text:
\section{Introduction}
The problem of defining a second-order stochastic process with a given spectrum and a given marginal distribution is of practical importance especially in control theory, signal processing and other engineering applications. A usual approach for obtaining realizations with the desired spectral and distributional properties is by using   linear time-invariant filters to white noise and then  nonlinear transformations to the filter's output, \cite{bib:Grigoriu_book}. The difficulty in the method arises from the interconnection between  the spectral density and the marginal distribution, see for example \cite{bib:BaxevaniP}. Changing the one changes the other, and the relation that links them together is not always simple or tractable.

\cite{bib:Kay} solves this problem by proposing a model in which the distributional and spectral properties are decoupled.  The model extends the classical spectral representation by allowing frequencies to be random. More specifically he considers
\begin{equation}
\label{eq:sinusoidal}
X(t)=\frac{1}{\sqrt{m/2}}\sum_{i=1}^{m}\xi_{i}\cos(\lambda_i t +\phi_i),
\end{equation}
where the sequences of  independent and identically distributed (iid) random variables $(\xi_i)$, $(\lambda_i)$ and $(\phi_i)$ are mutually independent.  In  model (\ref{eq:sinusoidal}), the  phases $\phi_i$ need to be  uniformly distributed over any interval of length $2\pi$ to ensure the  stationarity of the  process.  On the other hand,  the distributions of   frequencies $\lambda_i$ and amplitudes $\xi_i$ can be chosen to control the spectral and marginal distribution of the signal $X$, respectively.

The role of random frequencies in controlling spectral densities has  been noticed already in earlier works, see \cite{bib:Priestley} and \cite{bib:Anderson}. However, these authors  considered   processes  consisting of a single harmonic and as such they were non-ergodic and hence of little use in practise. Summing a large number of harmonics  may to some extent alleviate this problem. For example,  \cite{bib:Kay} showed the  process (\ref{eq:sinusoidal})  to be  `nearly' ergodic in the autocorrelation for large $m$. Specifically he showed   the variance of the sample autocorrelation at time $t$ to converge as the observation time increases without bound, to
\begin{equation}
\label{mom}
\frac{\mathbb{E}[\xi^4]}{m}+\frac{\mathbb{E}[\xi^4]r_X(2t)}{m \mathbb{E}[\xi^2]}-\frac{r^2_X(t)}{m},
\end{equation}
where $\xi$ has the distribution of $\xi_i$'s and $r_X$ is the autocorrelation of process $X$.
It is tempting to conclude  ergodicity in the autocorrelation by letting $m$ go to infinity. However,  this would  lead to  conceptual inconsistency since process $X$  depends on $m$ and formally speaking there is no process corresponding to this limiting case of $m=\infty$.
In fact as shown in \cite{bib:Kay}, if the number of harmonics $m$ in (\ref{eq:sinusoidal})  is fixed and the distribution of amplitudes $\xi_i$ is such that it does not depend on  $m$, the   distribution of the process $X$ must depend on $m$. Similarly choosing the distribution of the process $X$ independently of $m$, which is more natural in applications, gives harmonic amplitudes with distributions that depend clearly on  $m$. In either case the limiting behavior of the distributions depending on $m$, when $m$  increases without bound, is non-trivial and  requires special care when formal arguments are presented.  We overcome these difficulties  by proposing a certain natural extension to model (\ref{eq:sinusoidal}) that allows  treating imbedding it within the class of harmonizable processes and thus avoiding the above difficulties following from dealing with  finite sums with a specific number of harmonics.

The ergodic behavior of a process is one of the important properties  for which correct analysis under large $m$ is critical.
In \cite{bib:Kay}, ergodicity in the autocorellation is studied. Under this assumption,  sample estimators of the mean and the covariance function are consistent and therefore  converge in probability to the estimated values. It happens that the argument goes through (\ref{mom}) only under the assumption of amplitudes  $\xi_i$ having distribution that does not depend on $m$, which is obviously not true when  the distribution of $X$ is independent of  $m$. We demonstrate that in most observed cases  the autocorrelation estimator is not consistent, with its variance  that converges to a  random limit rather than to zero (which would grant the consistency).
By  using the framework of harmonizable processes, we apply Birkoff's ergodic theorem to obtain convergence with probability one for time averages of any functional of the process.  It is important to realize that the strong consistency does not hold, as was already shown in \cite{bib:Wright} for the case of sample variance.     Specifically, it was shown   that the sample variance converges with probability one  to a random quantity that equals the sum of squares of the jumps of the sample process. Thus, it was demonstrated that  signals consisting of harmonics with random frequencies are never ergodic according to the  mathematical definition of ergodicity.

Moreover, we show that the observed ergodicity for large $m$ for the case when distribution of  the amplitudes of the harmonics not depending on $m$ can be interpreted as the convergence of the underlying process $X$ to a Gaussian process. We observe a trade-off here, either we want the process not to be gaussian but then one needs to accept non-ergodicity or if one demands ergodicity we return to the Gaussian domain.

To summarize, in this work, we propose the proper mathematical framework so that all the spectral, distributional and ergodic  properties of signals of the form in (\ref{eq:sinusoidal})  can be treated in a unified manner and with mathematical rigor. At the same time, the aforementioned  processes obtain a simple and interpretable stochastic representation that can be utilized for simulation purposes  and numerical studies. Moreover, the separation between the spectral density and the marginal distribution is preserved, with the former being controlled by the distribution of the frequencies and the latter by the distribution of the amplitudes.

