# Codifference as a practical tool to measure interdependence

Agnieszka Wyłomańska  Aleksei Chechkin  Janusz Gajda  Igor M. Sokolov  Hugo
Steinhaus Center, Institute of Mathematics and Computer Science,
Wroclaw University of Technology, Wroclaw, Poland
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
Akhiezer Institute for Theoretical Physics, National Science Center "Kharkov
Institute of Physics and Technology"
Kharkov 61108, Ukraine, and
Max Planck Institute for Physics of Complex Systems, Noethnitzer Str 38,
D-01187 Dresden, Germany
<EMAIL_ADDRESS>Institut f r Physik, Humboldt-Universit t zu
Berlin, Newtonstrasse 15, D-12489 Berlin, Germany
<EMAIL_ADDRESS>
###### Abstract

Correlation and spectral analysis represent the standard tools to study
interdependence in statistical data. However, for the stochastic processes
with heavy-tailed distributions such that the variance diverges, these tools
are inadequate. The heavy-tailed processes are ubiquitous in nature and
finance. We here discuss codifference as a convenient measure to study
statistical interdependence, and we aim to give a short introductory review of
its properties. By taking different known stochastic processes as generic
examples, we present explicit formulas for their codifferences. We show that
for the Gaussian processes codifference is equivalent to covariance. For
processes with finite variance these two measures behave similarly with time.
For the processes with infinite variance the covariance does not exist,
however, the codifference is relevant. We demonstrate the practical importance
of the codifference by extracting this function from simulated as well as from
real experimental data. We conclude that the codifference serves as a
convenient practical tool to study interdependence for stochastic processes
with both infinite and finite variances as well.

###### keywords:

PACS:

##  1 Introduction

Stochastic processes with diverging variance are ubiquitous in nature and
finance. A remarkable example is an alpha-stable Lévy motion, or Lévy flights,
that is the class of non-Gaussian Markovian random processes whose stationary
independent increments are distributed according to Lévy stable distributions
[1]. Lévy stable laws are important for three fundamental properties: (i)
according to generalized central limit theorem, they form the basin of
attraction for sums of random variables with diverging variance [2]; (ii) the
probability density functions of Lévy stable laws decay in asymptotic power-
law form and thus appear naturally in the description of many fluctuation
processes with largely scattering statistics characterized by bursts or large
outliers; (iii) Lévy flights are statistically self-affine, a property used
for the description of random fractal processes. Examples of Lévy flights
range from light propagation in fractal medium called Lévy glass [3] and
plasma fluctuations in fusion devices [4, 5, 6, 7] to circulation of dollar
bills [8] and behavior of the marine verterbrates in response to patchy
distribution of food resources [9]; more examples can be found, e.g., in [10,
11]. The Lévy flight dynamics can also stem from a simple Brownian random walk
in systems whose operational time typically grows superlinearly with physical
time $t$ [12, 13].

Other prominent examples of the processes with heavy-tailed distributions are
Lévy Ornstein-Uhlenbeck (OU) process describing the overdamped harmonic
oscillator driven by alpha-stable Lévy noise, and fractional Lévy stable
motion. The Lévy OU process is a natural generalization of the Gaussian OU
process; such generalization has gained popularity, e.g., in finance [14, 15,
16]. The weakly damped harmonic oscillator driven by Lévy noise was discussed
in [17, 18]. The Lévy OU process accounts for interdependence (association) of
exponential type. On the contrary, fractional motions and fractional noises
have an infinite span of interdependence [19, 20]. Fractional processes are
also widely spread in applications [21, 22, 23]. Indeed, in a large class of
many-particle systems whose overall dynamics is Markovian, the probe particle
coupled with the rest of the system through space correlations exhibit
fractional motion with long-ranged non-Markovian memory effects [24, 25, 26].
Fractional Lévy stable motion with long-range dependence was detected in beat-
to-beat heart rate fluctuations [27], in solar flare time series [28], and was
shown to be a model qualitatively mimicking self-organized criticality
signatures in data [29].

What is the measure of interdependence for the processes with infinite
variance? Apparently, correlation or spectral power analysis, strictly
speaking, can not be used. The alternative measures of dependence are rarely
discussed in application-oriented literature.

The notion of covariance (CV) used in correlation analysis, can be generalized
for the alpha-stable Lévy process, leading to the notion of covariation [20,
30]. Its definition is based on the Lévy measure, and its practical usefulness
is limited. Some results related to the covariation of autoregressive process
with Lévy stable distribution are presented in [31]. The other measure is the
Lévy correlation cascade [32]. It is defined for infinitely divisible
processes, and the properties of that measure are discussed in [33] in detail,
see also [34]. However, similar to covariation, the Lévy correlation cascade
is of limited practical value because of complicated definition based on the
Lévy measure of the underlying process.

Our paper deals with another measure of interdependence called codifference
(CD). It is based on the characteristic function of a given process, therefore
it can be used not only for alpha-stable processes. Moreover, the codifference
in the Gaussian case reduces to the classical covariance, so it can be treated
as the natural extension of the well-known measure. On the other hand,
according to the definition, it is easy to evaluate the empirical codifference
which is based on the empirical characteristic function of the analyzed data.
It is worth to mention that the codifference is closely related to the so-
called dynamical functional used to study ergodic properties of stochastic
processes [35, 36, 20]. In our paper we aim to give an introduction to the
concept of codifference and to show its usefulness for analyzing
interdependence not only for the processes with infinite variance but for
those with finite variance, as well.

The rest of the paper is structured as follows: In Section 2 we give
definitions of codifference. In Section 3 we compare autocovariance and
autocodifference for Gaussian processes and for non-Gaussian processes with
finite variance. In Section 4 we present the autocodifference for processes
with infinite variance. The method how to estimate autocodifference from
experimental data is described in Section 5 together with the examples taken
from simulations. In Section 6 we present the results of real data analysis
for two processes with infinite variance. The conclusions are presented in
Section 7, and several technical details of calculating autocodifference are
collected in the Appendix.

