Proof.

For $H=1/2$ the right hand side of (2.1) is $\mathrm{e}^{-\lambda t}/2\lambda$
wich is equal to the outocovariance of the classical Ornstein–Uhlenbeck
process with respect to the standard Brownian motion. For $H>1/2$, we obtain
(2.1) from (2.1) via integration by part. To prove it for $H<1/2$, we will
apply the same approach of the proof of Lemma B.1 in [2]

$\displaystyle\rho_{\lambda,H}(t)=$ $\displaystyle\
\mathbb{E}[U^{\lambda,H}_{t}U^{\lambda,H}_{0}]$ $\displaystyle=$
$\displaystyle\ \mathbb{E}\left[\int_{-\infty}^{0}\mathrm{e}^{\lambda
u}\,\mathrm{d}B^{H}_{u}\int_{-\infty}^{t}\mathrm{e}^{-\lambda(t-v)}\,\mathrm{d}B^{H}_{v}\right]$
$\displaystyle=$ $\displaystyle\ \mathrm{e}^{-\lambda
t}\bigg{\\{}\mathbb{V}ar(U^{\lambda,H}_{0})+\mathbb{E}\bigg{[}\int_{-\infty}^{0}\mathrm{e}^{\lambda
u}\,\mathrm{d}B^{H}_{u}\int_{0}^{t}\mathrm{e}^{\lambda
v}\,\mathrm{d}B^{H}_{v}\bigg{]}\bigg{\\}}.$

To obtain the term $\mathbb{V}ar(U^{\lambda,H}_{0})$ in a close form, [2]
referred to Lemma 5.2 in [13]; however, it was only obtained for $H\geq 1/2$,
and so we need to extend their result for $H<1/2$.

Since

$U^{\lambda,H}_{0}=\int_{-\infty}^{0}\mathrm{e}^{\lambda
u}\,\mathrm{d}B^{H}_{u}=-\lambda\int_{-\infty}^{0}\mathrm{e}^{\lambda
u}B^{H}_{u}\,\mathrm{d}u,$

we have

$\displaystyle\mathbb{V}ar(U^{\lambda,H}_{0})=$
$\displaystyle\mathbb{V}ar\left[-\lambda\int_{-\infty}^{0}\mathrm{e}^{\lambda
u}B^{H}_{u}\,\mathrm{d}u\right]$ $\displaystyle=$ $\displaystyle\
\lambda^{2}\mathbb{V}ar\left[\int_{0}^{\infty}\mathrm{e}^{-\lambda
u}B^{H}_{u}\,\mathrm{d}u\right]$ $\displaystyle=$ $\displaystyle\
\lambda^{2}\mathbb{E}\left[\left(\int_{0}^{\infty}\mathrm{e}^{-\lambda
u}B^{H}_{u}\,\mathrm{d}u\right)^{2}\right]$ $\displaystyle=$ $\displaystyle\
\lambda^{2}\mathbb{E}\left[\int_{0}^{\infty}\int_{0}^{\infty}\mathrm{e}^{-\lambda(u+v)}B^{H}_{u}B^{H}_{v}\,\mathrm{d}u\mathrm{d}v\right]$
$\displaystyle=$ $\displaystyle\
\frac{\lambda^{2}}{2}\int_{0}^{\infty}\int_{0}^{\infty}\mathrm{e}^{-\lambda(u+v)}\cdot\Big{\\{}u^{2H}+v^{2H}-|u-v|^{2H}\Big{\\}}\,\mathrm{d}u\mathrm{d}v$
$\displaystyle=$ $\displaystyle\
\frac{\lambda^{2}}{2}\bigg{\\{}2\left(\int_{0}^{\infty}\mathrm{e}^{-\lambda
u}\,\mathrm{d}u\right)\left(\int_{0}^{\infty}\mathrm{e}^{-\lambda
v}v^{2H}\,\mathrm{d}v\right)-\int_{0}^{\infty}\int_{0}^{\infty}\mathrm{e}^{-\lambda(u+v)}|u-v|^{2H}\,\mathrm{d}u\mathrm{d}v\bigg{\\}},$

Now choosing $x=\lambda u,\,y=\lambda v$ and Lemma 2.2 we have

$\displaystyle\mathbb{V}ar(U^{\lambda,H}_{0})$
$\displaystyle=\frac{\lambda^{-2H}}{2}\bigg{\\{}2\int_{0}^{\infty}\mathrm{e}^{-y}y^{2H}\,\mathrm{d}y-\int_{0}^{\infty}\int_{0}^{\infty}\mathrm{e}^{-(x+y)}|x-y|^{2H}\,\mathrm{d}x\mathrm{d}y\bigg{\\}}$
$\displaystyle=\frac{\lambda^{-2H}}{2}\Big{[}2\Gamma(2H+1)-\Gamma(2H+1)\Big{]}$
(2.13) $\displaystyle=\lambda^{-2H}H\Gamma(2H).$

On the other hand, as in Lemma 2.1 in [7] and the proof of Lemma B.1 in [2],
using formula

$\gamma_{\ell}(z,x)=\frac{\gamma_{\ell}(z+1,x)+x^{z}\mathrm{e}^{-x}}{z},$

where $\gamma_{\ell}$ is the well-known lower Gamma function, for $H<1/2$ we
have

(2.14) $\displaystyle\mathbb{E}\left[\int_{-\infty}^{0}\mathrm{e}^{\lambda
u}\,\mathrm{d}B^{H}_{u}\int_{0}^{t}\mathrm{e}^{\lambda
v}\,\mathrm{d}B^{H}_{v}\right]$
$\displaystyle=H(2H-1)\int_{-\infty}^{0}\int_{0}^{t}\mathrm{e}^{-\lambda(u+v)}|u-v|^{2H-2}\,\mathrm{d}u\mathrm{d}v$
$\displaystyle=\mathbb{V}ar(U^{\lambda,H}_{0})\bigg{\\{}\frac{\mathrm{e}^{2\lambda
t}-1}{2}-\frac{\lambda}{\Gamma(2H-1)}\int_{0}^{t}\mathrm{e}^{2\lambda
v}\int_{0}^{\lambda
v}\mathrm{e}^{-s}s^{2H-2}\,\mathrm{d}s\mathrm{d}v\bigg{\\}}$
$\displaystyle=\mathbb{V}ar(U^{\lambda,H}_{0})\bigg{\\{}\frac{\mathrm{e}^{2\lambda
t}-1}{2}-\frac{\lambda}{\Gamma(2H-1)}\int_{0}^{t}\mathrm{e}^{2\lambda
v}\gamma_{\ell}(2H-1,\lambda v)\,\mathrm{d}v\bigg{\\}}$
$\displaystyle=\mathbb{V}ar(U^{\lambda,H}_{0})\bigg{\\{}\frac{\mathrm{e}^{2\lambda
t}-1}{2}-\frac{\lambda}{\Gamma(2H)}\int_{0}^{t}\mathrm{e}^{2\lambda
v}\gamma_{\ell}(2H,\lambda
v)\,\mathrm{d}v-\frac{\lambda^{2H}}{\Gamma(2H)}\int_{0}^{t}\mathrm{e}^{\lambda
v}v^{2H-1}\,\mathrm{d}v\bigg{\\}}$
$\displaystyle=\mathbb{V}ar(U^{\lambda,H}_{0})\bigg{\\{}\frac{\mathrm{e}^{2\lambda
t}-1}{2}-\frac{\lambda}{\Gamma(2H)}\int_{0}^{t}\mathrm{e}^{2\lambda
v}\int_{0}^{\lambda
v}\mathrm{e}^{-s}s^{2H-1}\,\mathrm{d}s\mathrm{d}v-\frac{\lambda^{2H}}{\Gamma(2H)}\int_{0}^{t}\mathrm{e}^{\lambda
v}v^{2H-1}\,\mathrm{d}v\bigg{\\}}.$

Using (2.13) and (2.14), with similar arguments as we did for (2.1) we obtain
(2.1). ∎

##  3\. Long-Range Dependence

The increments of fBm, the fGn, is a well-known stationary process, that is
long-range dependent (LRD) if $H>1/2$, and short-range dependent (SRD) in
$H<\frac{1}{2}$ Motivated with this, we consider the LRD of the increments the
of mmfBm.

For a lag $\delta$ and a process $X$ we denote
$\Delta_{\delta}X_{t}=X_{t+\delta}-X_{t}.$ Then

$\Delta_{\delta}M_{t}=\sum_{k=1}^{\infty}\sigma_{k}\Delta_{\delta}B^{H_{k}}_{t}$

is stationary and its autocovariance function is denoted by

$\varrho(\delta;t)=\mathbb{E}\big{[}\Delta_{\delta}M_{s+t}\Delta_{\delta}M_{s}\big{]},$

######  Theorem 3.1.

For $t\to\infty$

(3.1)
$\varrho(\delta;t)\sim\delta^{2}\sum_{k=1}^{\infty}\sigma_{k}^{2}H_{k}(2H_{k}-1)t^{2H_{k}-2}=O(t^{2H_{\sup}-2}).$

So the mmfBm increment process $\Delta_{\delta}M_{t}$ is LRD if and only if
$H_{k}>1/2$ for some $k\geq 0$.

###### Proof.

By using the generalized binomial theorem

(3.2) $\displaystyle\varrho(\delta;t)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{k=1}^{\infty}\sigma_{k}^{2}\Big{\\{}(t+\delta)^{2H_{k}}+(t-\delta)^{2H_{k}}-2t^{2H_{k}}\Big{\\}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{k=1}^{\infty}\sigma_{k}^{2}t^{2H_{k}}\Bigg{\\{}\Big{(}1+\frac{\delta}{t}\Big{)}^{2H_{k}}+\Big{(}1-\frac{\delta}{t}\Big{)}^{2H_{k}}-2\Bigg{\\}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{k=1}^{\infty}\sigma_{k}^{2}t^{2H_{k}}\Bigg{\\{}\sum_{r=0}^{\infty}\binom{2H_{k}}{r}\Big{(}\frac{\delta}{t}\Big{)}^{r}+\sum_{r=0}^{\infty}\binom{2H_{k}}{r}(-1)^{r}\Big{(}\frac{\delta}{t}\Big{)}^{r}-2\Bigg{\\}}$
$\displaystyle\sim$
$\displaystyle\delta^{2}\sum_{k=1}^{\infty}\sigma_{k}^{2}H_{k}(2H_{k}-1)t^{2H_{k}-2}.$

Since

$\sigma_{k}^{2}H_{k}(2H_{k}-1)t^{2H_{k}-2}\leq\sigma_{k}^{2},$

the series (3.2) is uniformly convergent. So we have

$\lim_{t\to\infty}\sum_{k=1}^{\infty}\sigma_{k}^{2}H_{k}(2H_{k}-1)t^{2H_{k}-2}=\sum_{k=1}^{\infty}\lim_{t\to\infty}\sigma_{k}^{2}H_{k}(2H_{k}-1)t^{2H_{k}-2}.$

This yields (3.1). ∎

To investigate LRD for the mmfOU process, we first need some lemmas.

The following theorem shows that similar to the mmfBm increment process, the
long-range dependence of the mmfOU is governed by the long-range dependence of
the largest Hurst index in the driving mmfBm.

######  Theorem 3.2.

For $t\to\infty$ and each $N=1,2,\ldots$

(3.3)
$\rho_{\lambda}(t)=\frac{1}{2}\sum_{k=1}^{\infty}\sum_{n=1}^{N}\sigma^{2}_{k}\lambda^{-2n}\left(\prod_{j=0}^{2n-1}(2H_{k}-j)\right)t^{2H_{k}-2n}+O(t^{2H_{\sup}-2N-2}).$

So the mmfOU process $U$ is LRD if and only if $H_{k}>1/2$ for some $k\geq 0$.