\section{Signal harmonics with random frequencies}
\label{sec:prob_framework}
We provide a proper mathematical framework for representing signals made of harmonics featuring random frequencies.
Following \cite{bib:Kay}, who extended the investigation of such signals  beyond the single harmonic case that was treated earlier in \cite{bib:Anderson,bib:Priestley}, let us consider
\begin{equation}
\label{eq:sinusoidal_sum}
X_{m}(t)=\sum_{i=1}^{m}\xi_{i,m}\cos(\lambda_it+\phi_i).
\end{equation}
All three sequences $(\xi_{i,m})$, $(\lambda_i)$, and $(\phi_i)$ are considered to be random, the first two made of positive variables while the $\phi_i$'s are uniformly distributed over $[0,2\pi)$, and $m$ is, at least for now, a fixed positive integer.
Often one is interested in considering a large number of frequencies  $m$  and studying the convergence of time averages over  large periods  $T$, which in mathematical terms means studying asymptotic properties of the process with both $m$ and $T$ converging to infinity.
However, the order of passing to the  limits with $m$ and $T$ should be treated with care as it may produce apparent formal inconsistencies, as pointed out in the introduction.
Thus studying the asymptotic characteristics of the signal in the  form of the series in  (\ref{eq:sinusoidal_sum}),  is not necessarily the best option since it may lead to some misinterpretation of the resulting properties. However there is an elegant and simple framework that alleviates these shortcomings by representing such signals as the so-called harmonizable processes, which have been introduced in  \cite{bib:Loeve} and their integral representation properties have been investigated in \cite{bib:Loeve, bib:Rosanov} amongst others.
In a sense, harmonizable processes represent the limiting form of $X_m$  when the number of harmonics $m$ increases without bound.
In mathematical literature these processes are conveniently described through L\'evy processes, see  \cite{bib:Sato, bib:Kyprianou} and references therein.   Next we give a brief review of the latter and a simple intuitive construction of a  harmonizable process in which the finite sum (\ref{eq:sinusoidal_sum}) is naturally embedded.

The processes we consider have the form
\begin{equation}
    \label{process_series}
    X(t)=\sum_{i=1}^{\infty}\xi_i \cos(\lambda_it+\phi_i),~t\in \mathbb R,
    \end{equation}
where $(\lambda_i)$ and $(\phi_i)$ are as before, i.e. they are two independent sequences of independent and identically distributed (i.i.d) random variables, the first defined on the half-line with some distribution function (cdf) $F_\lambda$ while the second  are uniformly distributed over $[0,2\pi)$.
The sequence of amplitudes $(\xi_i)$ is still independent of the phases and the frequencies.
However, deviating from (\ref{eq:sinusoidal_sum}), the non-negative amplitudes $(\xi_i)$ are no longer i.i.d but  have the form
$$
\xi_i=
\sigma_0\sqrt{\Lambda^{-1}(\Gamma_i)}R_i,
$$
where  $\sigma_0>0$ is non-random, $\Gamma_i$ are arrivals of the standard Poisson process, and  $(R_i)$ has i.i.d elements  distributed according to the standard Rayleigh distribution, i.e. according to density $re^{-r^2/2}$, and independent of everything else. Measure $\Lambda$ is defined   on $[0,\infty)$, and is the L\'evy measure   satisfying
\begin{equation}
\label{Levy}
\Lambda(\{0\})=0, \quad \int_{0}^\infty u^2~\Lambda(du) <\infty,
\end{equation}
 \cite{bib:Sato, bib:Kyprianou},
The generalized inverse $\Lambda^{-1}$  of the tail of $\Lambda$ is defined by
\begin{equation}
\label{Levy_inverse}
\Lambda^{-1}(u)=\inf\{x>0 :\Lambda\left[x,\infty\right)<u\}, u>0.
\end{equation}
For normalization purposes, that will be clear in the next section,  additionally to (\ref{Levy}), we assume
\begin{equation}
\label{Levy_restriction}
\int_{1}^\infty u~\Lambda(du)=1.
\end{equation}

We also note that for the variance mixtures of a standard Gaussian vector
$$
\mathbf W_i=(W_{1i},W_{2i})=\sqrt{
\Lambda^{-1}(\Gamma_i)}( Z_{1i},Z_{2i}),
$$
with $Z_{1i}$ and $Z_{2i}$  mutually independent standard normal, that are also independent of $\Gamma_i$, we have
$$
\xi_i=\sigma_0 \|\mathbf W_i\|=\sigma_0\sqrt{W_{1i}^2+W^2_{2i}}.
$$

As we will see in the next section,   $\Lambda$ denotes the L\'evy measure of an infinitely divisible distribution with finite second moment defined  on the positive half-line.
We shall also argue for the correctness of the infinite series in (\ref{process_series}) using its spectral representation in the form of a second order harmonizable process.
We also point out that process $X$ is uniquely defined, up to a scale, by $(\sigma_0, F_\lambda, \Lambda)$, with the process spectrum   defined by $F_\lambda$ and $\Lambda$  responsible for the distribution of $X$.