##  2 Definitions

We start from the definition of codifference for the symmetric alpha-stable
(S$\alpha$S) random variables. Let us remind that the random variable $X$
which is S$\alpha$S with parameter $0<\alpha\leq 2$ and scale parameter
$\sigma_{X}>0$ has the following characteristic function $\Phi_{X}(k)$ [2]:

$\displaystyle\Phi_{X}(k)=<\exp(ikX)>=\exp\\{-\sigma_{X}^{\alpha}|k|^{\alpha}\\},$
(1)

where $<...>$ denote ensemble average or average over realizations. The
codifference of two jointly S$\alpha$S, $0<\alpha\leq 2$, random variables $X$
and $Y$ is defined as follows [20]:

$CD(X,Y)=\sigma_{X-Y}^{\alpha}-\sigma_{X}^{\alpha}-\sigma_{Y}^{\alpha},$ (2)

where $\sigma_{X},\sigma_{Y}$ and $\sigma_{X-Y}$ denote, respectively the
scale parameters of $X,Y$ and $X-Y$. The codifference can be also defined in
the language of the characteristic function [30, 31, 33]

$CD(X,Y)=\ln(<exp\\{iX\\}>)+\ln(<exp\\{-iY\\}>)-\ln(<exp\\{i(X-Y)\\}>).$ (3)

Thus, the definition given in Eq.(2) can be extended to a more general class
of random variables, and in the further analysis we use the representation
given in (3). We note, that one of the reasons to take the minus sign, i.e.,
$-iY$ in the second term of the right-hand side of Eq.(3) is to ensure that CD
is reduced to covariance in the Gaussian case (with the minus sign); see
below.
It is worth to mention that the codifference shares useful properties [20]

  * 1.

it is always well-defined, since the definition of $CD(X,Y)$ is based on the
characteristic functions of appropriate random variables;

  * 2.

if the random variables are symmetric, then $CD(X,Y)=CD(Y,X)$;

  * 3.

if $X$ and $Y$ are independent and jointly S$\alpha$S, then $CD(X,Y)=0$.
Conversely, if $CD(X,Y)=0$ and $0<\alpha<1$, then $X$ and $Y$ are independent.
When $1<\alpha<2$, $CD(X,Y)=0$ does not imply that $X$ and $Y$ are
independent. We note that in case of two Gaussian random variables (i.e.
S$\alpha$S with $\alpha=2$) the independence of $X$ and $Y$ is equivalent to
zero covariance.

The above properties confirm that the codifference is an appropriate
mathematical tool for measuring the dependence between alpha-stable random
variables as well as random variables from more general class of distributions
(e.g., infinitely divisible). In the literature one can also find the
generalized codifference which is defined as [20]:

$GCD(X,Y;\theta_{1},\theta_{2})=\ln(<exp\\{i\theta_{1}X\\}>)+\ln(<exp\\{i\theta_{2}Y\\}>)-\ln(<exp\\{i(\theta_{1}X+\theta_{2}Y)\\}>),$
(4)

where $\theta_{1},\theta_{2}\in R$. The generalized codifference is useful in
cases when the random variables considered have different scales or different
units [18]. Thus, the proper choice of $\theta_{1}$ and $\theta_{2}$
parameters allows one to analyze the interdependence at the same scale. Also,
to make the measure of dependence invariant against the change of units, it is
reasonable to take $\theta_{1}$ equal to $\sigma_{X}^{-1}$ and $\theta_{2}$
equal to $\sigma_{Y}^{-1}$. Equation (4) will then define the "cosum" of non-
dimensional, normalized variables [18]. The usage of "cosum" has an additional
advantage when e.g., asymmetric alpha-stable Lévy variables are discussed,
since the sum of two such variables has a distribution which differ only by a
scaling factor, whereas the difference has a distribution which belongs to a
different class. In the present paper we however stick to the definition (3).
For a stochastic process $\\{X(t)\\}$, the measure $CD(X(t),X(s))$ called
autocodifference is defined as 111In this paper we denote a stochastic process
as $\\{X(t)\\}$ and the value of the process at time $t$ as $X(t)$.

$CD(X(t),X(s))=\ln(<exp\\{iX(t)\\}>)+\ln(<exp\\{-iX(s)\\}>)-\ln(<exp\\{i(X(t)-X(s))\\}>).$
(5)

For the stationary process the autocodifference $CD(X(t),X(s))$ depends on
$|t-s|$.

##  3 Autocodifference versus autocovariance for processes with finite second
moment

In this Section we consider autocodifference for the several processes with
finite variance, which are widely used in applications. We restrict ourselves
with stationary processes and processes with stationary increments.

###  3.1 Gaussian processes

For the Gaussian processes the autocodifference is simply reduced to
autocovariance with the minus sign. Indeed, let $\\{X(t)\\}$ be a Gaussian
process with mean $f(t)$ and variance $g(t)$, then the characteristic function
reads:

$\displaystyle\Phi_{X(t)}(k)=<e^{ikX(t)}>=e^{ikf(t)-g(t)k^{2}/2}.$ (6)

Then, for fixed $t$ and $s$ ($s<t$) the increments $X(t)-X(s)$ have also
Gaussian distribution with the mean $f(t)-f(s)$ and variance equal to
$g(t)+g(s)-2cov(Y(t),Y(s))$. Therefore, from Eq.(5) we get:

$\displaystyle CD(X(t),X(s))$ $\displaystyle=$
$\displaystyle+if(t)-\frac{g(t)}{2}-if(s)-\frac{g(s)}{2}$ $\displaystyle-$
$\displaystyle
i(f(t)-f(s))+\frac{g(t)+g(s)-2cov(X(t),X(s))}{2}=-cov(X(t),X(s)).$

Below we give some examples of the autocodifference for Gaussian processes.

####  3.1.1 Gaussian white noise process

The process $\\{b(t)\\}$ is called white Gaussian noise if for each $t$ the
random variable $b(t)$ has Gaussian distribution with zero mean and finite
variance $\sigma^{2}$, and for each $t\neq s$ the values $b(t)$ and $b(s)$ are
uncorrelated. Thus, the autocodifference is given by:

$\displaystyle
CD(b(t),b(s))=-cov(b(t),b(s))=\left\\{\begin{array}[]{lll}-\sigma^{2},&\mbox{if}&t=s,\\\
0,&&\mbox{otherwise.}\end{array}\right.$ (9)

####  3.1.2 Ordinary Brownian motion

The process is called ordinary Brownian motion if it has stationary
independent increments possessing Gaussian distribution. The standard ordinary
Brownian motion $\\{B(t)\\}$ has zero mean and variance equal to $t$. In this
case we have:

$\displaystyle CD(B(t),B(s))=-cov(B(t),B(s))=-\min\\{t,s\\}.$ (10)

####  3.1.3 Gaussian Ornstein-Uhlenbeck process

The Gaussian Ornstein-Uhlenbeck process is defined as a stationary solution of
the Langevin equation for the overdamped oscillator:

$\displaystyle\frac{dY(t)}{dt}+\lambda Y(t)=b(t),~{}~{}\lambda>0$ (11)

where $\\{b(t)\\}$ is the Gaussian white noise (heuristically
$b(t)=dB(t)/dt$). The Gaussian Ornstein-Uhlenbeck process has the following
moving average representation:

$\displaystyle Y(t)=\int_{-\infty}^{t}e^{-\lambda(t-s)}d\tilde{B}(s),$ (12)

where $\tilde{B}(t)$ is an extension of the Brownian motion for the negative
axis, that is

$\displaystyle\tilde{B}(t)=\left\\{\begin{array}[]{lll}B(t),&\mbox{when}&t\geq
0,\\\ B(-t),&&\mbox{otherwise.}\end{array}\right.$ (15)

We remind that $\\{Y(t)\\}$ is the only stationary and Markovian - Gaussian
stochastic process, according to the Doob’s theorem [38]. The autocodifference
is given by:

$\displaystyle
CD(Y(t),Y(s))=-cov(Y(t),Y(s))=-\frac{e^{-\lambda|t-s|}}{2\lambda}.$ (16)

####  3.1.4 Fractional Brownian motion

The fractional Brownian motion is a zero mean Gaussian process
$\\{B_{H}(t)\\}$ defined as follows [19]:

$\displaystyle
B_{H}(t)=\int_{-\infty}^{0}\left\\{(t-u)^{H-1/2}-(-u)^{H-1/2}\right\\}dB(u)+\int_{0}^{t}(t-u)^{H-1/2}dB(u),$
(17)

where $\\{B(t)\\}$ is the classical Brownian motion. The $H$ parameter called
Hurst exponent is a real number from the interval $(0,1)$. The process is
self-similar and reduces to ordinary Brownian motion, if $H=1/2$. The
parameter $H$ controls the type of diffusion, namely the process is
superdiffusive for $H>1/2$ while subdiffusive for $H<1/2$. The
autocodifference takes the following form:

$\displaystyle
CD(B_{H}(t),B_{H}(s))=-cov(B_{H}(t),B_{H}(s))=-\frac{k(H)}{2}(|t|^{2H}+|s|^{2H}-|t-s|^{2H}),$
(18)

where
$k(H)=Var(B_{H}(1))=\int_{0}^{\infty}\left|(1+x)^{H-1/2}-x^{H-1/2}\right|^{2}dx+\frac{1}{2H}$.

####  3.1.5 Fractional Gaussian noise

The fractional Gaussian noise $\\{b_{H}(t)\\}$ is defined heuristically as the
derivative of the fractional Brownian motion, i.e. $b_{H}(t)=dB_{H}(t)/dt$.
Therefore, the autocodifference of the process takes the following form [39]:

$\displaystyle
CD(b_{H}(t),b_{H}(s))=-cov(b_{H}(t),b_{H}(s))=\frac{k(H)}{2}(|t-s+1|^{2H}-2|t-s|^{2H}+|t-s-1|^{2H}),$
(19)

where
$k(H)=Var(b_{H}(t))=\int_{0}^{\infty}\left|(1+x)^{H-1/2}-x^{H-1/2}\right|^{2}dx+\frac{1}{2H}$.
Detailed derivation of covariance structure for both fractional Brownian
motion and fractional Gaussian noise is presented in [39, 40].

###  3.2 Non-Gaussian processes

####  3.2.1 Poisson process

The Poisson process with intensity $\lambda$ is a continuous-time counting
process $\\{P(t)\\}$ which has stationary, independent increments [41, 42].
The increment $P(t)-P(s)$ (for $t>s$) has Poisson distribution with parameter
$\lambda(t-s)$. The covariance function of the Poisson process takes the
following form:

$\displaystyle cov(P(t),P(s))=\min\\{t,s\\}\lambda.$ (20)

The characteristic function of the Poisson process is given by
$\Phi_{P(t)}(k)=e^{t\lambda(e^{ik}-1)}$, therefore the autocodifference takes
the following form:

$\displaystyle CD(P(t),P(s))=-2\lambda\min\\{t,s\\}(1-\cos 1).$ (21)

Some interesting extensions of Poisson process one can find in [43].

####  3.2.2 Tempered stable Lévy process

The tempered stable distributions were suggested in [44] and studied in [45]
in more detail. The general mathematical description of this class of
processes was established in [46]. In our paper we consider the tempered
stable distribution with the Lévy triplet $(\kappa^{2},\nu,\gamma)$, which is
defined as follows [47]:

$\displaystyle\kappa=0$
$\displaystyle\nu(dx)=(\tilde{C}_{+}e^{-\lambda_{+}x}1_{x>0}+\tilde{C}_{-}e^{-\lambda_{-}|x|}1_{x<0})\frac{dx}{|x|^{\alpha+1}}$
$\displaystyle\gamma=m-\int_{|x|>1}x\nu(dx),$

where $\tilde{C}_{+},\tilde{C}_{-},\lambda_{+},\lambda_{-}>0$,
$\alpha\in(0,2)$ and $m\in R$.
A Lévy process (i.e. process with independent stationary increments) having
tempered stable distribution is called a tempered stable process with
parameters $\alpha,\tilde{C}_{+},\tilde{C}_{-},\lambda_{+},\lambda_{-},m$.

For simplicity we consider the special case, namely we assume here $m=0$ and
$\lambda_{+}=\lambda_{-}=\lambda$, $\alpha>1$. Moreover if we substitute

$C_{+}=\frac{1}{\Gamma(-\alpha)}\tilde{C}_{+},~{}~{}C_{-}=\frac{1}{\Gamma(-\alpha)}\tilde{C}_{-},$

then it can be shown that in this case the characteristic function
$\phi_{T(t)}(k)=Ee^{ikT(t)}$ of the tempered stable process $\\{T(t)\\}$ takes
the following form:

$\displaystyle\phi_{T(t)}(k)=\exp\left\\{t\left[i\alpha
k\lambda^{\alpha-1}(C_{+}-C_{-})+C_{+}(\lambda-
ik)^{\alpha}+C_{-}(\lambda+ik)^{\alpha}-\lambda^{\alpha}(C_{+}+C_{-})\right]\right\\}.$
(22)

The tempered stable Lévy process (called also truncated Lévy flight) is a
process $\\{T(t)\\}$ with independent stationary increments possessing
tempered stable distribution described above. As an example, we analyze the
symmetric tempered stable process $\\{T(t)\\}$, i.e. such that the
characteristic function of $T(t)$ is given by:

$\displaystyle\Phi_{T(t)}(k)=\exp\left\\{t\left[(\lambda+ik)^{\alpha}+(\lambda-
ik)^{\alpha}-2\lambda^{\alpha}\right]\right\\}.$ (23)

The tempered stable process is an extension of the alpha-stable Lévy process,
and for $\lambda=0$ the random variable $T_{\alpha,\lambda}(t)$ reduces to the
appropriate alpha-stable random variable, see [48].
The autocovariance of the analyzed tempered stable process has the following
form:

$\displaystyle
cov(T(t),T(s))=2\alpha(\alpha-1)\lambda^{\alpha-2}\min\\{s,t\\}.$ (24)

The autocodifference is given by

$\displaystyle
CD(T(t),T(s))=2[(\lambda-i)^{\alpha}+(\lambda+i)^{\alpha}-2\lambda^{\alpha}]\min\\{t,s\\}.$
(25)

For more details see Appendix, Section A1.