###### Proof.

By the proof of Lemma 2.2 and Theorem 2.3 in [7]

$\displaystyle\rho_{\lambda}(t)=$
$\displaystyle\mathbb{E}\left[\int_{-\infty}^{0}\mathrm{e}^{\lambda
u}\,\mathrm{d}M_{u}\int_{-\infty}^{t}\mathrm{e}^{-\lambda(t-v)}\,\mathrm{d}M_{v}\right]$
$\displaystyle=$ $\displaystyle\mathrm{e}^{-\lambda
t}\mathbb{E}\left[\int_{-\infty}^{0}\mathrm{e}^{\lambda
u}\,\mathrm{d}M_{u}\int_{-\infty}^{1/\lambda}\mathrm{e}^{\lambda
v}\,\mathrm{d}M_{v}\right]$ $\displaystyle+$
$\displaystyle\mathrm{e}^{-\lambda
t}\sum_{i=1}^{\infty}\sigma_{i}^{2}H_{i}(2H_{i}-1)\int_{-\infty}^{0}\mathrm{e}^{\lambda
u}\left(\int_{1/\lambda}^{t}\mathrm{e}^{\lambda
v}(v-u)^{2H_{i}-2}\,\mathrm{d}v\right)\mathrm{d}u$ $\displaystyle=$
$\displaystyle O(\mathrm{e}^{-\lambda
t})+\frac{1}{2}\sum_{i=1}^{\infty}\sigma_{i}^{2}\frac{H_{i}(2H_{i}-1)}{\lambda^{2H_{i}}}\Bigg{\\{}\mathrm{e}^{-\lambda
t}\int_{1}^{\lambda
t}\mathrm{e}^{y}y^{2H_{i}-2}\,\mathrm{d}y+\mathrm{e}^{\lambda t}\int_{\lambda
t}^{\infty}\mathrm{e}^{-y}y^{2H_{i}-2}\,\mathrm{d}y\Bigg{\\}}$
$\displaystyle\leq$ $\displaystyle O(\mathrm{e}^{-\lambda
t})+\frac{1}{2}\sum_{k=1}^{\infty}\sum_{n=1}^{N}\sigma^{2}_{k}\lambda^{-2n}\left(\prod_{j=0}^{2n-1}(2H_{k}-j)\right)t^{2H_{k}-2n}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\sum_{k=1}^{\infty}\sigma_{k}^{2}\frac{\Big{|}H_{k}(2H_{k}-1)\cdots(2H_{k}-2-2N)\Big{|}}{\lambda^{2H_{k}}}\left[\mathrm{e}^{-\frac{\lambda
t}{2}}+(1+2^{2H_{k}-2N-3})(\lambda t)^{2H_{k}-2N-3}\right].$

Now, for $t\in[1,\infty)$

$\displaystyle\sigma_{k}^{2}$
$\displaystyle\frac{\Big{|}H_{k}(2H_{k}-1)\cdots(2H_{k}-2-2N)\Big{|}}{\lambda^{2H_{k}}}\,\mathrm{e}^{-\frac{\lambda
t}{2}}<\Lambda_{N}\sigma_{k}^{2}$ $\displaystyle\sigma_{k}^{2}$
$\displaystyle\frac{\Big{|}H_{k}(2H_{k}-1)\cdots(2H_{k}-2-2N)\Big{|}}{\lambda^{2H_{k}}}\,(1+2^{2H_{k}-2N-3})(\lambda
t)^{2H_{k}-2N-3}<\Pi_{N}\sigma_{k}^{2},$

where

$\displaystyle\Lambda_{N}$ $\displaystyle=$ $\displaystyle
H_{\sup}\frac{\max\Big{(}|2H_{\inf}-1|,|2H_{\sup}-1|\Big{)}}{\max\Big{(}\lambda^{2H_{\inf}},\lambda^{2H_{\sup}}\Big{)}}\Big{|}(2H_{\inf}-2)\cdots(2H_{\inf}-2-2N)\Big{|},$
$\displaystyle\Pi_{N}$ $\displaystyle=$ $\displaystyle
H_{\sup}\frac{\max\Big{(}|2H_{\inf}-1|,|2H_{\sup}-1|\Big{)}}{\lambda^{2N+3}}\Big{|}(2H_{\inf}-2)\cdots(2H_{\inf}-2-2N)\Big{|}(1+2^{2H_{\sup}-2N-3}).$

So, as $\sum_{k=1}^{\infty}\sigma_{k}^{2}<\infty$, the series in the right-
hand side of the inequality (3) is uniformly convergent on $t\in[1,\infty)$.
Hence

$\displaystyle\lim_{t\to\infty}\sum_{k=1}^{\infty}\sigma_{k}^{2}\frac{\Big{|}H_{k}(2H_{k}-1)\cdots(2H_{k}-2-2N)\Big{|}}{\lambda^{2H_{k}}}\left[\mathrm{e}^{-\frac{\lambda
t}{2}}+(1+2^{2H_{k}-2N-3})(\lambda t)^{2H_{k}-2N-3}\right]$
$\displaystyle=\sum_{k=1}^{\infty}\sigma_{k}^{2}\frac{\Big{|}H_{k}(2H_{k}-1)\cdots(2H_{k}-2-2N)\Big{|}}{\lambda^{2H_{k}}}\lim_{t\to\infty}\left[\mathrm{e}^{-\frac{\lambda
t}{2}}+(1+2^{2H_{k}-2N-3})(\lambda t)^{2H_{k}-2N-3}\right].$

This proves (3.3). ∎

##  4\. Continuity

######  Theorem 4.1.

Both mmfBm and mmfOU have Hölder index $H_{\inf}$.

###### Proof.

For $\epsilon>0$ and $|t-s|<1$, the mmfBm satisfies

$\mathbb{E}\Big{[}(M_{t}-M_{s})^{2}\Big{]}=\sum_{k=1}^{\infty}\sigma_{k}^{2}|t-s|^{2H_{k}}\leq\bigg{(}\sum_{k=1}^{\infty}\sigma_{k}^{2}\bigg{)}|t-s|^{2H_{\inf}-\epsilon}=C_{0}|t-s|^{2H_{\inf}-\epsilon},$

where $C_{0}:=\sum_{k=1}^{\infty}\sigma_{k}^{2}>0$. Thus, Hölder continuity
with exponent $H_{\inf}-\varepsilon$ follows from Theorem 1 of [1]. On the
other hand, for some $j\geq 1$ we have $H_{\inf}\leq H_{j}<H_{\inf}+\epsilon$
and so the fBm $B^{H_{j}}$ is not $(H_{\inf}+\epsilon)$-Hölder continuous.
Hence the process $M=\sigma_{j}B^{H_{j}}+\sum_{k\neq j}\sigma_{k}B^{H_{k}}$ is
not $(H_{\inf}+\epsilon)$-Hölder continuous. This proves the claim for mmfBm.

For the mmfOU, we apply the Corollary 2 of [1]. That states that the
stationary process $U$ is Hölder-continuous with any exponent $0<a<H_{\inf}$
if and only if for each $0<\epsilon<2H_{\inf}$, there is some $0<\delta<1$
that

(4.1)
$\int_{0}^{\infty}(1-\cos(sx))f_{\lambda}(x)dx<C_{\epsilon}s^{2H_{\inf}-\epsilon},\quad
s\in(0,\delta).$

This is equivalent to have

$\int_{0}^{\infty}\frac{(1-\cos(sx))}{s^{2H_{\inf}-\epsilon}}f_{\lambda}(x)dx<C_{\epsilon},\quad
s\in(0,\delta).$

To show this, here for $s<1$ we have

$\displaystyle\int_{0}^{\infty}\frac{(1-\cos(sx))}{s^{2H_{k}-\epsilon}}f_{\lambda,H_{k}}(x)dx$
$\displaystyle=s^{\epsilon}c_{H_{k}}\int_{0}^{\infty}(1-\cos(sx))\frac{x\cdot(sx)^{-2H_{k}}}{\lambda^{2}+x^{2}}dx$
$\displaystyle=s^{\epsilon}c_{H_{k}}\int_{0}^{\infty}(1-\cos
u)\frac{u^{1-2H_{k}}}{s^{2}\lambda^{2}+u^{2}}du\quad(u=sx)$ $\displaystyle\leq
c_{H_{k}}\int_{0}^{\infty}(1-\cos
u)\frac{u^{1-2H_{k}}}{s^{2}\lambda^{2}+u^{2}}du\quad(0<s<1)$
$\displaystyle\leq c_{H_{k}}\Big{\\{}\int_{0}^{\epsilon}(1-\cos
u)\frac{u^{1-2H_{k}}}{u^{2}}du+\int_{\epsilon}^{\infty}\frac{u^{1-2H_{k}}}{u^{2}}du\Big{\\}}$
$\displaystyle=c_{H_{k}}\Big{\\{}\int_{0}^{\epsilon}\frac{2\sin^{2}(\frac{u}{2})}{u^{2}}u^{1-2H_{k}}du+\int_{\epsilon}^{\infty}u^{-1-2H_{k}}du\Big{\\}}$
$\displaystyle\leq
c_{H_{k}}\Big{\\{}\int_{0}^{\epsilon}\frac{1}{2}u^{1-2H_{k}}du+\int_{\epsilon}^{\infty}u^{-1-2H_{k}}du\Big{\\}}$
$\displaystyle=c_{H_{k}}\Big{\\{}\frac{\epsilon^{2-2H_{k}}}{4(1-H_{k})}+\frac{\epsilon^{-2H_{k}}}{2H_{k}}\Big{\\}}=:C_{\epsilon,H_{k}}<\infty.$

Therefore,

(4.2) $\int_{0}^{\infty}(1-\cos(sx))f_{\lambda,H_{k}}(x)dx\leq
C_{\epsilon,H_{k}}s^{2H_{k}-\epsilon}\leq
C_{\epsilon,H_{k}}s^{2H_{\inf}-\epsilon}.$

Also, we have

$\displaystyle\sum_{k=1}^{\infty}\sigma_{k}^{2}C_{\epsilon,H_{k}}$
$\displaystyle=\sum_{k=1}^{\infty}\sigma_{k}^{2}\,\frac{\sin(\pi
H_{k})\Gamma(1+2H_{k})}{2\pi}\Big{\\{}\frac{\epsilon^{2-2H_{k}}}{4(1-H_{k})}+\frac{\epsilon^{-2H_{k}}}{2H_{k}}\Big{\\}}$
(4.3)
$\displaystyle\leq\frac{\Gamma(3)}{2\pi}\Big{\\{}\frac{\epsilon^{2-2H_{sup}}}{4(1-H_{sup})}+\frac{\epsilon^{-2H_{inf}}}{2H_{inf}}\Big{\\}}\Big{(}\sum_{k=1}^{\infty}\sigma_{k}^{2}\Big{)}=:C_{\epsilon}<\infty,$

if and only if $0<H_{\inf}\leq H_{\sup}<1$ and
$\sum_{k=1}^{\infty}\sigma_{k}^{2}<\infty$. Now, (4.2) and (4.3) yield (4.1).
Moreover, for some $j\geq 1$ we have $H_{\inf}\leq H_{j}<H_{\inf}+\epsilon$
and so the fOU $U^{H_{j}}$ is not $(H_{\inf}+\epsilon)$-Hölder continuous.
Hence the process $U=\sigma_{j}U^{H_{j}}+\sum_{k\neq j}\sigma_{k}U^{H_{k}}$ is
not $(H_{\inf}+\epsilon)$-Hölder continuous. This proves the claim for mmfOU.
∎

##  5\. $p$-Variation

######  Theorem 5.1.

For $p>0$, the equidistant $p$-variations of the mmfBm $M$ and the mmfOU $U$
on the time-interval $[0,T]$ are equal and

(5.1)
$V^{p}_{T}(M)=V^{p}_{T}(U)=\left\\{\begin{array}[]{lll}\infty&;&pH_{\inf}<1\\\
\\\
T\Bigg{(}\displaystyle\underset{H_{i}=H_{\inf}}{\sum}\sigma_{i}^{2}\Bigg{)}^{p/2}\mu_{p}&;&pH_{\inf}=1\\\
\\\ 0&;&pH_{\inf}>1\end{array}\right.$

###### Proof.

For the mmfBm $M$, We have

$\displaystyle v^{p}_{\pi_{n}}(M)$ $\displaystyle:=$
$\displaystyle\sum_{t_{k}\in\pi_{n}}|\Delta M_{t_{k}}|^{p}$ $\displaystyle=$
$\displaystyle\sum_{t_{k}\in\pi_{n}}\left|\sum_{i=1}^{\infty}\sigma_{i}^{2}(\Delta
t_{k})^{2H_{i}}\right|^{p/2}\left|\frac{\Delta
M_{t_{k}}}{\Big{[}\sum_{i=1}^{\infty}\sigma_{i}^{2}(\Delta
t_{k})^{2H_{i}}\Big{]}^{1/2}}\right|^{p}$ $\displaystyle\overset{d}{=}$
$\displaystyle\left(\sum_{i=1}^{\infty}\sigma_{i}^{2}T^{2H_{i}}n^{2/p-2H_{i}}\right)^{p/2}\cdot\frac{1}{n}\sum_{k=1}^{n}|Z_{k}|^{p}.$

as $|\pi_{n}|\to 0$ or equivalently $n\to\infty$. Here $(Z_{k})$ is a
stationary Gaussian process and so by the proof of Lemma 3.7 in [23]

$\frac{1}{n}\sum_{k=1}^{n}|Z_{k}|^{p}\to\mu_{p},$

as $n\to\infty$, where $\mu_{p}$ is the $p$th absolute moment of the standard
Gaussian process. Now, if $pH_{\inf}<1$ then $H_{\inf}<1/p$, and so there
exists some $j\geq 1$ that $H_{j}<1/p$, and so $2/p-2H_{j}>0$. Therefore

$v^{p}_{\pi_{n}}(M)\geq\left(\sigma_{j}^{2}T^{2H_{j}}n^{2/p-2H_{j}}\right)^{p/2}\cdot\frac{1}{n}\sum_{k=1}^{n}|Z_{k}|^{p}\to\infty.$

On the other hand, if $pH_{\inf}\geq 1$ for $x\in(1,\infty)$

$\sigma_{i}^{2}T^{2H_{i}}x^{2/p-2H_{i}}\leq\sigma_{i}^{2}T^{2},$

and because $\sum_{i=1}^{\infty}\sigma_{i}^{2}<\infty$, the
$\sum_{i=1}^{\infty}\sigma_{i}^{2}T^{2H_{i}}x^{2/p-2H_{i}}$ is uniformly
convergent on $x\in[1,\infty)$. So for $pH_{\inf}\geq 1$