\subsection{The finite sum with independent amplitudes}
\label{fsia}
Now, we demonstrate that the finite sums (\ref{eq:sinusoidal_sum}) embed in our model.
Consider the case of a L\'evy measure $\Lambda$ with  support separated from zero, i.e.   for some fixed $u_0 >0$,
\begin{equation*}
\label{eq:truncated_measure}
\Lambda[0,u_0]=0,
\end{equation*}
and let $u_*$ be the supremum over such $u_0$'s.
It is easy to see that non-increasing $u\mapsto \Lambda[u,\infty)$ is constant and equal to $L=\Lambda[u_*,\infty)$ for $0< u\le u_*$.
Notice that $\Lambda^{-1}(\Gamma_i)=0$ for $\Gamma_i>L$ and for $\Gamma_i < L$, the generalized inverse $\Lambda^{-1}(\Gamma_i)$ is different than zero. Hence, there exists $ N : \Lambda^{-1}(\Gamma_{N})\neq 0$ and $\Lambda^{-1}(\Gamma_{N+1})=0$.
It is easy to see that $N$ is a random variable depending on $L$ and Poisson distributed with  mean $L$, i.e. $N=N(L)$, where $N(t)$, $t>0$ is the standard Poisson process given by $\Gamma_i$, $i\in \mathbb N$.
By truncating the series (\ref{process_series}), the infinite sum becomes a finite one with a random number of terms
\begin{equation}
\label{eq:series_Poisson}
X(t)=\sigma_0\sum_{i=1}^{N(L)}\sqrt{\Lambda^{-1}(\Gamma_i)}R_i \cos(\lambda_it+\phi_i).
\end{equation}
Conditioning on $N(L)=m$, the arrival times $\Gamma_i$, $i\le m$ of the Poisson process, are distributed as the order statistics of variables that are uniformly distributed on  $[0,L]$, yielding the following representation for the $X$ process
\begin{equation}
\label{eq:series_Poisson_2}
(X(t)\mcond N(L)=m)\stackrel{d}{=}\sum_{i=1}^{m}\xi_{i,L}\cos(\lambda_it+\phi_i),
\end{equation}
with i.i.d variables $\xi_{i,L}=\sigma_0\sqrt{\Lambda^{-1}(L{U_i})}R_i$, where ${U_i}$  uniformly distributed on $[0,1]$.
Here and in what follows $\stackrel{d}{=}$ stands for the equality in distribution.
We observe that the above form is almost identical to (\ref{eq:sinusoidal_sum}) that was discussed in  \cite{bib:Kay}.
In this sense, the processes discussed there are embedded in our model as the processes conditional on $N(L)=m$.
The limiting argument of $m$ increasing without bound that was used there in studying convergence of sample correlation can be now properly stated as the case when $L$, and thus $N(L)$, increases without bound.

 More specifically, if we start with general $\Lambda$ that is not separated from zero, by considering a truncated measure
$$
\Lambda_{L}(A)=\Lambda\left(\mathbb I_{[u,\infty)}\cap A\right),
$$
where $u=\Lambda^{-1}(L)$, we obtain the approximation of $X(t)$ by $X_L(t)$ defined for $\Lambda_L$, i.e. by  the right hand side in (\ref{eq:series_Poisson}).
For $L$ increasing without bound $X_L$ clearly converges almost surely to $X$.
\subsection{Harmonizable Laplace process and Gamma L\'evy measure}
Although any   measure $\Lambda$  that satisfies  (\ref{Levy}) can be used in the presented construction, here we discuss in full detail the important case of the L\'evy measure characterizing the Gamma process. Recall that a stochastic process  $G(u)$ is called a Gamma process if it starts at zero, has independent and stationary increments, and the  increments  are distributed as   Gamma  with some scale and shape parameters.
The corresponding L\'evy measure  $\Lambda$ is defined on $[0,\infty)$  in terms of the exponential integral function
$$
\Lambda\left([u,\infty)\right)=E_1(u)/\nu=
\frac{1}{\nu}\int_{u}^\infty \frac{e^{-x}}x ~dx,
$$
see also \cite{bib:Abramowitz}.
Thus   $\Lambda^{-1}(\Gamma_i)=E_1^{-1}(\nu \Gamma_i)$.
Since inverting the integral function can be computationally costly, it is better to utilize the shot noise series expansion of the gamma process,  given in  \cite{bib:Bondesson} and \cite{bib:Rosinski}:
\begin{equation}
\label{eq:Gamma_L}
G(u)= \nu\sum_{i=1}^{\infty}  V_i e^{-\nu \Gamma_i} \mathbb I_{[U_i,1]}(u), u\in [0,1]
\end{equation}
where $V_i$ are i.i.d standard exponentially distributed random variables independent of both $\Gamma_i$ and $U_i$. Both sequences $\Gamma_i$ and $U_i$  are distributed as before. The increments $G(u+h)-G(u)$ have been chosen to follow the Gamma distribution with scale $\nu$ and shape $h/\nu $, so that $\mathbb{E}(G(1))=1$.

Using this shot noise expansion of   $G$, process $X(t)$ in (\ref{eq:series_Poisson}),  attains the following alternative and more explicit series representation
\begin{equation}
    \label{eq:Gamma_series2}
    X(t)\stackrel{d}{=}\sigma_0 \nu^{1/2} \sum_{i=1}^{\infty}  e^{-\nu\Gamma_i/2}V_i^{1/2} R_i\cos(\lambda_i t+\phi_i).
\end{equation}
As it will be seen in section \ref{sec:distrib}, this  harmonizable process has  generalised Laplace distribution as marginal and for this reason will be referred to as \emph{harmonizable Laplace process}.