####  3.2.3 Laplace motion

The Laplace motion (called also variance gamma process) is a process
$\\{\Lambda(t)\\}$, $t\geq 0$ defined as follows [49]:

$\Lambda(t)=B(G(t)),$ (26)

where $\\{B(\cdot)\\}$ is an ordinary Brownian motion, and $\\{G(t)\\}$ is a
Gamma process with parameters $\gamma,\lambda$ defined as a pure jump Lévy
increasing process with increments having Gamma distribution. It has the
following characteristic function:

$\Phi_{G(t)}(k)=<\exp(ikG(t))>=(1-ik/\lambda)^{-\gamma t},k\in R.$ (27)

The Laplace motion has been successfully applied in the modeling of credit
risk in structural models [50, 51, 52]. Some extensions of this process one
can find, e.g., in [53].
The process $\\{\Lambda(t)\\}$ has a zero mean and stationary independent
increments. The covariance function reads

$\displaystyle cov(\Lambda(t),\Lambda(s))=\frac{\Gamma(\gamma
s+1)}{\lambda\Gamma(\gamma s)},\mbox{$s<t$},$ (28)

where $\Gamma(a)$ is a gamma function, while the autocodifference reads

$\displaystyle CD(\Lambda(t),\Lambda(s))=-2\gamma
s\ln(1+1/(2\lambda)),\mbox{$s<t$}.$ (29)

See Appendix, Section A2, for more details.

##  4 Autocodifference for processes with infinite variance

In this section we analyze the processes for which the autocovariance function
is not defined, and then the autocodifference is the main measure of
dependence.

###  4.1 White Lévy noise

The white Lévy noise $\\{l_{\alpha}(t)\\}$, $t\geq 0$, is a process such that
for each $s\neq t$ the random variables $l_{\alpha}(s)$ and $l_{\alpha}(t)$
are independent, and for each $t$ the random variable $l_{\alpha}(t)$ has the
following characteristic function:

$\displaystyle\Phi_{l_{\alpha}(t)}(k)=\left\\{\begin{array}[]{lll}\exp\\{-\sigma^{\alpha}|k|^{\alpha}(1-i\beta
sgn(k)\tan\frac{\pi\alpha}{2})+ik\mu\\}&\mbox{if}&\alpha\neq 1,\\\
\exp\\{-\sigma|k|(1-i\beta\frac{2}{\pi}sgn(k)\ln|k|)+ik\mu)\\}&\mbox{if}&\alpha=1,\end{array}\right.$
(32)

where $0<\alpha\leq 2$ is the stability index, $\sigma>0$ the scale parameter,
$-1\leq\beta\leq 1$ the skewness parameter, and $\mu\in R$ is the shift
parameter.
The autocodifference reads as

$\displaystyle
CD(l_{\alpha}(t),l_{\alpha}(s))=\left\\{\begin{array}[]{lll}-2\sigma^{\alpha},&\mbox{if}&t=s,\\\
0,&&\mbox{otherwise.}\end{array}\right.$ (35)

It is worth to mention that the above formula is valid not only for symmetric
Lévy noise.

###  4.2 Lévy flights

The Lévy flight (called also alpha-stable Lévy motion) is the process
$\\{L_{\alpha}(t)\\}$, $t\geq 0$, with independent stationary increments
possessing alpha-stable distribution, i.e. in general case for each $t$ the
random variable $L_{\alpha}(t)$ has the stable distribution with index of
stability $\alpha$, scale parameter $t^{1/\alpha}\sigma$, skewness $\beta$ and
shift parameter $\mu=0$. The autocodifference for the analyzed process takes
the form

$\displaystyle
CD(L_{\alpha}(t),L_{\alpha}(s))=-2\sigma^{\alpha}\min\\{t,s\\}.$ (36)

Let us note, for $\alpha=2$ and $\sigma=1/\sqrt{2}$ (parameters of symmetric
alpha-stable distribution corresponding to the standard Gaussian one) the
above formula reduces to (10).

###  4.3 Lévy Ornstein-Uhlenbeck process

The Lévy Ornstein-Uhlenbeck process $\\{Y_{\alpha}(t)\\}$ is defined via the
following Langevin equation:

$\displaystyle\frac{dY_{\alpha}(t)}{dt}+\lambda
Y_{\alpha}(t)=l_{\alpha}(t),~{}~{}\lambda>0,$ (37)

where $\\{l_{\alpha}(t)\\}$ is white Lévy noise. The process was examined, for
example, in [54]. The stationary solution of equation (37) reads

$\displaystyle
Y_{\alpha}(t)=\int_{-\infty}^{t}e^{-\lambda(t-u)}d\tilde{L}_{\alpha}(u),$ (38)

where

$\displaystyle\tilde{L}_{\alpha}(t)=\left\\{\begin{array}[]{lll}L_{\alpha}(t),&\mbox{when}&t\geq
0,\\\ L_{\alpha}(-t),&&\mbox{otherwise.}\end{array}\right.$ (41)

For simplicity, we assume that in the considered case $\\{L_{\alpha}(t)\\}$ is
symmetric alpha-stable Lévy process i.e. process, for which the increments
have alpha-stable distribution with the $\mu=0$, $\beta=0$ and scale parameter
$\sigma$. Then the autocodifference takes the form

$\displaystyle
CD(Y_{\alpha}(t),Y_{\alpha}(s))=-\sigma^{\alpha}\frac{1+e^{-\lambda\alpha|t-s|}-|1-e^{-\lambda|t-s|}|^{\alpha}}{\lambda\alpha}.$
(42)

For more details see Appendix, Section A3. We note that for $\alpha=2$ and
$\sigma=1/\sqrt{2}$ (parameters of symmetric alpha-stable distribution
corresponding to the standard Gaussian one) the above formula reduces to (16).