$\lim_{n\to\infty}\sum_{i=1}^{\infty}\sigma_{i}^{2}T^{2H_{i}}n^{2/p-2H_{i}}=\sum_{i=1}^{\infty}\lim_{n\to\infty}\sigma_{i}^{2}T^{2H_{i}}n^{2/p-2H_{i}}.$

This yields the values mentioned in (5.1) are correct for the $p$-variation of
$M$. For the mmfOU $U$, as it is stationary we have

$v^{p}_{\pi_{n}}(U):=\sum_{t_{k}\in\pi_{n}}|\Delta
U_{t_{k}}|^{p}\overset{d}{=}\sum_{k=1}^{n}\Big{(}\mathbb{V}ar[\Delta
U_{t_{1}}]\Big{)}^{p/2}|Z_{k}|^{p}=n\Big{(}\mathbb{V}ar[U_{\frac{T}{n}}-U_{0}]\Big{)}^{p/2}\cdot\frac{1}{n}\sum_{k=1}^{n}|Z_{k}|^{p}.$

As $\frac{1}{n}\sum_{k=1}^{n}|Z_{k}|^{p}\to\mu_{p}$ for $n\to\infty$, problem
vanish to find

$\lim_{n\to\infty}n\Big{(}\mathbb{V}ar[U_{\frac{T}{n}}-U_{0}]\Big{)}^{p/2}.$

To find it, again because $U$ is stationary, and using the proof of Theorem
2.1 we have

$\displaystyle\mathbb{V}ar[U_{\frac{T}{n}}-U_{0}]$ $\displaystyle=$
$\displaystyle\mathbb{V}arU_{\frac{T}{n}}+\mathbb{V}arU_{0}-2\,\mathrm{C}ov\Big{(}U_{\frac{T}{n}},U_{0}\Big{)}$
$\displaystyle=$ $\displaystyle
2\,\mathbb{V}arU_{0}-2\,\mathrm{C}ov\Big{(}U_{\frac{T}{n}},U_{0}\Big{)}$
$\displaystyle=$ $\displaystyle
2\sum_{i=1}^{\infty}\sigma_{i}^{2}\lambda^{-2H_{i}}H_{i}\Gamma(2H_{i})-2\sum_{i=1}^{\infty}\sigma_{i}^{2}\frac{\Gamma(2H_{i}+1)}{2\lambda^{2H_{i}}}\bigg{\\{}\cosh\Big{(}\frac{\lambda
T}{n}\Big{)}$ $\displaystyle-\frac{1}{\Gamma(2H_{i})}\int_{0}^{\frac{\lambda
T}{n}}s^{2H_{i}-1}\cosh\Big{(}\frac{\lambda
T}{n}-s\Big{)}\,\mathrm{d}s\bigg{\\}}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{\infty}\sigma_{i}^{2}\frac{\Gamma(2H_{i}+1)}{\lambda^{2H_{i}}}\bigg{\\{}1-\cosh\Big{(}\frac{\lambda
T}{n}\Big{)}+\frac{1}{\Gamma(2H_{i})}\int_{0}^{\frac{\lambda
T}{n}}s^{2H_{i}-1}\cosh\Big{(}\frac{\lambda
T}{n}-s\Big{)}\,\mathrm{d}s\bigg{\\}}.$

For the large values of $n$ the final series in the right hand side above, is
uniformly convergent. So, the $\displaystyle\lim_{n\to\infty}$ and
$\sum_{i=1}^{\infty}$ could change places. This yields

$\displaystyle\lim_{n\to\infty}n\Big{(}\mathbb{V}ar[U_{\frac{T}{n}}-U_{0}]\Big{)}^{p/2}$
$\displaystyle=$
$\displaystyle\lim_{n\to\infty}\Big{(}n^{2/p}\,\mathbb{V}ar[U_{\frac{T}{n}}-U_{0}]\Big{)}^{p/2}$
$\displaystyle=$
$\displaystyle\Bigg{(}\sum_{i=1}^{\infty}\sigma_{i}^{2}\frac{\Gamma(2H_{i}+1)}{\lambda^{2H_{i}}}\cdot\lim_{n\to\infty}n^{2/p}\bigg{\\{}1-\cosh\Big{(}\frac{\lambda
T}{n}\Big{)}+\,\frac{1}{\Gamma(2H_{i})}\int_{0}^{\frac{\lambda
T}{n}}s^{2H_{i}-1}\cosh\Big{(}\frac{\lambda
T}{n}-s\Big{)}\,\mathrm{d}s\bigg{\\}}\Bigg{)}^{p/2}.$

Now for $t\to 0$, by the Taylor expansion

$1-\cosh t=-\sum_{r=1}^{\infty}\frac{t^{2r}}{(2r)!},$

and via integration by parts

$\int_{0}^{t}s^{2H_{i}-1}\cosh(t-s)\,\mathrm{d}s=\frac{t^{2H_{i}}}{2H_{i}}+\frac{1}{2H_{i}}\int_{0}^{t}s^{2H_{i}}\sinh(t-s)\,\mathrm{d}s.$

Again for $t\to 0$, by the Taylor expansion

$\int_{0}^{t}s^{2H_{i}}\sinh(t-s)\,\mathrm{d}s\leq\int_{0}^{t}t^{2H_{i}}\sinh
t\,\mathrm{d}s=t^{2H_{i}+1}\sinh
t=\sum_{r=1}^{\infty}\frac{t^{2r+2H_{i}}}{(2r-1)!}.$

These yield for $t\to 0$

$1-\cosh
t+\frac{1}{\Gamma(2H_{i})}\int_{0}^{t}s^{2H_{i}-1}\cosh(t-s)\,\mathrm{d}s\sim\frac{t^{2H_{i}}}{2H_{i}+1}.$

Therefore

$\lim_{n\to\infty}n\Big{(}\mathbb{V}ar[U_{\frac{T}{n}}-U_{0}]\Big{)}^{p/2}=\bigg{(}\sum_{i=1}^{\infty}\sigma_{i}^{2}T^{2H_{i}}\lim_{n\to\infty}n^{2/p-2H_{i}}\bigg{)}^{p/2},$

this proves (5.1). ∎

##  6\. Conditional Full Support

As explained in [4], in mathematical finance models one of the must require
the so-called Conditional Full Support (CFS) to avoid simple kind of
arbitrage. This means that, given the information up to any stopping time
$\tau\in[0,T]$, the process is inherently free enough to go anywhere after the
stopping time $\tau$ with positive probability. This motivates us to study the
CFS property of the mmfBm and mmfOU processes but first we restate the precise
definition of CFS from [10].

######  Definition 6.1.

Let $X=(X_{t})_{0\leq t\leq T}$ be a continuous stochastic process defined on
a probability space $(\Omega,\mathscr{F},\mathbb{P})$, and $(\mathscr{F}_{t})$
be its natural filtration. The process $X$ is said to have CFS if, for all
$(\mathscr{F}_{t})$-stopping times $\tau$, the conditional law of
$(X_{u})_{\tau\leq u\leq T}$ given $\mathscr{F}_{\tau}$, almost surely has
support $C_{X_{\tau}}[\tau,T]$, where $C_{x}[\tau,T]$ is the space of
continuous functions $f$ on $[\tau,T]$ satisfying $f(t)=x$. Equivalently, this
means that, for all $t\in[0,T]$, $f\in C_{0}[\tau,T]$, and $\varepsilon>0$,

$\mathbb{P}\left(\sup_{\tau\leq u\leq
T}|X_{u}-X_{\tau}-f(u)|<\varepsilon\bigg{|}\mathscr{F}_{\tau}\right)>0,$

almost surely.

######  Theorem 6.1.

Both the mmfBm and the mmfOU have conditional full support.

###### Proof.

It is easy to check that

$f(x)=\sum_{k=1}^{\infty}\sigma_{k}^{2}f_{H_{k}}(x)\geq\left\\{\begin{array}[]{lll}\frac{\varepsilon_{H}\Gamma(1)}{2\pi}\biggl{(}\displaystyle\sum_{k=1}^{\infty}\sigma_{k}^{2}\biggl{)}|x|^{1-2H_{\inf}}&:&|x|\leq
1\\\ \\\
\frac{\varepsilon_{H}\Gamma(1)}{2\pi}\biggl{(}\displaystyle\sum_{k=1}^{\infty}\sigma_{k}^{2}\biggl{)}|x|^{1-2H_{\sup}}&:&|x|\geq
1\end{array}\right.=:h(x),$

where

$\varepsilon_{H}:=\inf\Big{\\{}\sin(\pi H_{k})\Big{\\}}_{k\geq
1}=\inf\Big{\\{}\sin(\pi H_{\inf}),\sin(\pi H_{\sup})\Big{\\}}$

Since $0<H_{\inf}\leq H_{\sup}<1$, $\varepsilon_{H}>0$. Thus $h(x)>0$ for
$x\neq 0$. Thus, for any $x_{0}>1$ we have

$\displaystyle\int_{x_{0}}^{\infty}\frac{\log f(x)}{x^{2}}\,\mathrm{d}x$
$\displaystyle\geq$ $\displaystyle\int_{x_{0}}^{\infty}\frac{\log
h(x)}{x^{2}}\,\mathrm{d}x$ $\displaystyle=$
$\displaystyle\log\left\\{\frac{\varepsilon_{H}}{2\pi}\biggl{(}\sum_{k=1}^{\infty}\sigma_{k}^{2}\biggl{)}\right\\}\int_{x_{0}}^{\infty}\frac{\mathrm{d}x}{x^{2}}+(1-2H_{\sup})\int_{x_{0}}^{\infty}\frac{\log
x}{x^{2}}\,\mathrm{d}x$ $\displaystyle>$ $\displaystyle-\infty,$

and by Theorem 2.1 of [10] this proves that $M$ has conditional full support.

For mmfOU it is easy to check that

$f_{\lambda}(x)=\sum_{k=1}^{\infty}\sigma_{k}^{2}f_{\lambda,H_{k}}(x)\geq\left\\{\begin{array}[]{lll}\frac{\varepsilon_{H}\Gamma(1)}{2\pi}\biggl{(}\displaystyle\sum_{k=1}^{\infty}\sigma_{k}^{2}\biggl{)}\frac{|x|^{1-2H_{\inf}}}{\lambda^{2}+x^{2}}&:&|x|\leq
1\\\ \\\
\frac{\varepsilon_{H}\Gamma(1)}{2\pi}\biggl{(}\displaystyle\sum_{k=1}^{\infty}\sigma_{k}^{2}\biggl{)}\frac{|x|^{1-2H_{\sup}}}{\lambda^{2}+x^{2}}&:&|x|\geq
1\end{array}\right.=:h(x),$

where

$\varepsilon_{H}:=\inf\Big{\\{}\sin(\pi H_{k})\Big{\\}}_{k\geq
1}=\inf\Big{\\{}\sin(\pi H_{\inf}),\sin(\pi H_{\sup})\Big{\\}}$

Since $0<H_{\inf}\leq H_{\sup}<1$, we have $\varepsilon_{H}>0$. Consequently,
$h(x)>0$ for $x\neq 0$. Therefore, for any $x_{0}>1$ we have that

$\displaystyle\int_{x_{0}}^{\infty}\frac{\log
f_{\lambda}(x)}{x^{2}}\,\mathrm{d}x$ $\displaystyle\geq$
$\displaystyle\int_{x_{0}}^{\infty}\frac{\log h(x)}{x^{2}}\,\mathrm{d}x$
$\displaystyle=$
$\displaystyle\log\left\\{\frac{\varepsilon_{H}}{2\pi}\biggl{(}\sum_{k=1}^{\infty}\sigma_{k}^{2}\biggl{)}\right\\}\int_{x_{0}}^{\infty}\frac{\mathrm{d}x}{x^{2}}+(1-2H_{\sup})\int_{x_{0}}^{\infty}\frac{\log
x}{x^{2}}\,\mathrm{d}x-\int_{x_{0}}^{\infty}\frac{\log(\lambda^{2}+x^{2})}{x^{2}}\,\mathrm{d}x$
$\displaystyle>$ $\displaystyle-\infty.$

The claim follows now from Theorem 2.1 of [10]. ∎

##  7\. Sample Paths

We illustrate the mmfBm and the mmfOU by simulating their sample paths.

Each of the sample paths, both for the mmfBm and for the mmfOU is given on
$N=1000$ equidistant points $t_{k}=k/(N-1)$ of the time-interval $[0,1]$, with
$n=10$ equidistant Hurst exponents
$H_{i}=H_{\inf}+(i-1)(H_{\sup}-H_{\inf})/(n-1)$ on the Hurst interval
$[H_{\inf},H_{\sup}]$. Also, the volatility coefficients
$\sigma_{i}=i^{-1},i!^{-1},\mathrm{e}^{-i}$ are used in the sample paths. In
all mmfOU paths $\lambda=1$.