Although the above series expression resembles the  representation given in (\ref{process_series}), the two series are not exactly the same since while $\sqrt{\Lambda^{-1}(\Gamma_i)}$'s in (\ref{process_series}) are non-increasing, in  (\ref{eq:Gamma_series2}) the non-increasing  $\exp(-\nu \Gamma_i)$ are multiplied by the i.i.d standard exponential variables $V_i$.
However, the above explicit form allows for the following elegant representation when conditioned on the Poisson process corresponding to $\Gamma_i$'s. Let $N(L)$ as before denote  the random number of arrivals of the  Poisson process associated with $\Gamma_{i}$'s, where $\Gamma_{N(L)}\le L <\Gamma_{N(L)+1}$. Truncating the series to $N=N(L)$ and then conditioning on $N=m$ yields
\begin{equation}
    \label{eq:Gamma_series_trunc2}
    (X(t)\mcond N=m)\stackrel{d}{=

    \sigma_0 \nu^{1/2}
    \sum_{i=1}^{m}
    \left(
     \left(e^{-U_{i}}\right)^{L\nu }V_i
     \right)^{1/2}
     R_i\cos(\lambda_i t+\phi_i
    =
        \sum_{i=1}^{m}
 \xi_{i,L}\cos(\lambda_i t+\phi_i)

   \end{equation}
where $\xi_{i,L}= \sigma_0 \nu^{1/2}\left(
     \left(e^{-U_{i}}\right)^{L\nu }V_i
     \right)^{1/2}
     R_i$ which yields yet another approximation of $X$ that has the form (\ref{eq:sinusoidal_sum}).
 Similarly, as before by defining for $L>0$
 $$
 X_{L}(t){=}
    \sigma_0 \nu^{1/2}
    \sum_{i=1}^{N(L)}
     e^{-\nu\Gamma_i/2}V_i^{1/2} R_i\cos(\lambda_i t+\phi_i),
 $$
 we obtain an almost sure approximation of $X$ by the finite sum  $X_{L}$ with Poisson number of terms.
 We note that this approximation has different distribution of the i.i.d amplitudes than the one based on direct inversion of the L\'evy measure and thus illustrating the importance of the choice of asymptotic approach in the approximation of $X$.
\section{Spectral Representation}
\label{spre}
In this section we  discuss how the  infinite sum of harmonics in (\ref{process_series}) can be represented as a harmonizable process
\begin{equation}
\label{har}
X(t)=\int_{-\infty}^{\infty} e^{i\lambda t} d\zeta(\lambda),
\end{equation}
with $\zeta$ a  complex-valued stochastic measure,  symmetric about zero, i.e. $\zeta(-A)=\overline{\zeta(A)}$,  the overline indicates the complex conjugate, and independently scattered on the  real line. The terms spectral measure and spectral process for $\zeta$ will be used indistinguishably.  Harmonizable processes  have been introduced in \cite{bib:Loeve} and further studied in \cite{bib:CambanisHardinWerron, bib:Yaglom} and \cite{bib:Cramer} amongst others, as a first step generalization of weakly stationary mean-square continuous stochastic processes. While harmonizable processes  do not need to  satisfy the second-order property, we consider only this case originally treated in \cite{bib:Loeve}. So from  now on, the  stochastic measure $\zeta$ has finite second order moment.  Bochner's theorem for the covariance function of  $X$ yields  a spectral measure $F$ that controls the spectral process $\zeta$, which from  now on  will be denoted by $\zeta_F$. Without losing generality we assume that $F$ does not have an atom at zero.

For the purposes of this paper we are only interested in symmetric around zero spectral distributions. Hence, $\zeta_F$  can be viewed as the zero concatenation of two processes defined for $\lambda>0$, that  have necessarily  the form
\begin{equation}
 \label{eq:spectral_meas}
\begin{split}
&\zeta_F(0,\lambda]=\frac{\sqrt{2}}{2}\bigg(B(G(F(0,\lambda]))+i\tilde{B}(G(F(0,\lambda]))\bigg),\\
&\zeta_F[-\lambda,0)=\overline{\zeta_F(0,\lambda]},
\end{split}
\end{equation}
where $B$ and $\tilde{B}$ are two independent standard Brownian motions. $G: [0,\infty)\mapsto [0, \infty)$ is a non-decreasing pure jump L\'evy process   with  $\mathbb{E}(G(1))=1$, which is exactly property (\ref{Levy_restriction}),  and independent of both $B$ and $\tilde{B}$. Finally $F: \mathbb R  \mapsto [0,F(\mathbb R)]$  is the  spectral distribution function controlling the spectral process $\zeta_F$.

The  \emph{Inverse L\'evy Measure Method} \cite{bib:Rosinski}, yields  the following  series representation for the process $G$ with $ u\in [0,F(0,\infty)]$ :
\begin{equation}
    \label{LevyPr}
    G(u)=F[0,\infty)\sum_{i=1}^{\infty} \Lambda^{-1}(\Gamma_i)~ \mathbb I_{\displaystyle [U_i,1]}(u/ F(0,\infty)),\\
\end{equation}
where   $U_i$  are  i.i.d  random variables distributed as uniform  on  $[0,1]$, and $\Gamma_i$ are  the arrival times  of a standard Poisson process. Sequences ($U_i$)  and  $(\Gamma_i)$ are mutually independent.  $\Lambda$ denotes the  L\'evy measure of the infinitely divisible distribution of the random variable $Y=G(F(0,\infty))/F(0,\infty)$ that  satisfies (\ref{Levy}) and (\ref{Levy_restriction})  and its generalized inverse is defined as in (\ref{Levy_inverse}).
The explicit connection between $G$ and $\Lambda$ is through the characteristic function of $Y$ that is given in terms of the  L\'evy-Khinchine representation
$$
\phi_Y(t)=\exp\left(\int_0^\infty \big(e^{ixt}-1-it x\mathbb I_{(0,1)}(x)\big) ~ d\Lambda(x)\right).
$$
It is easy to see, using (\ref{Levy_restriction}) that $\phi_Y'(t)|_{t=0}=i$ so that $\mathbb{E}(Y)=-i\cdot \phi_Y'(0)=1$. Then, from the definition of the random variable $Y$, it is straightforward to deduce that  $\mathbb{E}(G(1))=1$.