###  4.4 Fractional Lévy motion

The fractional Lévy motion is a process $\\{L_{\alpha,H}(t)\\}$ defined for
any $0<H<1$ as follows [20, 55, 56, 57]:

$\displaystyle
L_{\alpha,H}(t)=\int_{-\infty}^{0}\left\\{(t-u)^{H-1/\alpha}-(-u)^{H-1/\alpha}\right\\}dL_{\alpha}(u)+\int_{0}^{t}(t-u)^{H-1/\alpha}dL_{\alpha}(u),$
(43)

where $\\{L_{\alpha}(t)\\}$ is the symmetric alpha-stable Lévy process with
index of stability $\alpha\in(0,2)$. For simplicity we assume $\sigma=1$. The
process $\\{L_{\alpha,H}(t)\\}$ is self-similar, stationary-increment process
with infinite second moment. Similar to fractional Brownian motion, the
parameter $H$ controls the diffusion law, namely for $H<1/\alpha$ and
$H>{1}/{\alpha}$ the fractional Lévy motion exhibits sub- and superdiffusive
behavior, respectively [58]. The autocodifference for fractional Lévy motion
has the following form:

$\displaystyle
CD(L_{\alpha,H}(t),L_{\alpha,H}(s))=k(H,\alpha)\left(|t-s|^{\alpha
H}-|t|^{\alpha H}-|s|^{\alpha H}\right),$ (44)

where
$k(H,\alpha)=\int_{0}^{\infty}\left|(1+u)^{H-1/\alpha}-u^{H-1/\alpha}\right|^{\alpha}du+\frac{1}{H\alpha}$.
See Appendix, Section A4, for more details.
Let us remind that for $H=1/\alpha$ the fractional Lévy motion reduces to Lévy
flights. In this case the autocodifference (44) takes the form
$-2\min\\{s,t\\}$ which is consistent with formula (36) under the assumption
$\sigma=1$.

###  4.5 Fractional Lévy noise

The fractional Lévy noise $\\{l_{\alpha,H}(t)\\}$ is defined heuristically as
the derivative of the fractional Lévy motion $\\{L_{\alpha,H}(t)\\}$. For
large $t$ the autocodifference has a power law form [59]:
If either $0<\alpha\leq 1$, $0<H<1$ or $1<\alpha<2$,
$1-\frac{1}{\alpha(\alpha-1)}<H<1$, $H\neq 1/\alpha$, then for
$t\rightarrow\infty$

$\displaystyle CD(l_{\alpha,H}(t),l_{\alpha,H}(0))\sim Ct^{\alpha H-\alpha}.$
(45)

If $1<\alpha<2$ and $0<H<1-\frac{1}{\alpha(\alpha-1)}$, then for
$t\rightarrow\infty$

$\displaystyle CD(l_{\alpha,H}(t),l_{\alpha,H}(0))\sim Dt^{H-1/\alpha-1},$
(46)

where the constants $C$ and $D$ are time-independent.
Note that, for $\alpha=2$ the fractional Lévy noise reduces to fractional
Gaussian noise. In this case, the autocodifference given in (19) behaves as
$t^{2H-2}$ for $t\rightarrow\infty$ which is consistent with (45) [59].

###  4.6 Superdiffusion continuous time random walk-like process

We consider superdiffusive continuous time random walk-like process which is
defined as follows [12]:

$\displaystyle Z(t)=B(L_{\alpha}(t)),$ (47)

where $\\{B(t)\\}$ is the ordinary Brownian motion and $\\{L_{\alpha}(t)\\}$
the Lévy flight process described in Section 4.2, with $0<\alpha<1$,
$\beta=1$, $\mu=0$ and scale parameter $\sigma>0$, i.e. the process for which
the Laplace transform is given by:

$\displaystyle\mathcal{L}_{L_{\alpha}(t)}(k)=<\exp\\{-kL_{\alpha}(t)\\}>=\exp\\{-t(\sigma|k|)^{\alpha}\\}.$
(48)

The processes $\\{B(t)\\}$ and $\\{L_{\alpha}(t)\\}$ are independent.
The process $\\{L_{\alpha}(t)\\}$ in definition (47) plays the role of time
(i.e. is should be non-negative increasing process), therefore the above
restrictions on the parameters have to be assumed.

Since Brownian motion and Lévy flight processes have independent stationary
increments, then the process $\\{Z(t)\\}$ also possesses this property [60].
The possible applications of superdiffusion continuous time random walk-like
process are transport in heterogeneous catalysis, micelle systems, reactions
and transport in polymer systems under conformational motion and dynamical
systems [13].

Since $<B^{2}(L_{\alpha}(t))>=<L_{\alpha}(t)>$ and the mean of $L_{\alpha}(t)$
does not exists, then the process $\\{Z(t)\\}$ has infinite second moment. The
autocodifference for the process $\\{Z(t)\\}$ is given by:

$\displaystyle CD(Z(t),Z(s))=-2^{-\alpha+1}\sigma^{\alpha}\min\\{t,s\\}.$ (49)

See Appendix, Section A5, for more details.

##  5 How to estimate codifference from data

We define an estimator of autocodifference in the form:

$\displaystyle\widehat{CD}(X(t),X(s))=\ln(\hat{\phi}(1,0,X(t),X(s)))+\ln(\hat{\phi}(0,-1,X(t),X(s)))$
(50) $\displaystyle-\ln(\hat{\phi}(1,-1,X(t),X(s))),$

where $\hat{\phi}(u,v,X(t),X(s))$ is an estimator of the characteristic
function:

$\displaystyle\hat{\phi}(u,v,X(t),X(s))=<\exp\\{i(uX(t)+vX(s))\\}>.$ (51)

In [61] an efficient methodology is introduced for estimating the codifference
from a single trajectory of stationary process. Namely, if
$\\{x_{k},k=1,...,N\\}$ is realization of a stationary process $\\{X(t)\\}$,
then the estimator of the characteristic function takes the form:

$\displaystyle\hat{\phi}(u,v,X(t),X(s))=\frac{1}{N}\sum_{k=1}^{N-|t-s|}\exp(i(ux_{k+|t-s|}+vx_{k})).$
(52)

For a nonstationary process we are not able to estimate empirical
autocodifference by using only a single trajectory, therefore the above
estimator requires modification. Suppose, we have $M$ trajectories of a
nonstationary process $\\{X(t)\\}$. Let us take a sample
$\\{x^{t}_{k},k=1,...,M\\}$ being a realization of a random variable $X(t)$,
that is the values of the process $\\{X(t)\\}$ taken at a fixed time $t$, and
a sample $\\{x^{s}_{k}:k=1,...,M\\}$ composed from the values of the process
$\\{X(t)\\}$ taken at a fixed time $s$. By construction, both samples consist
of independent identically distributed random variables. Thus, in the
nonstationary case the estimator of characteristic function
$\phi(u,v,X(t),X(s))$ is defined as:

$\displaystyle\hat{\phi}(u,v,X(t),X(s))=\frac{1}{M}\sum_{k=1}^{M}\exp(i(ux^{t}_{k}+vx^{s}_{k})).$
(53)

In Fig. 1 we show a single trajectory of the tempered stable Lévy motion and
compare estimator of the autocodifference obtained from Eqs.(50) and (53) with
theoretical value given by Eq.(25). One observes perfect agreement between
theoretical and empirical results. In Fig. 2 we present a trajectory of Lévy
stable Ornstein-Uhlenbeck process together with the theoretical
autocodifference given by Eq.(42) and its estimator obtained from Eqs.(50) and
(52). In Fig. 3 we plot the path of fractional Lévy noise and also compare the
theoretical autocodifference, Eq.(46), and its estimator. In all three cases
one observes almost perfect agreement between the empirical and theoretical
results.

Figure 1: Top panel: single trajectory of the tempered stable Lévy motion
$\\{T_{\alpha,\lambda}(t)\\}$ with parameters $\alpha=0.9,\lambda=0.01$.
Bottom panel: Estimator of autocodifference
$CD(T_{\alpha,\lambda}(t),T_{\alpha,\lambda}(20))$ calculated with $1000$
trajectories, and the theoretical value given by Eq.(25).  Figure 2: The
trajectory of stable Lévy Ornstein-Uhlenbeck process $\\{Y_{\alpha}(t)\\}$
with parameters $\alpha=1.6382,\lambda=0.0045$ (top panel), together with
theoretical autocodifference $CD(Y_{\alpha}(t),Y_{\alpha}(0))$ and its
estimator (bottom panel).  Figure 3: The trajectory of fractional Lévy noise
$l_{\alpha,H}(t)$ with parameters $\alpha=1.95,H=0.8$ (top panel), together
with the estimator of autocodifference $CD(l_{\alpha,H}(t),l_{\alpha,H}(0))$
and fitted power function $t^{\alpha(H-1)}$ (bottom panel).

##  6 Real data analysis

###  6.1 Plasma data

In this section we investigate the data obtained in experiment on the
controlled fusion device. The important characteristics of edge plasma
turbulence, such as fluctuation amplitudes, spectra, and turbulence-induced
transport are investigated in the Uragan-3M (U-3M) stellarator torsatron by
the use of high resolution measurements of density (ion saturation current)
and potential (floating potential) fluctuations with the help of movable
Langmuir probe arrays. We address the reader to [62] for the details of
experimental set-up and description of the data base. Here we present the
analysis of the ion saturation current fluctuations (in mA) measured at the
small torus radial position $r=9.9cm$. Similar data were analyzed in [7]. In
Fig. 4 we present the examined time series (top panel) and corresponding
estimator of the autocodifference (bottom panel).

Figure 4: The examined time series of the ion saturation current fluctuations
(in mA) registered in the U-3M torsatron at the small torus radial positions
$r=9.9cm$ (top panel), and the estimator of autocodifference (bottom panel).

The estimator of autocodifference has non-zero value at $t=0$ and then drops
to zero. Therefore, we infer that the data can be considered as a white noise.
Moreover, we also observe non-Gaussian behavior of the underlying series.
Thus, we propose to model the process by using white Lévy noise. We confirm
our assumption by using the Anderson-Darling goodness-of-fit test for stable
distribution [63], which indicates that the data come from Lévy stable
distribution (p-value is equal to $0.88$). Then, by using the regression
method [63] we estimate the index of stability. As a result we get
$\hat{\alpha}=1.95$.

On the basis of autocodifference we can also estimate the scale parameter
$\sigma$ of the analyzed series. In order to do this we compare the value of
the estimator of autocodifference for $t=0$ with the theoretical value
$-2\sigma^{\alpha}$. As the result we obtain $\hat{\sigma}=1.05$. We then
compare the value obtained with the estimates of $\sigma$ parameter made by
other methods, namely, regression [63] and McCulloch [64] methods. The
obtained values are $\hat{\sigma}=1.06$ and $\hat{\sigma}=1.03$, respectively,
which is in good agreement with the value obtained from the autocodifference.

###  6.2 Financial data

As a second example we analyze time series that describes closing prices of
the investment holding company Cosco Pacific Ltd. The data are quoted daily in
the period 04.01.2000-14.01.2013, [65]. In Fig. 5 (top panel) we present the
examined time series.

Figure 5: The examined time series that describes closing prices (in USD) of
Cosco Pacific Ltd. in the period 04.01.2000-14.01.2013 (top panel), and the
estimator of autocodifference and theoretical function given by Eq.(42)
(bottom panel).

We recall that the Gaussian Ornstein-Uhlenbeck process originally used by
Vasicek for the analysis of interest rates [14], was successfully applied to
many other data from financial markets [15]. In line with these findings, we
intend to describe the data analyzed by using Ornstein-Uhlenbeck process.
However, in contrast to the classical model, significant jumps in the examined
data are clearly observed, which may indicate non-Gaussian behavior of the
process. We thus suggest the Lévy Ornstein-Uhlenbeck process defined in (37)
as a candidate for fitting the data after removing the mean. At the first step
of our analysis we estimate the relaxation parameter $\lambda$ that enters
Eq.(37). Here we employ the Whittle estimation method described in [66] (also
used in [67]), which is based on the sample periodogram of the analyzed time
series. Once the $\hat{\lambda}$ estimate is found, the residuals of the
process can be derived (recall that in our case the residual of the Lévy
Ornstein-Uhlenbeck model is a process $l_{\alpha}(t)$ in (37)). On the basis
of the residual series, by using the regression method, we can estimate the
index of stability $\alpha$. We also confirmed the stable distribution of the
residual series by using the Anderson-Darling goodness of fit test. As the
result, we obtain $\hat{\lambda}=0.0045$ and $\hat{\alpha}=1.64$. In Fig. 5 we
present the estimator of autocodifference along with the theoretical value
given by Eq.(42).

Similar to the plasma data analysis, we estimate the scale parameter $\sigma$
from the autocodifference function by using estimated values of $\lambda$ and
$\alpha$ in Eq.(42). As the result we obtain $\hat{\sigma}=0.17$, which is
exactly the same value that the regression and McCulloch methods give for
these data.