Figure 1. Sample paths of mmfBm with equidistant time points and equidistant
Hurst parameters. Figure 2. Sample paths of mmfOU with equidistant time points
and equidistant Hurst parameters.

## References

  * [1] E. Azmoodeh, T. Sottinen, L. Viitasaari, and A. Yazigi, Necessary and sufficient conditions for Hölder continuity of Gaussian processes, Statist. Probab. Lett., 94 (2014), pp. 230–235.
  * [2] L. A. Barboza and F. G. Viens, Parameter estimation of Gaussian stationary processes using the generalized method of moments, Electron. J. Stat., 11 (2017), pp. 401–439.
  * [3] C. Bender, T. Sottinen, and E. Valkeila, Arbitrage with fractional Brownian motion?, Theory Stoch. Process., 13 (2007), pp. 23–34.
  * [4] , Pricing by hedging and no-arbitrage beyond semimartingales, Finance Stoch., 12 (2008), pp. 441–468.
  * [5] F. Biagini, Y. Hu, B. Øksendal, and T. Zhang, Stochastic calculus for fractional Brownian motion and applications, Probability and its Applications (New York), Springer-Verlag London, Ltd., London, 2008.
  * [6] P. Cheridito, Mixed fractional Brownian motion, Bernoulli, 7 (2001), pp. 913–934.
  * [7] P. Cheridito, H. Kawaguchi, and M. Maejima, Fractional Ornstein-Uhlenbeck processes, Electron. J. Probab., 8 (2003), pp. no. 3, 14 pp. (electronic).
  * [8] R. M. Dudley and R. Norvaiša, An introduction to p-variation and Young integrals: With emphasis on sample functions of stochastic processes, Centre for Mathematical Physics and Stochastics, University of Aarhus, 1998.
  * [9] R. M. Dudley and R. Norvaiša, Differentiability of six operators on nonsmooth functions and $p$-variation, vol. 1703 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1999.  With the collaboration of Jinghua Qian.
  * [10] D. Gasbarra, T. Sottinen, and H. Van Zanten, Conditional full support of gaussian processes with stationary increments, Journal of Applied Probability, 48 (2011), pp. 561–568.
  * [11] I. Gelfand and G. Shilov, Generalized functions vol 1 (new york: Academic), (1964).
  * [12] C. Houdré and J. Villa, An example of infinite dimensional quasi-helix, in Stochastic models (Mexico City, 2002), vol. 336 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2003, pp. 195–201.
  * [13] Y. Hu and D. Nualart, Parameter estimation for fractional ornstein–uhlenbeck processes, Statistics & probability letters, 80 (2010), pp. 1030–1038.
  * [14] J. Kalemkerian and A. Sosa, Long-range dependence in the volatility of returns in uruguayan sovereign debt indices, Journal of Dynamics & Games, 7 (2020), p. 225.
  * [15] A. N. Kolmogoroff, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum, C. R. (Doklady) Acad. Sci. URSS (N.S.), 26 (1940), pp. 115–118.
  * [16] J. Lévy-Véhel, Fractal approaches in signal processing, vol. 3, 1995, pp. 755–775.  Symposium in Honor of Benoit Mandelbrot (Curaçao, 1995).
  * [17] H. Maleki Almani, S. M. Hosseini, and M. Tahmasebi, Fractional Brownian motion with two-variable Hurst exponent, J. Comput. Appl. Math., 388 (2021), pp. Paper No. 113262, 23.
  * [18] B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), pp. 422–437.
  * [19] Y. S. Mishura, Stochastic calculus for fractional Brownian motion and related processes, vol. 1929 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2008.
  * [20] E. Perrin, R. Harba, C. Berzin-Joseph, I. Iribarren, and A. Bonami, nth-order fractional brownian motion and fractional gaussian noises, IEEE Transactions on Signal Processing, 49 (2001), pp. 1049–1059.
  * [21] E. Perrin, R. Harba, I. Iribarren, and R. Jennane, Piecewise fractional Brownian motion, IEEE Trans. Signal Process., 53 (2005), pp. 1211–1215.
  * [22] G. Samorodnitsky and M. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Stochastic Modeling Series, Taylor & Francis, 1994.
  * [23] T. P. Sottinen, Fractional Brownian motion in finance and queueing, 2003.  Thesis (Ph.D.)–Helsingin Yliopisto (Finland).


Conversely, assume that $ ent (\mathcal{A}) \ge k $. This means that there exist distinct states $ u_1, \dots, u_k $ and there exists a monotone sequence  $ (\alpha_i)_{i \in \mathbb{N}} $ that goes through each state infinitely many times. Since the alphabet $ \Sigma $ is finite, up to removing a finite number of initial elements, we can assume that all $ \alpha_n $'s end with the same $ m = |Q|^k $ letters (in particular, all $ \alpha_n $'s have at least length $ m $), so we can write $ \alpha_n = \alpha'_n \theta $, for some $ \theta \in \Sigma^m $. Since $ |Q| $ is finite, up to taking a subsequence we can assume that the sequence still goes through each state infinitely many times, and all the $ \alpha_i $'s ending in the same state share the sorted sequence of the last $ m + 1 $ states of their path on $ \mathcal{A} $ (that is, the states through which we read $ \theta $). Finally, up to take a subsequence we can assume that $ \alpha_i \in I_{u_j} $ if and only if $ j - i $ is a multiple of $ k $, that is, $ \alpha_1 $, $ \alpha_{k + 1} $, $ \alpha_{2k + 1} $, $ \dots $ are in $ I_{u_1} $, $ \alpha_2 $, $ \alpha_{k + 2} $, $ \alpha_{2k + 2} $, $ \dots $ are in $ I_{u_2} $, and so on.

Let $ x^j_0, x^j_1, \dots, x^j_m $, with $ u_j = x^j_m $ the last $ m + 1 $ states of the path labeled $ \alpha_j $ starting from the initial state, for $ j = 1, \dots, k $. By the properties of the monotone sequence, these are also the last $ m + 1 $ states of the path labeled $ \alpha_i $, for every $ i $ such that $ j - i $ is a multiple of $ k $.

Notice the for every $ 0 \le s \le m $ states $ x^j_s $'s are pairwise distinct, otherwise some $ u_j $'s would be equal (because all $ \alpha_n $'s end with the same $ m $ letters). Moreover, for every $ 0 \le s \le m $ the tuple $ (x^1_s, \dots, x^k_s) $ consists of $ k $ states, and we have $ m + 1 = |Q|^k + 1 $ such tuples, so two tuples must be equal. In other words, there exist $ h, h' $, with $ 0 \le h < h' \le m $, such that $ x^j_h = x^j_{h'} $, for every $ j =1, \dots, k $. This means that between $ x^j_h $ and $ x^j_{h'} $ we read a cycle for every $ j $, and this cycle is the same for every $ j $ (because $ \theta $ is the same for every $ j $). Call this cycle $ \gamma' $, write $ \theta = \phi \gamma' \psi $ for some strings $ \phi $ and $ \psi $, and define $ \beta_i = \alpha'_i \phi $. The sequence $ (\beta_i)_{i \in \mathbb{N}} $ is monotone as well, so from the first $ 2k $ elements of this sequence we can pick $ 2k - 1 $ elements $ \delta_1, \dots, \delta_{2k - 1} $ such that:

* $ \delta_1 \prec \dots \prec \delta_{2k - 1} $;

* $ \delta_i $ and $ \delta_{k + i} $ end in the same state $ u_i $, for $ i = 1, \dots, k - 1 $.

* $ u_1, \dots, u_{k - 1} $ and the state $ u_k $ where $ \delta_k $ ends are pairwise distinct.

* For every $ i = 1, \dots, k $ there is a cycle labeled $ \gamma' $ starting at $ u_i $.

Let $ r $ be an integer such that $ |(\gamma')^r| > |\delta_i| $ for every $ i = 1, \dots, 2k - 1 $. Then $ \gamma = (\gamma')^r $ is again a cycle starting at $ u_i $, for every $ i = 1, \dots, k $, and $ \gamma $ is not a suffix of $ \delta_i $, for every $ i = 1, \dots, 2k - 1 $. We distinguish two cases:

* $ \delta_{k} \prec \gamma $. In this case, let $ \mu_1, \dots, \mu_k $ be equal to $ \delta_1, \dots, \delta_k $.

* $ \gamma \prec \delta_{k} $. In this case, let $ \mu_1, \dots, \mu_k $ be equal to $ \delta_k, \dots, \delta_{2k - 1} $.

The conclusion follows with these choices for $ \mu_1, \dots, \mu_k $, $ \gamma $, $ u_1, \dots, u_{k} $.
Consider a   regular language $\mathcal L=\mathcal L(\mathcal A)$, where   $\mathcal A$ is  the  minimum   DFA recognizing   $\mathcal L$ with   initial state $s$. Then  $\text{width}(\mathcal L)\geq k$   if and only if there exist words $\mu_0\prec \mu_1\prec \ldots \prec\mu_{k-1}\prec \mu'_0 \prec \mu'_1\prec \ldots \prec\mu'_{k-1}$  and  $\gamma$  such that:

* for all $i=0, \ldots, k-1$, the strings $\mu_i,\mu'_i$ label  paths from $s$ to a state $u_i$;

* the states    $u_0, \ldots, u_{k-1}$ are pairwise different;

* $\gamma$  labels $k$   cycles    starting from  $u_0, \ldots, u_{k-1}$.

le condizioni di cui sopra sono equivalenti a richiedere l'esistenza di  parole $\nu_0\prec \nu_1\prec \ldots \prec\nu_{k-1}$ e $\gamma'$  per cui  le parole  $\nu_i$ arrivano a stati diversi, in ognuno di questi stati cicla $\gamma'$, $\gamma'$ non e' un suffisso di nessuno dei $\nu_i$ e
$\gamma'$ è minore di tutti i $\nu_i$ o maggiore di tutti i $\nu_i$. Come nel lemma seguente, possiamo poi limitare la lunghezza di tutte queste parole  polinomialmente nel numero degli stati dell'automa. Se queste condizioni sono migliori per l'algoritmo di programmazione dinamica, posso rifrasare il teorema;
se invece per la programmazione dinamica non cambia molto,  nella attuale forma la dimostrazione del Teorema  è leggermente piu' semplice
In the following proof,  for every $n\in \mathbb N$ we denote by   $r(n), q(n)$   the reminder and the quotient of the integer division of $n$ by $k$.

If  the   conditions stated in the theorem  hold, we prove that $\text{ent}(\mathcal A)\geq k$, so that
$\text{width}(\mathcal L) \geq k$ follows from  Theorem <ref>.

W.l.o.g.,  we may suppose   that $\gamma$ is not a suffix of any $\mu_i$'s or $\mu_i'$'s, because  if so   we could  consider a string $\gamma^h$ instead of $\gamma$, for $h$  such that  the length of $\gamma^h$ is greater than the length of all $\mu_i, \mu_i'$.

We consider two cases,

* $\gamma \preceq  \mu_{k-1}$:  then
$\gamma \prec  \mu'_0 \prec \mu'_1\prec \ldots \prec\mu'_{k-1}$.  Since $\gamma$ is not a suffix of $\mu_0'$,
from $\gamma\prec\mu_0' $ it follows $\mu'_{k-1}\gamma \prec  \mu_0' $.  Define

\[
\alpha_n=\mu'_{k-1-r(n)}\gamma^{q(n)}.
\]

\[\alpha_0=\mu'_{k-1}\succ \alpha_1=\mu'_{k-2} \succ \ldots \succ \alpha_{k-1}=\mu'_{0}\succ \alpha_k=\mu'_{k-1}\gamma \succ  \ldots \succ \alpha_{k+(k-1) }=\mu'_{0}\gamma \succ \ldots \]

It follows that  the sequence $(\alpha_n)_{n\in \mathbb N}$ is monotone decreasing.   Moreover   the sequence $(\alpha_n)_{n\in \mathbb N}$ goes through each $u_i$ infinitely often, for each $i=1, \ldots, k-1$. Hence, $\text{ent}(\mathcal A)\geq k$.

* $\mu_{k-1} \prec \gamma $: then
$   \mu_0 \prec \mu_1\prec \ldots \prec\mu_{k-1}\prec \gamma $ and  $\mu_{k-1}\prec \mu_0\gamma$.  Then, by defining

\[
\alpha_n=\mu_{r(n)}\gamma^{q(n)},
\]

we obtain a    monotone increasing  sequence which   goes through each $u_i$ infinitely often, for each
$i=0, \ldots, k-1$. Hence, $\text{ent}(\mathcal A)\geq k$.