\begin{Proposition}
\label{prop:spectral}
The stochastic process $X(t)$ defined in (\ref{process_series}) has the spectral representation
\begin{equation}
    \label{process_real}
    X(t)\stackrel{d}{=} \int_{-\infty}^{\infty} e^{i\lambda t}~d\zeta_F(\lambda).
\end{equation}
with spectral process $\zeta_F$ defined as in (\ref{eq:spectral_meas} ) with the L\'evy process $G$ in (\ref{LevyPr})  defined so that $\Lambda$ is the L\'evy measure of $Y=G(F(0,\infty))/F(0,\infty)$, and
$$
F(0,\lambda]=\frac{\sigma_0^2}{2} F_\lambda(\lambda), ~\lambda\ge 0.
$$
\end{Proposition}
\begin{proof}
Replacing $u$  in (\ref{LevyPr}) by $F(0,\lambda]$ and using the facts that $F(0,\infty)=\frac{\sigma_0^2}{2}$ and  $F_{\lambda}$ is a cumulative distribution function, we obtain
\begin{equation}
\label{eq:G_series}
G(F(0,\lambda])=F(0,\infty)\sum_{i=1}^{\infty} \Lambda^{-1}(\Gamma_i) \mathbb I_{[F_\lambda^{-1}(U_i),\infty]}(\lambda).
\end{equation}

Embedding the  series expansion of  $G$, given  in (\ref{eq:G_series}), in  the spectral process $\zeta_F$ in   (\ref{eq:spectral_meas}) and denoting by $\sigma_0^2/2$ the $F(0,\infty)$,  we obtain the following expression for the process $X$ defined in (\ref{process_real})
\begin{eqnarray}
    \label{process_series_new}
    X(t)&\stackrel{d}{=}\sigma_0\sum_{i=1}^{\infty}\sqrt{\Lambda^{-1}(\Gamma_i)}\cdo

     \left( Z_i^{(1)}\cos(F_\lambda^{-1}(U_i)t)- Z_i^{(2)}\sin(F_\lambda^{-1}(U_i)t)\right)\\ \nonumber
    &\stackrel{d}{=}\sigma_0\sum_{i=1}^{\infty}\sqrt{\Lambda^{-1}(\Gamma_i)}R_i \cos(F_\lambda^{-1}(U_i)t+\phi_i)\\ \nonumber
   & \stackrel{d}{=}\sigma_0\sum_{i=1}^{\infty}\sqrt{\Lambda^{-1}(\Gamma_i)}R_i \cos(\lambda_it+\phi_i) \nonumber
\end{eqnarray}
where $(Z_i^{(j)}), j=1,2$, are sequences of independent standard normal random variables. Moreover,  $(R_i)$ and $(\phi_i)$ are   sequences of independent standard Rayleigh and uniform on $[0,2\pi)$  random variables respectively. $R_i$ and $\phi_i$ are also jointly independent. The first equivalence in (\ref{process_series_new}) follows   by noticing  that the increments of a Brownian motion are stationary, independent and distributed as normal random variables with mean zero and variance $\sigma_0^2\Lambda^{-1}(\Gamma_i)$. The second equivalence follows from $\left(Z_i^{(1)},Z_i^{(2)}\right)\stackrel{d}{=}R_i \left(\cos \phi_i, \sin \phi_i\right)$. Finally $(\lambda_i)$ is a sequence of random variables distributed according to the distribution $F_\lambda$.
\end{proof}
\begin{Remark}
 Notice that the series representation of $X(t)$ in (\ref{process_series_new}) decomposes the process into a series of real harmonics with phases and amplitudes that are random and independent of each other but also, departing from the classical case, frequencies that are also random and distributed according  to $F_\lambda$. Hence this is a method that can be used to produce stochastic processes with given marginal density and given spectral distribution function.
\end{Remark}

\subsection{The discrete spectrum }
We finish this section  by having a closer look to the case of discrete  frequency distribution $F_{\lambda}$. It turns out,  following from spectral representation in (\ref{process_real}), that in this case the randomness of the frequencies  is only apparent.  More precisely, let us consider a spectrum that us concentrated at the frequencies $\pm l_k, k=1,\dots$  with masses $\nu_k=F_{\lambda}(\{l_k\}), \forall k$. We  have used   the standard convention $-l_k=l_{-k}$. Then,   process $X$ in (\ref{process_real}) reduces to the following sum of harmonics with non-random frequencies
\begin{equation*}
    \label{process_series_discrete}
    X(t)\stackrel{d}{=}\sum_{k=1}^{\infty}\sqrt{G_k}\cdot \left( Z_k^{(1)}\cos(l_kt)- Z_k^{(2)}\sin(l_k t)\right)\stackrel{d}{=}\sum_{k=1}^{\infty}\sqrt{G_k}R_k \cos(l_kt+\phi_k),
\end{equation*}
where $G_k$ are independent random variables  with distribution uniquely identified by the  L\'evy measure $\Lambda$ of the process $G$ through $\mathbb{E} (G_k)=\nu_k$. The variables $Z_k^{(1)}, Z_k^{(2)}, R_k$ and $\phi_k$ are distributed as before.
\section{Distributional Properties}
\label{sec:distrib}
In this section we study the distributional properties of the harmonizable process  $X$ in (\ref{process_real}). We start with a short discussion of the distribution of the different components in the series representation of the process before we turn to the marginal  distribution of the process  and the joint distribution of the process and its derivative. The latter is important when one is interested in the level crossing sampling distributions of the process which are expressed by means of the Rice formula, \cite{bib:Rice1944,bib:Rice1945,bib:AbergP}.