##  7 Conclusions

In this paper we have examined the codifference, a general measure of
interdependence, that can be considered as an alternative to covariance
function. We have indicated the importance of codifference especially for
processes with non-Gaussian distribution, for which the correlation function
is not defined. For a class of Gaussian processes the autocodifference is
simply reduces to autocovariance with negative sign. We then analyzed this
measure for several well-known processes with finite variance and show close
similarity between autocodifference and autocovariance. We furthermore present
the generic examples of the processes, for which the autocovariance does not
exist, therefore the codifference becomes of particular importance. After
giving a simple practical recipe how to estimate the autocodifference from
experimental data for both stationary and nonstationary processes, we
estimated codifference from the surrogate data and analyze two real data sets
representing random fluctuations observed in laboratory plasma and prices on
financial market. For both examples we demonstrated how the estimated
autocodifference can be used as a tool for recognition a proper stochastic
model. Also, we have shown that on the basis of codifference it is possible to
estimate parameters of the fitted model. Summarizing, we conclude that the
codifference serves as a convenient practical tool to study interdependence
for stochastic processes with infinite and finite variances as well.

## Acknowledgements

The research of Agnieszka Wyłomańska is co-financed by the National Science
Center, Poland, under the contract No. UMO-20127/B/ST8/03031.

## Appendix

A1.

Tempered stable Lévy process
Since the process $\\{T(t)\\}$ has stationary independent increments, zero
mean, and $<T^{2}(t)>=2t\alpha(\alpha-1)\lambda^{\alpha-2}$, then for $s<t$ we
obtain

$cov(T(t),T(s))=<T(t)(T(s)>=<T(s)(T(t)-T(s))>+<T^{2}(s)>=2s\alpha(\alpha-1)\lambda^{\alpha-2}.$

Using the same reasoning for $t<s$ we get
$cov(T(t),T(s))=2\alpha(\alpha-1)\lambda^{\alpha-2}\min\\{s,t\\}$.
Now let us calculate characteristic function of $T(t)-T(s)$ for $s<t$. We have

$<\exp\\{i(T(t)-T(s))\\}>=<\exp\\{iT(t-s)\\}>=\exp\left\\{(t-s)\left[(\lambda+i)^{\alpha}+(\lambda-i)^{\alpha}-2\lambda^{\alpha}\right]\right\\}.$

Moreover,

$<\exp\\{iT(t)\\}>=\exp\left\\{t\left[(\lambda+i)^{\alpha}+(\lambda-i)^{\alpha}-2\lambda^{\alpha}\right]\right\\},$
$<\exp\\{-iT(s)\\}>=\exp\left\\{s\left[(\lambda+i)^{\alpha}+(\lambda-i)^{\alpha}-2\lambda^{\alpha}\right]\right\\}.$

As a final result, we obtain:

$CD((T(t),T(s))=-(t-s)\left[(\lambda+i)^{\alpha}+(\lambda-i)^{\alpha}-2\lambda^{\alpha}\right]+t\left[(\lambda+i)^{\alpha}+(\lambda-i)^{\alpha}-2\lambda^{\alpha}\right]+s\left[(\lambda+i)^{\alpha}+(\lambda-i)^{\alpha}-2\lambda^{\alpha}\right]$
$=2s[(\lambda-i)^{\alpha}+(\lambda+i)^{\alpha}-2\lambda^{\alpha}].$

Therefore, we have

$CD((T(t),T(s))=2\min\\{t,s\\}[(\lambda-i)^{\alpha}+(\lambda+i)^{\alpha}-2\lambda^{\alpha}].$

A2.

Laplace motion
The gamma process $\\{G(t)\\}$ with parameters $\gamma,\lambda$ is a pure jump
Lévy increasing process with the moments

$E(G^{n}(t))=\lambda^{-n}{\Gamma}(\gamma t+n)/{\Gamma}(\gamma t),n\geq 0,$
(54)

and ${\Gamma}(\cdot)$ is a Gamma function. If $\Lambda(t)=B(G(t))$, then
$<\Lambda(t)>=0$, and we have

$\displaystyle cov(\Lambda(t),\Lambda(s))=<\Lambda(t)\Lambda(s)>$
$\displaystyle=$
$\displaystyle<(\Lambda(t)-\Lambda(s)+\Lambda(s))\Lambda(s)>=<(\Lambda(t)-\Lambda(s))\Lambda(s)+\Lambda^{2}(s)>$
$\displaystyle=$
$\displaystyle<\Lambda^{2}(s)>=<B^{2}(u)|G(s)=u>=<G(s)>=\frac{{\Gamma}(\gamma
s+1)}{\lambda{\Gamma}(\gamma s)}.$

Let us calculate the autocodifference of the process $\\{\Lambda(t)\\}$. For
$s<t$ we have:

$\displaystyle\ln<\exp(i(\Lambda(t)-\Lambda(s))>$ $\displaystyle=$
$\displaystyle\ln<\exp(i(\Lambda(t-s))>=\ln<\exp(iB(u))|G(t-s)=u>$
$\displaystyle=$
$\displaystyle\ln<\exp(-1/2\Gamma(t-s))>=(-\gamma(t-s))\ln(1+1/(2\lambda)).$

Thus the autocodifference for $s<t$ is equal to

$\begin{split}CD(\Lambda(t),\Lambda(s))=-\gamma t\ln(1+1/(2\lambda))-\gamma
s\ln(1+1/(2\lambda))+\gamma(t-s)\ln(1+1/(2\lambda))\\\ =-2\gamma
s\ln(1+1/(2\lambda)).\end{split}$

A3.

Lévy Ornstein-Uhlenbeck process
We note that the random variable $Y_{\alpha}(t)$ given by (38) for $\lambda>0$
is S$\alpha$S with $\sigma^{\alpha}=\frac{\sigma^{\alpha}}{\alpha\lambda}$.
This fact follows directly from the Propositions 3.4.1 and 3.5.2 in [20]. To
obtain the formula for the autocodifference, we use the relation between the
scale parameters and codifference given in (2). Then we obtain:

$\displaystyle\sigma_{Y_{\alpha}(t),Y_{\alpha}(t)}^{\alpha}=\sigma_{Y_{\alpha}(s),Y_{\alpha}(s)}^{\alpha}=\frac{\sigma^{\alpha}}{\lambda\alpha}.$

From Eq.(38) for $s<t$ we get:

$Y_{\alpha}(t)-Y_{\alpha}(s)=\int_{-\infty}^{\infty}f(t-x)-f(s-x)dL_{\alpha}(x),$

where $f(t-x)=e^{-\lambda(t-x)}1_{\\{x<t\\}}$. Using Proposition 3.5.2 in [20]
we obtain the scale parameter of random variable $Y_{\alpha}(t)-Y_{\alpha}(s)$
for $s<t$:

$\sigma_{Y_{\alpha}(t)-Y_{\alpha}(s)}^{\alpha}=\sigma^{\alpha}\int_{-\infty}^{\infty}|f(t-x)-f(s-x)|^{\alpha}dx.$

Therefore we have:

$\displaystyle\sigma_{Y_{\alpha}(t)-Y_{\sigma}(s)}^{\alpha}$ $\displaystyle=$
$\displaystyle\sigma^{\alpha}\int_{-\infty}^{s}|f(t-x)-f(s-x)|^{\alpha}dx+\sigma^{\alpha}\int_{s}^{t}|f(t-x)|^{\alpha}dx$
$\displaystyle=$
$\displaystyle\sigma^{\alpha}\int_{-\infty}^{s}|e^{-\lambda(t-x)}-e^{-\lambda(s-x)}|^{\alpha}dx+\sigma^{\alpha}\int_{s}^{t}e^{-\lambda\alpha(t-x)}dx$
$\displaystyle=$ $\displaystyle\sigma^{\alpha}|e^{-\lambda t}-e^{-\lambda
s}|^{\alpha}\int_{-\infty}^{s}e^{\lambda\alpha
x}dx+\sigma^{\alpha}\frac{1-e^{-\lambda\alpha(t-s)}}{\lambda\alpha}$
$\displaystyle=$
$\displaystyle\sigma^{\alpha}\frac{|1-e^{-\lambda(t-s)}|^{\alpha}+1-e^{-\lambda\alpha(t-s)}}{\lambda\alpha}.$

Then we obtain:

$\displaystyle CD(Y_{\alpha}(t),Y_{\alpha}(s))$ $\displaystyle=$
$\displaystyle-\sigma^{\alpha}\frac{1+e^{-\lambda\alpha(t-s)}-|1-e^{-\lambda(t-s)}|^{\alpha}}{\lambda\alpha}.$

A4.

Fractional Lévy motion
From Proposition 3.4 in [68] we infer that for the stable integrals
$\int_{S}f(x)dL_{\alpha}(x)$, where $\\{L_{\alpha}(t)\\}$ is a symmetric Lévy
motion with $0<\alpha<2$ and $S\subset R$, the following holds:

$\displaystyle<\exp\left\\{i\theta\int_{S}f(x)dL_{\alpha}(x)\right\\}>=\exp\left\\{-|\theta|^{\alpha}\int_{S}|f(x)|^{\alpha}dx\right\\}$
(55)

if $\int_{S}|f(x)|^{\alpha}dx<\infty$.
Using formula (55) and the fact that fractional Lévy motion
$\\{L_{\alpha,H}(t)\\}$ is self similar and has stationary increments, for
$t<s$ we obtain:

$CD(L_{\alpha,H}(t),L_{\alpha,H}(s))=\ln(<exp\\{it^{H}L_{\alpha,H}(1)\\}>)+\ln(<exp\\{-is^{H}L_{\alpha,H}(1)\\}>)-\ln(<exp\\{i(t-s)^{H}L_{\alpha,H}(1))\\}>).$

Therefore, we have:

$CD(L_{\alpha,H}(t),L_{\alpha,H}(s))=\left(|t-s|^{\alpha H}-|t|^{\alpha
H}-|s|^{\alpha H}\right)k(H,\alpha),$

where

$k(H,\alpha)=\int_{-\infty}^{\infty}\left|\max(1-u,0)^{H-1/\alpha}-\max(-u,0)^{H-1/\alpha}\right|^{\alpha}du.$

Now we show that the above integral converges. Indeed, we have

$\displaystyle k(H,\alpha)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{0}\left|(1-u)^{H-1/\alpha}-(-u)^{H-1/\alpha}\right|^{\alpha}du+\int_{0}^{1}|1-u|^{H\alpha-1}du=\int_{0}^{\infty}\left|(1+u)^{H-1/\alpha}-u^{H-1/\alpha}\right|^{\alpha}du+\frac{1}{H\alpha}$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\left|(1+u)^{H-1/\alpha}-u^{H-1/\alpha}\right|^{\alpha}du+\int_{1}^{\infty}\left|(1+u)^{H-1/\alpha}-u^{H-1/\alpha}\right|^{\alpha}du+\frac{1}{H\alpha}.$

The first integral converges, because

$\int_{0}^{1}\left|(1+u)^{H-1/\alpha}-u^{H-1/\alpha}\right|^{\alpha}du\leq\int_{0}^{1}(1+u)^{H\alpha-1}du=\frac{2^{H\alpha}}{H\alpha}.$

Note that

$(1+u)^{H-1/\alpha}-u^{H-1/\alpha}=(H-1/\alpha)\int_{0}^{1}(x+u)^{H-1/\alpha-1}du.$

Therefore, for $H\neq 1/\alpha$ we have

$\left|(1+u)^{H-1/\alpha}-u^{H-1/\alpha}\right|^{\alpha}=|(H-1/\alpha)|^{\alpha}\left|\int_{0}^{1}(x+u)^{H-1/\alpha-1}dx\right|^{\alpha}\leq|(H-1/\alpha)|^{\alpha}|1+u|^{\alpha(H-1)-1},$

which gives for $H<1$

$\int_{1}^{\infty}\left|(1+u)^{H-1/\alpha}-u^{H-1/\alpha}\right|^{\alpha}du\leq\int_{1}^{\infty}|(H-1/\alpha)|^{\alpha}|1+u|^{\alpha(H-1)-1}du=\frac{|(H-1/\alpha)|^{\alpha}}{\alpha(1-H)}.$

A5.

Supediffusion continuous time random walk-like process
Let us first calculate the characteristic function of $Z(t)$. We have:

$\Phi_{Z(t)}(k)=<\exp\\{-\frac{L_{\alpha}(t)k^{2}}{2}\\}>.$

Now taking the formula of the Laplace transform of the process $\\{Z(t)\\}$
given in (48), we obtain:

$\Phi_{Z(t)}(k)=\exp\\{-tk^{2}\sigma^{\alpha}2^{-\alpha}\\}.$

Since $Z(t)-Z(s)\sim Z(t-s)$ for $t>s$ ($"\sim"$ means equality in
distribution), then we obtain:

$CD(Z(t),Z(s))=-t\sigma^{\alpha}2^{-\alpha}-s\sigma^{\alpha}2^{-\alpha}+(t-s)\sigma^{\alpha}2^{-\alpha}=-\sigma^{\alpha}2^{-\alpha+1}s=-\sigma^{\alpha}2^{-\alpha+1}\min\\{t,s\\}.$

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