Suppose now that $\text{width}(\mathcal L)=\text{ent}(\mathcal A)\geq k$.
By definition of $\text{ent}(\mathcal A)$   there are pairwise different states $q_0, \ldots, q_{k-1}$  and a monotone sequence
$(\alpha_n)_{n\in \mathbb N}$ going through each $q_i$ infinitely often.
Assume that the sequence is increasing  (the decreasing case   is analogous).
By possibly  erasing  a finite number of initial elements in the sequence,  we can   assume that all $\alpha_n$'s  end with the same $m=|\mathcal A|^k$ letters. Let    $\theta \in \Sigma^*$ be such that   $|\theta|=m$ and $\alpha_n=\alpha'_n\theta$, so that  the sequence  $(\alpha'_n)_{n\in \mathbb N}$ is increasing.    Since $\alpha_i' \equiv_{\mathcal L}\alpha_j'$ implies  $\alpha_i\equiv_{\mathcal L}\alpha_j$,  the monotone sequence $(\alpha'_n)_{n\in \mathbb N}$  goes through more than $k$ states infinitely often, and
by possibly considering a subsequence of  $(\alpha'_n)_{n\in \mathbb N}$ we can further suppose that        $i,j$  have the same remainder modulo $k$  if and only if  $\alpha'_i,\alpha'_j$ end  in the same state $x_{r(i)}$.

For $i=0, \ldots,k-1$, consider the path
\[\pi^i=(x_{r(i)}=x^i_0, x^i_{2}, \ldots, x^i_{m})\]
consisting of   the  last $m+1$ states of the $ \alpha_{i}$-labelled path  from the initial state  $s$. Note that all $\alpha_j$'s with  $r(j)=r(i)$ share this path, that all paths $\pi^i$
are labelled by   $\theta$, and  that if $0\leq i<j <k$, then  $x^i_h\neq x^j_h$, for all $h=1, \ldots m$.

Since $|\theta|=m =|\mathcal A|^k $,    we can find $0\leq h<h'\leq m$ with $(x^0_{h}, x^1_{h}, \ldots,x^{k-1}_h)= (x^0_{h'}, x^1_{h'}, \ldots,x^{k-1}_{h'})$ (see table below).

$\pi^0$     =     $x^0_0$,     $x^0_2$,     …,    $x^0_h$,   …,   $x^0_j$,   …,    $x^0_{h'}$,   …,    $x^0_m$

$\pi^1$     =     $x^1_0$,     $x^1_2$,     …,    $x^1_h$,   …,   $x^1_j$,   …,    $x^1_{h'}$,   …,    $x^1_m$

⋮         ⋮

$\pi^{k-1}$     =       $x^{k-1}_0$,      $x^{k-1}_2$,     …,    $x^{k-1}_h$,   …,   $x^{k-1}_j$,   …,    $x^{k-1}_{h'}$,   …,    $x^{k-1}_m$

Hence,  for $i=0, \ldots, k-1$,   the  $k$ subpaths

\[ x^i_{h}, x^i_{h+1}, \ldots , x^i_{h'}=x^i_{h}, \]

are cycles of the same length  labelled by the same word, say $\gamma$.

Note that $|\gamma|\leq |\mathcal A|^k$. We will need this bound together with Lemma <ref> when proving  that $\text{width}(L)$ is calculable.

Since $ \gamma $ is a factor of $ \theta $,  there exist $\eta, \delta \in \Sigma^*$ such that  $\theta= \eta \gamma \delta$.
Moreover, for  $i=0, \ldots,k-1$,    all words $\alpha'_{j}\eta$, where $r(j)=i$ end in the state $x^i_{h}$; moreover, if $0\leq i<i'\leq k-1$, then
$x^i_h\neq x^{i'}_h$ while  $\gamma$ labels  $k$ cycles starting in $x^i_{h}$ for  $i=0, \ldots,  k-1$.

\[
\mu_0=\alpha'_{0}\eta,~~ \mu_1=\alpha'_{1}\eta,~ \ldots~, \mu_{k-1}=\alpha'_{k-1}\eta
\]

\[
\mu'_0=\alpha'_{k}\eta,~~ \mu'_1=\alpha'_{k+1}\eta,~ \ldots ~, \mu'_{k-1}=\alpha'_{2k-1}\eta
\]

then we have
\[\mu_0\prec \mu_1\prec \ldots \prec\mu_{k-1}\prec \mu'_0 \prec \mu'_1\prec \ldots \prec\mu'_{k-1}\]

and the $\mu_i, \mu'_i$ satisfy the conditions stated in the theorem.

To obtain   decidable conditions on the minimum automaton allowing us to calculate the width of a language, we now prove that the lengths of the words $\mu_i, \mu'_i, \gamma$ in Theorem <ref> can be made   polynomial  in   the number of states of the minimum automaton. This will follows from a    general results on    DFA's:

Consider a   regular language $\mathcal L=\mathcal L(\mathcal A)$, where   $\mathcal A$ is  the  minimum   DFA recognizing   $\mathcal L$ with   initial state $s$. Then  $\text{width}(\mathcal L)\geq k$   if and only if there exist words $\mu_0\prec \mu_1\prec \ldots \prec\mu_{k-1}\prec \mu'_0 \prec \mu'_1\prec \ldots \prec\mu'_{k-1}$  and  $\gamma$  such that all conditions in Theorem <ref> hold and moreover $|\mu_i|, |\mu'_i|\leq 2k-1+|\mathcal A|+\ldots |\mathcal A|^{2k}$, $|\gamma|\leq |\mathcal A|^k$.

Apply Remark <ref> and Lemma <ref>.

By using Lemma <ref> we can state a computable variant of Theorem <ref>.

Let $ \mathcal{L} $ be a regular language, and let $ \mathcal{A} $ the minimum DFA of $ \mathcal{L} $, with set of states $ Q $. Let $ k \ge 2 $ be an integer. Then, $ width(\mathcal{L}) \ge k $ if and only if there exist strings $ \mu_1, \dots, \mu_k $ and $ \gamma $ and there exist pairwise distinct $ u_1, \dots, u_k \in Q $ such that:

* $ \mu_j $ labels a path from the initial state $ s $ to $ u_j $, for every $ j = 1, \dots, k $;

* $ \gamma $ labels a cycle starting (and ending) at $ u_j $, for every $ j = 1, \dots, k $;

* either all the $ \mu_j $'s are smaller than $ \gamma $ or $ \gamma $ is smaller than all $ \mu_j $'s;

* $ |\mu_1|, \dots, |\mu_k| < |\gamma| \le 2k - 3 + |Q|^k + \sum_{t = 1}^{2k - 1} |Q|^t $.

$ (\leftarrow) $ follows from Theorem <ref> because 4) implies that $ \gamma $ is not a suffix of any $ \mu_j $. As for $ (\rightarrow) $, it suffices to push forward the proof of Theorem <ref>. Notice that by Lemma <ref> we can assume $ |\delta_i | \le 2k - 3 + \sum_{t = 1}^{2k - 1} |Q|^t $ for every $ i = 1, \dots, 2k - 1 $ (and so $ |\mu_j| \le 2k - 3 + \sum_{t = 1}^{2k - 1} |Q|^t $ for every $ j = 1, \dots, k $). Moreover, notice that $ |\gamma'| \le |Q|^k $ (because $ |\theta| = |Q|^k $), and if we pick the minimum $ r $ such that $ |(\gamma')^r| > |\delta_i| $ for every $ i = 1, \dots, 2k - 1 $, then we obtain $ |\gamma| \le 2k - 3 + |Q|^k + \sum_{t = 1}^{2k - 1} |Q|^t $.

*Statement of Theorem <ref>.
Let $\mathcal L$ be  a regular language, given as input by means of any DFA $\mathcal A = (Q, s, \delta,F)$ recognizing $ \mathcal{L}$.
Then, $p = \text{width}(\mathcal L)$ is  computable in time $|Q|^{O(p)}$.

We exhibit a dynamic programming algorithm based on Corollary <ref>. The exact value for $\text{width}(\mathcal L)$ is then found by exponential and binary searches on $k$,  testing the conditions of  Corollary <ref> for $O(\log (\text{width}(\mathcal L)))$ values of $k$.

Up to minimizing $ \mathcal{A} $ we can assume that $ \mathcal{A} $ is the minimum DFA recognizing $ \mathcal{L} $. Let $N' = 2k - 3 + |Q|^k + \sum_{t = 1}^{2k - 1} |Q|^t$ be the upper-bound to the lengths of the strings $\mu_i$ ($1\leq i \leq k$) and $\gamma$ that need to be considered, and let $N = N'+1$ be the number of states in a path labeled by a string of length $N'$. Asymptotically, note that $N \in O(|Q|^{2k})$.
The high-level idea of the algorithm is as follows. First, in condition (3) of Corollary <ref>, we focus on finding paths $\mu_j$'s smaller than $\gamma$, as the other case (all $\mu_j$'s larger than $\gamma$) can be solved with a symmetric strategy. Then:

* For each state $u$ and each length $2\leq \ell \leq N$, we compute the co-lexicographically smallest path of length (number of states) $\ell$ connecting $s$ with $u$.

* For each $k$-tuple $u_1,\dots, u_k$ and each length $\ell \leq N$, we compute the co-lexicographically largest string $\gamma$ labeling $k$ cycles of length (number of states) $\ell$ originating (respectively, ending) from (respectively, in) all the states $u_1,\dots, u_k$.

Steps (1) and (2) could be naively solved by enumerating the strings $\mu_1, \dots, \mu_k$, and $\gamma$ and trying all possible combinations of states $u_1, \dots, u_k$. Because of the string enumeration step, however, this strategy would be exponential in $N$, i.e. doubly-exponential in $k$. We show that a dynamic programming strategy is exponentially faster.

. This construction is identical to the one  used  in [2] for the Wheeler case $ (p = 1) $. For completeness, we report it here.
Let $\pi_{u,\ell}$, with $u\in Q$ and $2 \leq \ell \leq N$, denote the predecessor of $u$ such that the co-lexicographically smallest path of length (number of states) $\ell$ connecting the source $s$ to $u$ passes through $\pi_{u,\ell}$ as follows: $s \rightsquigarrow \pi_{u,\ell} \rightarrow u$.

The node $\pi_{u,\ell}$ coincides with $s$ if $\ell=2$ and $u$ is a successor of $s$; in this case, the path is simply $s \rightarrow u$.
If there is no path of length $\ell$ connecting $s$ with $u$, then we write $\pi_{u,\ell} = \bot$.
We show that the set
$\{\pi_{u,\ell}\ :\ 2\leq \ell \leq N,\ u\in Q\}$
stores in just polynomial space all co-lexicographically smallest paths of any fixed length $2 \leq \ell \leq N$ from the source to any node $u$. We denote such path — to be intended as a sequence $u_1 \rightarrow \dots \rightarrow u_\ell$ of states — with $\alpha_\ell(u)$.
The node sequence $\alpha_\ell(u)$ can be obtained recursively (in $O(\ell)$ steps) as $\alpha_\ell(u) = \alpha_{\ell-1}(\pi_{u,\ell}) \rightarrow u$, where $\alpha_{1}(s) = s$ by convention.
Note also that $\alpha_\ell(u)$ does not fully specify the sequence of edges (and thus labels) connecting those $\ell$ states, since two states may be connected by multiple (differently labeled) edges. However, the corresponding co-lexicographically smallest sequence $\lambda^-(\alpha_\ell(u))$ of $\ell-1$ labels is uniquely defined as follows:

\left\{
\begin{array}{ll}
\lambda^-(\alpha_\ell(u)) = \mathrm{min}\{a\in\Sigma\ |\ \delta(s,a)=u\} & \mathrm{if}\ \ell=2 ,\\
\lambda^-(\alpha_\ell(u)) = \lambda^-(\alpha_{\ell-1}(\pi_{u,\ell}) \rightarrow u) = \lambda^-(\alpha_{\ell-1}(\pi_{u,\ell})) \cdot \mathrm{min}\{a\in\Sigma\ |\ \delta(\pi_{u,\ell},a)=u\} & \mathrm{if}\ \ell > 2.
\end{array}\right.

It is not hard to see that each $\pi_{u,\ell}$ can be computed in $|Q|^{O(1)}$ time using dynamic programming. First, we set $\pi_{u,2} = s$ for all successors $u$ of $s$. Then, for $\ell = 3, \dots, N$:

\pi_{u,\ell} = \underset{v\in \text{Pred}(u)}{\mathrm{argmin}} \Big(
\lambda^-( \alpha_{\ell-1}(v) ) \cdot  \mathrm{min}\{a\in\Sigma\ |\ \delta(v,a)=u\} \Big)

where $\text{Pred}(u)$ is the set of all predecessors of $u$ and the $\mathrm{argmin}$ operator compares strings in co-lexicographic order. In the equation above, if none of the $\alpha_{\ell-1}(v)$ are well-defined (because there is no path of length $\ell-1$ from $s$ to $v$), then $\pi_{u,\ell} = \bot$.
Since there are $|Q|\times N = |Q|^{O(k)}$ variables $\pi_{u,\ell}$ and each can be computed in time  $|Q|^{O(1)}$, overall Step (1) takes $|Q|^{O(k)}$ times.
This completes the description of Step (1).

. Fix a $k$-tuple $u_1,\dots, u_k$ and a length $2\leq \ell \leq N$. Our goal is now to show how to compute the co-lexicographically largest string $\gamma$ of length $\ell -1$ labeling $k$ cycles of length (number of states) $\ell$  originating (respectively, ending) from (respectively, in) all the states $u_1,\dots, u_k$. Our final strategy will iterate over all such $k$-tuple of states (in time exponential in $k$) in order to find one satisfying the conditions of Corollary <ref>.