\subsection{Phase distribution}
In the previous sections, we have considered processes  with  the random amplitudes   and  frequencies having  arbitrary distributions, but with the  phases  being uniformly distributed over  some interval of length $2\pi$. The latter is a restriction that needs to be imposed  for  the process $X$  in (\ref{process_real}) to be strictly stationary. Next, we show  correctness of this statement for the case of a  single harmonic. Extension of the  argument for any number of harmonics is rather straightforward.

Consider the single harmonic
\begin{equation}
\label{eq:single_harmonic}
\xi\cos(\lambda t+\phi)
\end{equation}
and  condition on $\lambda=\lambda_0$ and $\xi=\xi_0$. Then  for each $n\in \mathbb N$, $a_i, t_i\in \mathbb R$, $i=1,\dots, n$, $s\in \mathbb R$ we have
\begin{align*}
\sum_{i=1}^n a_i \xi_0 \cos(\lambda_0(t_i+s)+\phi)&=\sum_{i=1}^n a_i \xi_0 \cos(\lambda_0t_i+\phi+\lambda_0s)\\
&=\sum_{i=1}^n a_i \xi_0 \cos(\lambda_0t_i+\tilde \phi)\\
&\stackrel{d}{=}\sum_{i=1}^n a_i \xi_0 \cos(\lambda_0t_i+\phi).
\end{align*}
The last equality holds if phases $\tilde \phi = (\phi+\lambda_0s)_{\mod 2 \pi}$ have the same distribution as $\phi$, which is true if and only if the latter are  uniformly distributed over $(0,2\pi]$ and thus independent of $\lambda_0$ and, of course, of $\xi_0$.
This proves conditional and thus also unconditional stationarity in the strict sense for the single harmonic, independently of the distribution of $\lambda$ and $\xi$.
\begin{figure}[!t]
 \centering
 \includegraphics[width=3.5in]{Fig2.pdf}
 \caption {Theoretical density of $X(0)$ defined in (\ref{eq:Gamma_series2}) and histogram of $X(t)$ from one realisation based on 2000 simulated values  for $\nu=1.5$ which corresponds to a process with standard Laplace marginal distribution.  The frequencies follow a Uniform distribution on $[0,1]$ and the rest of the random variables are as indicated in the text.}
\label{fig:hist1}
\end{figure}

\begin{figure}[!t]
 \centering
 \includegraphics[width=3.5in]{Fig3.pdf}
 \caption {Theoretical density of $X(0)$ defined in (\ref{eq:Gamma_series2}) and histogram of $X(t)$  50 realisations of 2000 simulated values  for $\nu=1.5$ which corresponds to a process with standard Laplace marginal distribution.  The frequencies follow a Uniform distribution on $[0,1]$ and the rest of the random variables are as indicated in the text.}
\label{fig:hist2}
\end{figure}
For the record, we formulate this in the form of a theorem.
\begin{Theorem}
A process of the form (\ref{eq:single_harmonic}) with  arbitrary amplitude and frequency distribution  is strictly stationary if and only if $\phi$ is uniform on $[0,2\pi)$.
\end{Theorem}

\subsection{Marginal distribution}
The next result lists basic facts about the distribution of the process $X$.
\begin{Proposition}
\label{prop:distribution}
The harmonizable process $X(t)$ defined in (\ref{process_real}) has
\begin{enumerate}
    \item one dimensional marginal distribution of the $G$-type given  by
    \begin{equation}
        \label{eq:1dim}
        X(t)\stackrel{d}{=}\sqrt{2\cdot G(F(0,\infty))}\cdot Z,
    \end{equation}
    where $Z$ is a standard normal variable while $G$ and $F$  are as in Proposition~\ref{prop:spectral},
    \item finite dimensional distributions defined by the characteristic function
\begin{eqnarray}
\label{eq:fdd}
&\mathbb{E}\bigg(\exp\big(i\sum_{j=1}^nu_jX(t_j)\big)\bigg)=\exp\bigg( \int_{0}^{\infty} \int_{0}^{\infty}
\bigg(\exp\big(- x\frac{\mathbf{u}\mathbf A\mathbf{u}^T}{2}\big)-\\
&-1-x\frac{\mathbf{u}\mathbf A\mathbf{u}^T}{2}\mathbb I_{(0,1)}(x)\bigg)~d\Lambda(x)~dF(\lambda)\bigg) \nonumber
\end{eqnarray}
where $\mathbf u=(u_1,\dots,u_n)$, the transpose of $u$ is denoted by $u^T$  and the entries of the matrix $\mathbf A$ are $\cos(\lambda(t_j-t_k))$,
\item  characteristic function of $(X(0),X'(0))$ is given by
\begin{eqnarray*}
 \label{eq:dim_der}