Our goal can again be solved by dynamic programming. Let $u_1,\dots, u_k$ and $u'_1,\dots, u'_k$ be two $k$-tuples of states, and let $2\leq \ell \leq N$. Let moreover $\pi_{u_1,\dots, u_k, u'_1,\dots, u'_k, \ell}$ be the $k$-tuple $\langle u''_1,\dots, u''_k \rangle$ of states such that there exists a string $\gamma$ of length $\ell-1$ with the following properties:

* For each $1\leq i \leq k$, there is a path $u_i \rightsquigarrow u''_i \rightarrow u'_i$ of length (number of nodes) $\ell$ labeled with $\gamma$, and

* $\gamma$ is the co-lexicographically largest string satisfying the above property.

If such a string $\gamma$ does not exist, then we set $\pi_{u_1,\dots, u_k, u'_1,\dots, u'_k, \ell} = \bot$.

Remember that we fix $u_1,\dots, u_k$.
For $\ell = 2$ and each $k$-tuple $u'_1,\dots, u'_k$, it is easy to compute $\pi_{u_1,\dots, u_k, u'_1,\dots, u'_k, \ell}$:  this $k$-tuple is $\langle u_1,\dots, u_k \rangle$ (all paths have length 2) if and only if there exists $c\in\Sigma$ such that $u'_i \in \delta(u_i,c)$ for all $1\leq i\leq k$. Then, $\delta$ is formed by one character: the largest such $c$.

For $\ell >2$, the $k$-tuple $\pi_{u_1,\dots, u_k, u'_1,\dots, u'_k, \ell}$ can be computed as follows. Assume we have computed those variables for all lengths $\ell'<\ell$.
Note that for each such $\ell'<\ell$ and $k$-tuple $u''_1, \dots, u''_k$, the variables $\pi_{u_1,\dots, u_k, u''_1,\dots, u''_k, \ell'}$ identify $k$ paths $u_i \rightsquigarrow u''_i$ of length (number of nodes) $\ell'$. Let us denote with $\alpha_{\ell'}(u''_i)$ such paths, for $1\leq i\leq k$.

Then, $\pi_{u_1,\dots, u_k, u'_1,\dots, u'_k, \ell}$ is equal to  $\langle u''_1,\dots, u''_k \rangle$ maximizing co-lexicographically the string $\gamma'\cdot c$ defined as follows:

* $u'_i \in \delta(u''_i,c)$ for all $1\leq i\leq k$,

* $\pi_{u_1,\dots, u_k, u''_1,\dots, u''_k, \ell-1} \neq \bot$, and

* $\gamma'$ is the co-lexicographically largest string labeling all the paths $\alpha_{\ell-1}(u''_i)$. Note that this string exists by condition (2), and it can be easily built by following those paths in parallel (choosing, at each step, the largest character labeling all the $k$ considered edges of the $k$ paths).

If no $c\in \Sigma $ satisfies condition (1), or condition (2) cannot be met, then $\pi_{u_1,\dots, u_k, u'_1,\dots, u'_k, \ell} = \bot$.

Note that $\pi_{u_1,\dots, u_k, u_1,\dots, u_k, \ell}$ allows us to identify (if it exists) the largest string $\gamma$ of length $\ell-1$ labeling $k$ cycles originating and ending in each $u_i$, for $1\leq i \leq k$.

Each tuple $\pi_{u_1,\dots, u_k, u'_1,\dots, u'_k, \ell}$ can be computed in $|Q|^{O(1)}$ time by dynamic programming (in order of increasing $\ell$), and there are $|Q|^{O(k)}$ such tuples to be computed (there are $|Q|^{O(k)}$ ways of choosing $u_1,\dots, u_k, u'_1,\dots, u'_k$, and $N \in O(|Q|^{2k}$). Overall, also Step (2) can therefore be solved in $|Q|^{O(k)}$ time.

To sum up, we can check if the conditions of Corollary <ref> hold as follows:

* We compute $\pi_{u_i,\ell}$ for each $u\in Q$ and $\ell \leq N$. This identifies a string $\mu_u^\ell$ for each such pair $u\in Q$ and $\ell \leq N$: the co-lexicographically smallest one, of length $\ell$, labeling a path connecting $s$ with $u$.

* For each $k$-tuple $u_1, \dots, u_k$ and each $\ell \leq N$, we compute $\pi_{u_1,\dots, u_k, u_1,\dots, u_k, \ell}$. This identifies a string $\gamma_{u_1,\dots, u_k}^\ell$ for each such tuple $u_1,\dots, u_k$ and $\ell \leq N$: the co-lexicographically largest one, of length $\ell$, labeling $k$ cycles  originating and ending in each $u_i$, for $1\leq i \leq k$.

* We identify the $k$-tuple $u_1, \dots, u_k$ and the lengths $\ell' < \ell \leq N$ (if they exist) such that $\mu_{u_i}^{\ell'} \prec \gamma_{u_1,\dots, u_k}^\ell$ for all $1\leq i \leq k$.

The conditions of Corollary <ref> hold if and only if step 3 above succeeds for at least one $k$-tuple $u_1, \dots, u_k$ and lengths $\ell' < \ell \leq N$. Overall, including the main exponential search on $k$, the algorithm terminates in $|Q|^{O(width(\mathcal L(\mathcal A)))}$ time.

§ PROOFS OF SECTION <REF>

Let $\mt L \subseteq \Sigma^* $ be a language, and let $ \sim $ be an equivalence relation on $ \pf L $. We say that $ \sim $ respects $ \pf L $ if:

\begin{equation*}
(\forall \alpha, \beta \in \pf L)(\forall \phi \in \Sigma^*)(\alpha \sim \beta \land \alpha \phi \in \pf L \to \beta \phi \in \pf L).
\end{equation*}

Now, let us define the right-invariant, $ \mathcal{P} $-consistent and $ \mathcal{P} $-convex refinements of an equivalence relation $ \sim $.

* Assume that $ \sim $ respects $ \pf  L $. For every $ \alpha, \beta \in \pf  L $, define:

\begin{equation*}
\alpha \sim^r \beta \iff (\forall \phi \in \Sigma^*)(\alpha \phi \in \pf L \to \alpha \phi \sim \beta \phi).
\end{equation*}

We say that $ \sim^r $ is the right-invariant refinement of $ \sim $.

* Let $ \mathcal{P} = \{U_1, \dots, U_p \} $ be a partition of $ \pf  L $. For every $ \alpha, \beta \in  \pf  L$, define:

\begin{equation*}
\alpha \sim^{cs} \beta \iff (\alpha \sim \beta) \land (U_\alpha = U_\beta)
\end{equation*}

We say that $ \sim^{cs} $ is the $ \mathcal{P} $-consistent refinement of $ \sim $.

* Let $ \mathcal{P} = \{U_1, \dots, U_p \} $ be a partition of $ \pf L $. Assume that $ \sim $ is $ \mathcal{P} $-consistent. For every $ \alpha, \gamma   \in \pf L $, define:

\begin{equation*}
\begin{split}
& \alpha \sim^{cv} \gamma \iff (\alpha \sim \gamma) \land \\
& \land (\forall \beta \in \pf L)(((U_{\alpha} = U_{\beta}) \land (\min \{\alpha, \gamma \} \prec \beta \prec \max \{\alpha, \gamma \}) \to \alpha \sim \beta).
\end{split}
\end{equation*}

We say that $ \sim^{cv} $ is the $ \mathcal{P} $-convex refinement of $ \sim $.

It is easy to check that $ \sim^r $ is the coarsest right-invariant equivalence relation refining $ \sim $, $ \sim^{cs} $ is the coarsest $ \mathcal{P} $-consistent equivalence relation refining $ \sim $ and $ \sim^{cv} $ is the coarsest $ \mathcal{P} $-convex equivalence relation refining $ \sim $.

We wish to prove that any equivalence relation that respects $ Pref (\mathcal{L}) $ admits a coarsest refinement being $ \mathcal{P} $-consistent, $ \mathcal{P} $-convex and right-invariant at once, because then we will be able to define an equivalence relation inducing the minimum ($ \mathcal{P} $-sortable) DFA.

Let $ \mathcal{L} \subseteq \Sigma^* $ be a language, and let $ \mathcal{P} $ be a partition of $ Pref (\mathcal{L}) $. If $\sim$ is a $ \mathcal{P} $-consistent and right-invariant equivalence relation on $ Pref (\mathcal{L}) $, then the relation $(\sim^{cv})^r$ is $ \mathcal{P} $-consistent, $ \mathcal{P} $-convex and right-invariant.

By definition $(\sim^{cv})^r$ is a right-invariant refinement. Moreover, $ \sim^{cv}$ and $(\sim^{cv})^r$ are $ \mathcal{P} $-consistent because they are refinements of the $ \mathcal{P} $-consistent equivalence relation $ \sim $. Let us prove that $ (\sim^{cv})^r $ is $\mathcal{P} $-convex. Assume that $ \alpha (\sim^{cv})^r \gamma $ and  $ \alpha \prec \beta \prec \gamma $ are such that $ U_\alpha  = U_\beta  $.  Being $(\sim^{cv})^r$ a $\mt P$-consistent relation, we have $ U_\alpha  = U_\beta =U_\gamma $. We must prove that $ \alpha (\sim^{cv})^r \beta $. Fix $ \phi \in \Sigma^* $ such that $ \alpha \phi \in \pf L $. We must prove that $ \alpha \phi \sim^{cv} \beta \phi $. Now, $ \alpha (\sim^{cv})^r \gamma $ implies $ \alpha \sim^{cv} \gamma $. Since $ \alpha \prec \beta \prec \gamma $ and $ U_\alpha = U_\beta = U_\gamma $, then the $ \mathcal{P} $-convexity of $ \sim^{cv} $ implies $ \alpha \sim^{cv} \beta $. In particular, $ \alpha \sim\beta $. Since $\sim$ is   right-invariant   we have  $ \alpha \phi \sim  \beta \phi $, and from the $\mt P$-consistency of $\sim$   we obtain    $ U_{\alpha \phi} = U_{\beta \phi} $. Moreover, $ \alpha (\sim ^{cv})^r \gamma $ implies $ \alpha \phi (\sim ^{cv})^r \gamma \phi $ by right-invariance, so $ \alpha \phi \sim ^{cv} \gamma \phi $. By $ \mathcal{P} $-convexity, from $ \alpha \phi \sim ^{cv} \gamma \phi $, $ U_{\alpha \phi} = U_{\beta \phi} $ and $ \alpha \phi \prec \beta \phi \prec \gamma \phi $ (since $ \alpha \prec \beta \prec \phi) $ we conclude $ \alpha \phi \sim ^{cv} \beta \phi $.

Let $ \mathcal{L} \subseteq \Sigma^* $ be a nonempty language, and let $ \mathcal{P} $ be a partition of $ Pref (\mathcal{L}) $. Let $ \sim $ be an equivalence relation that refines $ Pref (\mathcal{L}) $. Then, there exists a (unique) coarsest $ \mathcal{P} $-consistent, $ \mathcal{P} $-convex and right-invariant equivalence relation refining $ \sim $.

The equivalence relation $ (\sim^{cs})^r $ is $ \mathcal{P} $-consistent (because it is a refinement of the $ \mathcal{P} $-consistent equivalence relation $ \sim^{cs} $) and right-invariant (by definition it is a right-invariant refinement), so by Lemma <ref> the  equivalence relation $(((\sim^{cs})^r)^{cv})^r$ is $ \mathcal{P} $-consistent, $ \mathcal{P} $-convex and right-invariant. Moreover, every $ \mathcal{P} $-consistent, $ \mathcal{P} $-convex and right-invariant equivalence relation refining $ \sim $ must also refine $(((\sim^{cs})^r)^{cv})^r$, so $(((\sim^{cs})^r)^{cv})^r$ is the coarsest $ \mathcal{P} $-consistent, $ \mathcal{P} $-convex and right-invariant equivalence relation refining $ \sim $.

Corollary <ref> allows us to give the following definition.

Let $ \mt L \subseteq \Sigma^* $ be a language, and let $ \mathcal{P} = \{U_1, \dots, U_p \} $ be a partition of $ \pf L $. Denote by $ \equiv_\mathcal{L}^\mathcal{P} $ the coarsest $ \mathcal{P} $-consistent, $ \mathcal{P} $-convex and right-invariant equivalence relation refining the Myhill-Nerode equivalence $ \equiv_\mathcal{L} $.

Recall that, given a DFA $ \mathcal{A} = (Q, s, \delta, F) $, the equivalence relation $ \sim_\mathcal{A} $ on $ Pref (\mathcal{L(A)} $ is the one such that:

\begin{equation*}
\alpha \sim_\mathcal{A} \beta  \iff \delta (s, \alpha) = \delta (s, \beta).
\end{equation*}

Here are the key properties of $ \sim_\mathcal{A} $.

Let $ \mathcal{A} = (Q, s, \delta, F) $ be a $ \mathcal{P} $-sortable DFA, where $ \mathcal{P} = \{U_1, \dots, U_p \} $ is a partition of $ Pref (\mathcal{L(A)}) $. Then, $ \sim_\mathcal{A} $ refines $ \equiv_\mathcal{L} $, it respects $ \pf L $, it is $ \mathcal{P} $-consistent,$ \mathcal{P} $-convex, right-invariant, it has finite index and $ \mathcal{L(A)} $ is the union of some $ \sim_\mathcal{A} $-equivalence classes. In particular, $ \equiv_\mathcal{L}^\mathcal{P} $ is refined by $ \sim_\mathcal{A} $, it has finite index and $ \mathcal{L(A)} $ is the union of some $ \equiv_\mathcal{L}^\mathcal{P} $-equivalence classes.

Assume that $ \alpha \sim_\mathcal{A} \beta $. This means that if we read $ \alpha $ and $ \beta $ on $ \mathcal{A} $, then we reach the same state, which immediately implies that $ \mathcal{L(A)} $ is the union of some $ \sim_\mathcal{A} $-equivalence classes. Moreover, if $ \phi \in \Sigma^* $ satisfies $ \alpha \phi \in Pref (\mathcal{L(A)}) $, then it must be $ \beta \phi \in Pref (\mathcal{L(A)} $ and $ \delta(s, \alpha \phi) = \delta(s, \beta \phi) $, so proving that $ \sim_\mathcal{A} $ respects $ Pref (\mathcal{L(A)}) $, it is right-invariant and it refines $ \equiv_\mathcal{L} $. For every $ \alpha \in \pf L $ we have $ [\alpha]_\mathcal{A} = I_{\delta(s, \alpha)} $, which implies that $ \sim_\mathcal{A} $ is $ \mathcal{P} $-consistent and it has finite index. Moreover, $ \sim_\mathcal{A} $ is $ \mathcal{P} $-convex because for every $ \alpha \in Pref (\mathcal{L(A)}) $ we have that $ I_{\delta(s, \alpha)} $ is convex in $ U_\alpha $, because if $ q_1, \dots, q_k \in Q $ are such that $ U_\alpha = \cup_{i = 1}^k I_{q_i} $, then the $ I_{q_i} $'s are pairwise $ \le_\mathcal{A} $-comparable, being in the same $ \le_\mathcal{A} $-chain. Finally, $ \equiv_\mathcal{L}^\mathcal{P} $ is refined by $ \sim_\mathcal{A} $ because $ \equiv_\mathcal{L}^\mathcal{P} $ is the coarsest $ \mathcal{P} $-consistent, $ \mathcal{P} $-convex and right-invariant equivalence relation refining $ \equiv_\mathcal{L} $, so $ \equiv_\mathcal{L}^\mathcal{P} $ has finite index and $ \mathcal{L(A)} $ is the union of some $ \equiv_\mathcal{L}^\mathcal{P} $-equivalence classes because $ \sim_\mathcal{A} $ satisfies these properties.

We can now explain how to canonically build a $ \mathcal{P} $-sortable DFA starting from an equivalence relation.

Let $ L \subseteq \Sigma^* $ be a language, and let $ \mathcal{P} = \{U_1, \dots, U_p \} $ be a partition of $ \pf L $.  Assume that $ \mathcal{L} $ is the union of some classes of a $ \mathcal{P} $-consistent, $ \mathcal{P} $-convex, right invariant equivalence relation $ \sim $ on $ \pf L $ of finite index. Then, $ \mathcal{L} $ is recognized by a $ \mathcal{P} $-sortable DFA $ \mathcal{A_\sim} = (Q_\sim, s_\sim, \Sigma, E_\sim, F_\sim) $ such that:

* $ |Q_\sim| $ is equal to the index of $ \sim $;

* $ \sim_{A_\sim} $ and $ \sim $ are the same equivalence relation (in particular, $ |Q_\sim| $ is equal to the index of $ \sim_{A_\sim} $).

Moreover, if $ \mathcal{B} $ is a $ \mathcal{P} $-sortable DFA that recognizes $ \mathcal{L} $, then $ \mathcal{A_{\sim_\mathcal{B}}} $ is isomorphic to $ \mathcal{B} $.

Define the DFA $ \mathcal{A_\sim} = (Q_\sim, E_\sim, \Sigma, s_\sim, F_\sim) $ as follows.

* $ Q_\sim = \{[\alpha]_\sim | \alpha \in \pf L\} $;

* $ s_\sim = [\epsilon]_\sim $, where $ \epsilon $ is the empty string;

* $ E_\sim = \{([\alpha]_\sim, [\alpha a]_\sim, a) | \alpha \in \Sigma^*, a \in \Sigma, \alpha a \in \pf L \}   $;

* $ F_\sim = \{[\alpha] | \alpha \in \mathcal{L} \} $.

Since $ \sim $ is right-invariant, it has finite index and $ \mathcal{L} $ is the union of some $ \sim $-classes, then $ \mathcal{A_\sim} $ is a well-defined DFA (see also the proof of Theorem <ref>) and

\begin{equation}\label{eq13}
\alpha \in [\beta]_\sim \iff \delta_\sim (s_\sim, \alpha) = [\beta]_\sim.
\end{equation}

which implies that for every $ \alpha \in Pref(\mathcal{L}) $ it holds $ I_{[\alpha]_\sim} = [\alpha]_\sim $, and so $ \mathcal{L}(\mathcal{A_\sim}) = \mathcal{L} $.

For every $ i \in \{1, \dots, p \} $, define:

\begin{equation*}
Q_i = \{[\alpha]_\sim | U_\alpha = U_i \}.
\end{equation*}

Notice that each $ Q_i $ is well-defined because $ \sim $ is $ \mathcal{P} $-consistent, and each $ Q_i $ is a $ \preceq_{\mathcal{A_\sim}} $-chain because $ \sim $ is $ \mathcal{P} $-convex, so $ \{Q_i\}_{i = 1}^p $ is a $ \preceq_{\mathcal{A_\sim}} $-chain partition of $ Q_\sim $.

From equation <ref> we obtain:

\begin{equation*}
\begin{split}
Pref (\mathcal{L(A_\sim)})^i & = \{\alpha \in Pref (\mathcal{L(A_\sim)}) | \delta_\sim (s_\sim, \alpha) \in Q_i \} = \\
& = \{\alpha \in Pref (\mathcal{L(A_\sim)}) | (\exists [\beta]_\sim \in Q_i | \alpha \in [\beta]_\sim) \} = \\
& = \{\alpha \in Pref (\mathcal{L(A_\sim)}) | U_\alpha = U_i \} = U_i.
\end{split}
\end{equation*}

In other words, $ \mathcal{A}_\sim $ witnesses that $ \mathcal{L} $ is recognized by a $ \mathcal{P} $-sortable DFA. Moreover:

* The number of states of $ \mathcal{A}_\sim $ is clearly equal to the index of $ \sim $.

* By equation <ref>:

\begin{equation*}
\alpha \sim_{A_\sim} \beta \iff \delta_\sim (s_\sim, \alpha) = \delta_\sim (s_\sim, \beta) \iff [\alpha]_\sim = [\beta]_\sim \iff \alpha \sim \beta
\end{equation*}

so $ \sim_{A_\sim} $ and $ \sim $ are the same equivalence relation.

Finally, suppose $ \mathcal{B} $ is a $ \mathcal{P} $-sortable DFA that recognizes $ \mathcal{L} $. Notice that by lemma <ref> we have that $ \sim_{\mathcal{B}} $ is a $ \mathcal{P} $-consistent, $ \mathcal{P} $-convex, right invariant equivalence relation on $ Pref(\mathcal{L}) $ of finite index such that $ \mathcal{L} $ is the union of some $ \sim_\mathcal{B} $-classes, so $ \mathcal{A_{\sim_\mathcal{B}}} $ is well-defined. Call $ Q_\mathcal{B} $ the set of states of $ \mathcal{B} $, and let $ \phi: Q_{\sim_\mathcal{B}} \to Q_\mathcal{B} $ be the function sending $ [\alpha]_{\sim_\mathcal{B}} $ into the state in $ Q_\mathcal{B} $ reached by reading $ \alpha $. Notice that $ \phi $ is well-defined because by the definition of $ \sim_{\mathcal{B}} $ we obtain that all strings in $ [\alpha]_{\sim_\mathcal{B}} $ reach the same state of $ \mathcal{B} $. It is easy to check that $ \phi $ determines an isomorphism between $ \mathcal{A_{\sim_\mathcal{B}}} $ and $ \mathcal{B} $.

Here is the proof of our Myhill-Nerode theorem.

*Statement of Theorem <ref>.    Let $ \mathcal{L} $ be a language. Let $ \mathcal{P} $ be a partition of $ \pf L $. The following are equivalent:

* $ \mathcal{L} $ is recognized by a $ \mathcal{P} $-sortable DFA.

* $ \equiv_\mathcal{L}^\mathcal{P} $ has finite index.

* $ \mathcal{L} $ is the union of some classes of a $ \mathcal{P} $-consistent, $ \mathcal{P} $-convex, right invariant equivalence relation on $ \pf L $ of finite index.

Moreover, if one of the above statements is true (and so all the above statements are true), then there exists a unique minimum $ \mathcal{P} $-sortable DFA (that is, two $ \mathcal{P} $-sortable DFA having the minimum number of states must be isomorphic).

$ (1) \to (2) $ It follows from lemma <ref>

$ (2) \to (3) $ The desired equivalence relation is simply $ \equiv_\mathcal{L}^\mathcal{P} $.

$ (3) \to (1) $ It follows from lemma <ref>.

Now, let us prove that the minimum automaton is $ \mathcal{A_{\equiv_\mathcal{L}^\mathcal{P}}} $ as defined in Lemma <ref>. First, $ \mathcal{A_{\equiv_\mathcal{L}^\mathcal{P}}} $ is well-defined because $ \equiv_\mathcal{L}^\mathcal{P} $ is $ \mathcal{P} $-consistent, $ \mathcal{P} $-convex an right-invariant by definition, and it has finite index and $ \mathcal{L(A)} $ is the union of some $ \equiv_\mathcal{L}^\mathcal{P} $-equivalence classes by lemma <ref>. Now, the number of states of $ \mathcal{A_{\equiv_\mathcal{L}^\mathcal{P}}} $ is equal to the index of $ \equiv_\mathcal{L}^\mathcal{P} $, or equivalently, of $ \sim_{\mathcal{A_{\equiv_\mathcal{L}^\mathcal{P}}}} $. On the other hand, let $ \mathcal{B} $ be any $ \mathcal{P} $-sortable DFA recognizing $ \mathcal{L} $ non-isomorphic to $ \mathcal{A_{\equiv_\mathcal{L}^\mathcal{P}}} $. Then $ \sim_\mathcal{B} $ is a refinement of $ \equiv_\mathcal{L}^\mathcal{P} $ by Lemma <ref>, and it must be a strict refinement of $ \equiv_\mathcal{L}^\mathcal{P} $, otherwise $ \mathcal{A_{\equiv_\mathcal{L}^\mathcal{P}}} $ would be equal to $ A_{\sim_\mathcal{B}} $, which by Lemma <ref> is isomorphic to $ \mathcal{B} $, a contradiction. We conclude that the index of $ \equiv_\mathcal{L}^\mathcal{P} $ is smaller than the index of $ \sim_\mathcal{B} $, so again by Lemma <ref> the number of states of $ \mathcal{A_{\equiv_\mathcal{L}^\mathcal{P}}} $ is smaller than the number of states of $ A_{\sim_\mathcal{B}} $ and so of $ \mathcal{B} $.

[1]
Jarno Alanko, Giovanna D'Agostino, Alberto Policriti, and Nicola Prezza.
Regular languages meet prefix sorting.
In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM
Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA,
January 5-8, 2020, pages 911–930. SIAM, 2020.

[2]
Jarno Alanko, Giovanna D'Agostino, Alberto Policriti, and Nicola Prezza.
Wheeler languages.
CoRR, abs/2002.10303, 2020.

[3]
Jarno N. Alanko, Travis Gagie, Gonzalo Navarro, and Louisa Seelbach Benkner.
Tunneling on wheeler graphs.
In Ali Bilgin, Michael W. Marcellin, Joan Serra-Sagristà, and
James A. Storer, editors, Data Compression Conference, DCC 2019,
Snowbird, UT, USA, March 26-29, 2019, pages 122–131. IEEE, 2019.

[4]
Janusz A. Brzozowski and Faith E. Fich.
Languages of r-trivial monoids.
J. Comput. Syst. Sci., 20(1):32–49, 1980.

[5]
Arturs Backurs and Piotr Indyk.
Which regular expression patterns are hard to match?
CoRR, abs/1511.07070, 2015.

[6]
Michael Burrows and David J Wheeler.
A block-sorting lossless data compression algorithm.
Technical Report 124, Digital Equipment Corporation, 1994.

[7]
Nicola Cotumaccio, Giovanna D'Agostino, Alberto Policriti, and Nicola Prezza.
Which regular languages can be efficiently indexed?
CoRR, abs/2102.06798, 2021.