 &\phi(u_1,u_2)= \int_0^\infty \int_0^\infty \exp\bigg(-x(u_1^2+u_2^2\lambda)/2-1-x(u_1^2+u_2^2\lambda)/2 \cdot \mathbb I_{(0,1)}(x)\bigg)~d\Lambda(x)~dF(\lambda),

\end{eqnarray*}
where the derivative of $X(t)$, which is assumed to exist in the mean-square sense,  is given by
\begin{equation*}
X'(t)=-\int_{0}^\infty \lambda \sin(\lambda t)~dB(G(F(\lambda)))+\int_{0}^\infty \lambda \cos(\lambda t)~d\tilde{B}(G(F(\lambda))).
\end{equation*}

\end{enumerate}
\end{Proposition}
\begin{proof}

1) Relation (\ref{eq:1dim}) is immediate from noticing that since process $X(t)$ is stationary, it has the same marginal distribution as the random variable $X(0)$. To obtain the distribution of the latter notice that using the definition of the spectral measure in (\ref{eq:spectral_meas}), this is the same as the distribution of $\sqrt{2}B(G(F(0,\infty))$, which is  normal  with variance $2G(F(0,\infty))$.

2) Consider the random variable $T=\sum_{j=1}^nu_jX(t_j)$. Then
$$
T=\int_{0}^{\infty} f(\lambda)~dB(G(F(\lambda)))+ \int_{0}^{\infty} \tilde{f}(\lambda)~d\tilde{B}(G(F(\lambda))),
$$
with
$$
f(\lambda)=\sum_{j=1}^n u_j\cos(\lambda t_j), ~ \tilde{f}(\lambda)=\sum_{j=1}^n u_j\sin(\lambda t_j).
$$

Conditionally on $G$ we have
$$
T\mcond G\stackrel{d}{=}\int_{0}^{\infty} f(\lambda)~dB_{G\circ F}(\lambda)+ \int_{0}^{\infty} \tilde{f}(\lambda)~d\tilde{B}_{G\circ F}(\lambda).
$$
Since this  is the sum of two independent Gaussian random variables we get
\begin{equation}
    \label{eq:char_T_cond}
\mathbb{E} \big(e^{iuT}\mcond G\big)=\exp\bigg(-\frac{u^2}{2}\int_0^{\infty}\big(f^2(\lambda)+\tilde{f}^2(\lambda)\big)~dG_{F}(\lambda)\bigg),
\end{equation}
which we recognize as the Laplace transform evaluated at $u^2/2$ of the random variable  $V=\int_0^{\infty}\big(f^2(\lambda)+\tilde{f}^2(\lambda)\big)~d
G_{F}(\lambda)$.

Using  the well known form of characteristic function for a L\'evy random variable $G(1)$\cite{bib:Sato} with L\'evy measure $\Lambda$,
$$
\mathbb{E}(e^{iuG(1)})=\exp\bigg( \int_0^{\infty} \big(e^{i ux}-1-iu x \mathbb I_{(0,1)(x)}\big)~d\Lambda(x)\bigg)
$$

together with (\ref{eq:char_T_cond}), we obtain for the random variable $T$
\begin{eqnarray*}
\begin{aligned}
&\mathbb{E}\big(e^{iT}\big)= \mathbb{E}\bigg(\mathbb{E}\big(e^{iT}\mcond G\big)\bigg)=\\
&=\exp\bigg(\int_0^{\infty} \int_0^\infty \bigg(e^{-\frac{1}{2}(f^2(\lambda)+\tilde{f}^2(\lambda))x}- 1-\frac{1}{2}(f^2(\lambda)+\tilde{f}^2(\lambda))  x\mathbb I_{(0,1)}(x)\bigg)~d\Lambda(x)~dF(\lambda)\bigg).
\end{aligned}
\end{eqnarray*}
Now the proposition follows by observing that
$$
f^2(\lambda)+\tilde{f}^2(\lambda)=\sum_{j=1}^n\sum_{k=1}^nu_ju_k\cos(\lambda(t_j-t_k))=\mathbf u \mathbf A\mathbf u^T.
$$

3) The result is an immediate consequence of the second part if we  notice that
\begin{equation*}
u_1X(0)+u_2X'(0)= \int_0^\infty u_1~dB(G(F(\lambda)))+\int_0^\infty u_2\lambda ~d\tilde{B}(G(F(\lambda))),
\end{equation*}
which is the random variable $T$ for $f(\lambda)=u_1$ and $\tilde{f}(\lambda)=u_2\lambda$.
\end{proof}

\subsection{The harmonizable Laplace process} We conclude this section with the  special case of the harmonizable Laplace process given in (10).   In this case,  $L_F(\lambda):=B(G(F(\lambda))$ and $\tilde{L}_F(\lambda):=\tilde{B}(G(F(\lambda))$ with $G$ a Gamma process, are two Laplace processes. The distribution of $L$ and $\tilde{L}$  is an immediate consequence of the representation of the Laplace random variable as a Gaussian random variable subordinated to a gamma variable,    see \cite{bib:KotzKP}.  The resulting representation of the process in (10) is
\begin{equation}
\label{eq:Laplace}
X(t)\stackrel{d}{=}\int_0^\infty\cos(\lambda t)~dL_F(\lambda)+\int_0^\infty\sin(\lambda t)~d\tilde{L}_F(\lambda).
\end{equation}