[8]
Nicola Cotumaccio and Nicola Prezza.
On indexing and compressing finite automata.
In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM
Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10
- 13, 2021, pages 2585–2599. SIAM, 2021.

[9]
R. P. Dilworth.
A decomposition theorem for partially ordered sets.
Annals of Mathematics, 51(1):161–166, 1950.

[10]
Giovanna D'Agostino, Davide Martincigh, and Alberto Policriti.
Ordering regular languages: a danger zone.
CoRR, abs/2106.00315, 2021.

[11]
Massimo Equi, Roberto Grossi, and Veli Mäkinen.
On the complexity of exact pattern matching in graphs: Binary strings
and bounded degree.
CoRR, abs/1901.05264, 2019.

[12]
Massimo Equi, Veli Mäkinen, and Alexandru I. Tomescu.
Graphs cannot be indexed in polynomial time for sub-quadratic time
string matching, unless SETH fails.
In Tomás Bures, Riccardo Dondi, Johann Gamper, Giovanna
Guerrini, Tomasz Jurdzinski, Claus Pahl, Florian Sikora, and Prudence W. H.
Wong, editors, SOFSEM 2021: Theory and Practice of Computer Science -
47th International Conference on Current Trends in Theory and Practice of
Computer Science, SOFSEM 2021, Bolzano-Bozen, Italy, January 25-29, 2021,
Proceedings, volume 12607 of Lecture Notes in Computer Science, pages
608–622. Springer, 2021.

[13]
Daniel Gibney, Gary Hoppenworth, and Sharma V. Thankachan.
Simple reductions from formula-sat to pattern matching on labeled
graphs and subtree isomorphism.
In Hung Viet Le and Valerie King, editors, 4th Symposium on
Simplicity in Algorithms, SOSA 2021, Virtual Conference, January 11-12,
2021, pages 232–242. SIAM, 2021.

[14]
Travis Gagie, Giovanni Manzini, and Jouni Sirén.
Wheeler graphs: A framework for BWT-based data structures.
Theoretical Computer Science, 698:67 – 78, 2017.
Algorithms, Strings and Theoretical Approaches in the Big Data Era
(In Honor of the 60th Birthday of Professor Raffaele Giancarlo).

[15]
Daniel Gibney and Sharma V. Thankachan.
On the hardness and inapproximability of recognizing wheeler graphs.
In 27th Annual European Symposium on Algorithms, ESA 2019,
September 9-11, 2019, Munich/Garching, Germany., pages 51:1–51:16, 2019.

[16]
Leonid Libkin.
Elements of Finite Model Theory.
Springer, 2004.

[17]
Tomás Masopust and Markus Krötzsch.
Partially ordered automata and piecewise testability.
Log. Methods Comput. Sci., 17(2), 2021.

[18]
Nicola Prezza.
On locating paths in compressed tries.
In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM
Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10
- 13, 2021, pages 744–760. SIAM, 2021.

[19]
Aaron Potechin and Jeffrey O. Shallit.
Lengths of words accepted by nondeterministic finite automata.
Inf. Process. Lett., 162:105993, 2020.

[20]
H.-J. Shyr and G. Thierrin.
Ordered automata and associated languages.
Tamkang J. Math, 5:9–20, 1974.

[21]
Thomas Schwentick, Denis Thérien, and Heribert Vollmer.
Partially-ordered two-way automata: A new characterization of DA.
In Werner Kuich, Grzegorz Rozenberg, and Arto Salomaa, editors, Developments in Language Theory, 5th International Conference, DLT 2001,
Vienna, Austria, July 16-21, 2001, Revised Papers, volume 2295 of Lecture Notes in Computer Science, pages 239–250. Springer, 2001.


# Ab initio study of structural properties and dynamical stability of C3N2
encapsulating H guest atom

George S. Manyali  1Computational and Theoretical Physics Group, Department of
Physical Sciences, Kaimosi Friends University College P.O Box 385-50309,
Kaimosi Kenya

###### Abstract

We have investigated the structural properties and dynamical stability of C3N2
hosting hydrogen as a guest atom. The calculations were performed using the
density functional theory (DFT) as implemented in Quantum espresso code. The
generalized gradient approximation approach was used throughout the
calculations presented here. We find that the guest atom leaves the lattice
constant of the host almost unchanged. We, therefore, noted that C3N2 remains
dynamically stable when a single guest atom is inserted into its cage-like
structure. We also established that the guest vibrational modes are low-lying
modes that lowers the acoustic bandwidth of C3N2.

###### keywords:

C3N2, dynamical stability , H guest atom

††journal: Materials today communicationfn1fn1footnotetext:
<EMAIL_ADDRESS>
##  1 Introduction

The cage-like voids in crystal structures plays a key role in modifying the
behavior of materials. The voids are the guest spots where guest atoms are
inserted to form a stabilized compound or to improve material’s electrical
conductivity. A good example are the C60 and C20 fullerenes. They have a
number of very important physical and chemical properties that make them
desirable for special applications. It is reported that fullerene cages can
accommodate more than one guest atoms or molecules [1, 2]. Using C20 concept,
Tian et. al[3] combined carbon and nitrogen to predict C3N2, a novel hard
material. This followed other interesting works of Wang et.al [4], who
reported an unexpected high-pressure stabilization of cagelike diamondoid
nitrogen above 263 GPa. This findings provided a significant understanding of
behavior of nitrogen related material under extreme conditions. Previous
studies[5, 6, 7, 8, 9] on properties of C3N2, did cover the question of
whether the cage have inner cavity large enough to hold any atom and whether
it remains stable when it hosts a guest atom.

In the present study, atom insertion in the C3N2 host lattice has been
performed under zero-pressure conditions. The guest atoms occupied the body-
centered vacant site at (0,0,0) in the C3N2 host. We focused on the smallest
species with atomic radius(H=0.53Å). Investigation on guest atoms with atomic
radii of 1.12Å, 1.57Å, and 2.53Ådid not yield meaningful results and were
therefore not considered in this work.

The paper is outlined as follows: In Section 2 we will present the details of
our calculations and in Section 3 we will present our results. Finally in
Section 4 we will summarize our findings and present our conclusions.

##  2 Computational methodology

The first step in this work was to optimize the geometry of the lattice
structures and the determination of the equations of state. Therefore, the
density functional theory (DFT), which has been implemented in the QUANTUM
ESPRESSO code[10] was used. Thermo_pw package [11], a driver of the Quantum
ESPRESSO routines for the calculation of material properties was employed in
the calculations of phonons based on density-functional perturbation theory
[12]. The exchange correlation functional known as Generalized gradient
approximation (GGA) in the form of Perdew-Burke-Ernzerhof (PBE)[13] and
projector augmented wave (PAW) potentials[14] were used. A plane wave cutoff
energy of 40Ry, a density cutoff of 300Ry and a 6x6x6 Monkhorst-Pack[15] grid
for the electronic integrations were used throughout. Phonon frequencies were
calculated on a 3x3x3 q-point grid.

##  3 Results and discussion

###  3.1 Structural properties

The initial configuration of atoms in the unit cell of the simple cubic C3N2
is shown in figure 1. The guest atom occupied the body-centered vacant site at
(0,0,0) in the C3N2 host before the structural optimization of the lattice.
After the optimization, the lattice parameter of C3N2 was determined to be
5.095Å, which was in good agreement with previous studies[3, 6]. When H atom
was inserted in the C3N2 host lattice, the cage expands slightly as presented
in Table1. Such very small distortions may have very little impact on the
overall structural and elastic properties. It should be noted that the cage in
C3N2 is not large enough to hold atoms such as beryllium, lithium and barium,
which have atomic radii of 1.12Å, 1.57Å, and 2.53Å respectively. The guest
atom forces the cage to expand, just as the structure loses its dynamical
stability as the lattice parameter increases significantly.

(a) (b)

Figure 1: (a)Unit cell of the simple cubic C3N2 with space group Pm-3m(221).
The guest atoms occupied the body-centered vacant site at (0,0,0) in the C3N2
host as shown in (b).

Other parameters such as density also remained largely unchanged, indicating
none of the atoms were incorporated into C3N2 host.

Table 1: Lattice constant a(Å), ground-state energy (E0 in Ry) and density
$\rho$ (in g/cm3)

Property | C3N2 | HC3N2
---|---|---
a | 5.095,5.085a,5.049b | 5.103
E0 | -446.448096638 | -447.351685595
$\rho$ | 3.2159 | 3.2129

aRef.[6] bRef.[3]

###  3.2 Phonon dispersion

Our calculated dispersion curves are shown in the figure 2. The absence of
imaginary modes in the dispersion curves, clearly demonstrates that insertion
of H atom in C3N2 does not interfere with dynamical stability. If we Look at
the two regions, i.e one below 400cm-1 that encompasses the acoustic modes and
the region of optical modes above 1200cm-1, we observe very important
information.

Figure 2: The phonon-dispersion curves for C3N2 (left panel) and HC3N2 (right
panel).

We find that H guest atom in this C3N2 host, have low-lying vibrational modes,
so that their contributions to the heat capacity, the entropy, and the free
energy are negligible. It is observed that the guest vibrational modes lowers
the bandwidth of the acoustic range. We also note that above 1200cm-1, the
optic branches are flat, an indication that these modes do not contribute to
heat transport in C3N2.

###  3.3 Conclusion

In summary, the structural and dynamical properties of caged-C3N2
encapsulating hydrogen atoms were studied by first-principles calculations.
The voids in C3N2 are large enough to accommodate the H atoms, hence the host
remains dynamically stable after insertion of the guest atom. The phonon
frequencies of the guest atom are within the acoustic range, and are expected
to have little impact on the heat capacity and entropy.

## Acknowledgment

This work was financially supported by the Kenya Education Network (KENET)
through the Computational Modeling and Materials Science (CMMS) Research mini-
grants 2019. We acknowledge the Centre for High Performance Computing (CHPC),
Cape Town, South Africa, for providing us with computing facilities.

## References

  * [1] Y. Rubin, T. Jarrosson, G.-W. Wang, M. D. Bartberger, K. Houk, G. Schick, M. Saunders, R. J. Cross, Insertion of helium and molecular hydrogen through the orifice of an open fullerene, Angewandte Chemie International Edition 40 (8) (2001) 1543–1546.
  * [2] R. Zhang, W. Ma, K. Han, C. Lee, A thermodynamic and kinetic study of the formation of c 20 compounds encapsulating h, he and ne atoms, Theoretical Chemistry Accounts 109 (5) (2003) 278–283.
  * [3] F. Tian, J. Wang, Z. He, Y. Ma, L. Wang, T. Cui, C. Chen, B. Liu, G. Zou, Superhard semiconducting c 3 n 2 compounds predicted via first-principles calculations, Physical Review B 78 (23) (2008) 235431.
  * [4] X. Wang, Y. Wang, M. Miao, X. Zhong, J. Lv, T. Cui, J. Li, L. Chen, C. J. Pickard, Y. Ma, Cagelike diamondoid nitrogen at high pressures, Physical review letters 109 (17) (2012) 175502.
  * [5] Q. Wei, M. Zhang, H. Yan, R. Li, X. Zhu, Z. Lin, R. Yao, A new superhard phase of c3n2 polymorphs, Zeitschrift für Naturforschung A 70 (12) (2015) 1001–1005.
  * [6] H. Chenghua, W. Feng, Z. Zhou, Ab initio study of phase stability, thermodynamic and elastic properties of c3 n2 derived from cubic c20, Physica B, Condensed Matter 407 (2012) 3398–3404.
  * [7] J. Du, X. Li, Theoretical investigation on two novel high-pressure orthorhombic phases of superhard c3n2, Journal of Alloys and Compounds 815 (2020) 152324.
  * [8] F. Yuan, L. Shi, Y. Zhang, C. Jing, Y. Wang, X. Qian, First-principles calculation of the electronic properties and external force modulation of a new tetragonal structure c3n2 crystal, Solid State Communications 338 (2021) 114455\.
  * [9] C. Kou, X. Zhang, Y. Tian, S. Zhang, Y. Du, X. Cui, D. Zhang, M. Zhang, L. Gao, Adsorption of alkali metal atoms on predicted c3n2 sheet, Solid State Communications 334 (2021) 114367.
  * [10] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, et al., Quantum espresso: a modular and open-source software project for quantum simulations of materials, Journal of physics: Condensed matter 21 (39) (2009) 395502.
  * [11] C. Malica, A. Dal Corso, Quasi-harmonic temperature dependent elastic constants: applications to silicon, aluminum, and silver, Journal of Physics: Condensed Matter 32 (31) (2020) 315902.
  * [12] S. Baroni, S. De Gironcoli, A. Dal Corso, P. Giannozzi, Phonons and related crystal properties from density-functional perturbation theory, Reviews of Modern Physics 73 (2) (2001) 515.
  * [13] J. P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Physical review letters 77 (18) (1996) 3865.
  * [14] P. E. Blöchl, Projector augmented-wave method, Physical review B 50 (24) (1994) 17953.
  * [15] H. J. Monkhorst, J. D. Pack, Special points for brillouin-zone integrations, Physical review B 13 (12) (1976) 5188.