The marginal distributions of the process can be obtained in a similar way as in Proposition ~\ref{prop:distribution}, and are gathered in the next result.
\begin{Proposition}
\label{prop:Laplace}
The Laplace harmonizable process defined in (\ref{eq:Laplace}) has
\begin{enumerate}
    \item marginal distribution defined by the characteristic function
    \begin{equation}
        \label{eq:laplace_marg}
        \phi_{X(t)}(u)=\bigg(1+ \nu u^2\bigg)^{- F(0,\infty)/\nu},
    \end{equation}
    \item finite dimensional distributions defined by the following characteristic function
    \begin{equation}
        \label{eq:laplace_fdd}

        \mathbb{E}\exp\big(i\sum_{j=1}^n u_jX(t_j)\big)=\exp \bigg(-\int_0^\infty \ln\big(1+ \nu\mathbf{u}\mathbf A\mathbf u^T \big)~dF(\lambda)\bigg),

    \end{equation}
    with $\mathbf u$ and $\mathbf A$ as in Proposition~\ref{prop:distribution}.
\end{enumerate}
\end{Proposition}
\begin{proof}
1. Relation (\ref{eq:laplace_marg}) follows directly from (\ref{eq:1dim}) if we consider for $G$ a gamma process with scale $\nu$, shape $1/\nu$,  and use the expression for the Laplace transform  for a Gamma random variable.

2. The form of the characteristic function in (\ref{eq:laplace_fdd}) can be derived either from (\ref{eq:fdd}) for Gamma L\'evy measure $\Lambda$ or can be found in \cite{bib:AbergP}.
\end{proof}

\section{The time averages and ergodic properties}
The ergodicity of a signal is an important property as it allows to retrieve statistical characteristics of the model by applying time averages on a single realization.
Probably the most standard model exhibiting ergodicity is a moving average that represent a filter of a noise.
This model can feature both arbitrary spectrum and desirable distributional properties, see \cite{bib:AbergP,bib:BaxevaniP}.
On the other hand, if a model lacks ergodicity it may provide a framework for studying phenomena for which sample-to-sample stochastic variability is significant.

In \cite{bib:Kay}, it was suggested that the sum of $m$ harmonics with random frequencies and with the amplitudes distributed according to a random variable $\xi$ have the  ergodicity in autocovariance property and the key argument required that $m$ is increasing without bound and $\mathbb{E}(\xi^4)/m$ converges to zero,
see the middle of the first column at page 3450 and the bottom of the first column at page 3457 in the quoted work.
If the amplitude distribution of $\xi$ does not depend on $m$ this would reduce the requirement to the existence of a finite fourth moment for $\xi$.
However, typically the  distribution of $\xi$ does depend on $m$ or otherwise the distribution of $X(t)$ must depend on $m$ which is difficult to interpret when we assume that $m$ is changing.
Existence of the limiting distribution of such process has to be examined and it may lead to undesirable distribution as seen next.

We use our leading example of Laplace (double exponential) distribution to show that, the considered  amplitude has distribution of the form
$$
\xi\stackrel{d}{=}K \sqrt{m} R \sqrt{G_m},
$$
where $K>0$ is a numerical constant independent of $m$, $R$ is distributed as Rayleigh, and $G_m$ as gamma  with  shape parameter $a/m$, for some $a>0$ and scale 1/2.
For this distribution, the fourth moment is proportional to $m^2\cdot a/m(a/m+1)=a^2+am$, and thus, clearly, $\mathbb{E}(\xi^4)/m$ converges to $a$, which is non-zero. Thus the argument for ergodicity in covariance fails.

On the other hand if we keep the amplitude distribution to be independent of $m$, by considering for example
$$
\xi\stackrel{d}{=}K  R \sqrt{G_1},
$$
the limiting behavior of the distribution of the process $X_m$ becomes Gaussian.
In order to formulate this in mathematical terms, the passage to the limit with $m$ should be formulate in terms of $m=N(L)$, where $N$ is a Poisson process.
Recall that conditionally on $N(L)=m$ we have
\begin{equation}
\sigma_0\sum_{i=1}^{N(L)}\sqrt{\Lambda^{-1}(\Gamma_i)}R_i \cos(\lambda_it+\phi_i)=\sigma_0\sum_{i=1}^{N(L)}\sqrt{\Lambda^{-1}(L U_i)}R_i \cos(\lambda_it+\phi_i),
\end{equation}
where $U_i$ is uniform on $[0,1]$.
Thus, the case of the amplitude distribution not depending on $m$ requires that  $\Lambda^{-1}(L U_i)$ is not depending on $L$, which can be obtained by replacing $\Lambda$ by
$$\Lambda_L(\cdot)=L\cdot \Lambda(\cdot)
$$
and  consider instead the process
\begin{equation}
\label{eq:X_L}
X_L(t)=\sigma_0\sum_{i=1}^{\infty}\sqrt{\Lambda_L^{-1}(U_i)}R_i \cos(\lambda_it+\phi_i),
\end{equation}
with all random variables defined the same way as in the proof of Proposition~\ref{prop:spectral}.
Gaussianity of the limiting behavior of $X_L$ is then formulated as follows.
\begin{Proposition}
When $L$ increases without bound, the process $X_L/\sqrt{L}$ in (\ref{eq:X_L}) converges in distribution to the Gaussian process  $X(t)$ having spectral representation
$$
X(t)=\int_{-\infty}^{\infty} e^{i\lambda t} d\zeta_F(\lambda),
$$
with $\zeta_F(0,\lambda]=\frac{\sqrt{2}}{2}\bigg(B(F(0,\lambda])+i\tilde{B}(F(0,\lambda]))\bigg)$, $\zeta_F[-\lambda,0)=\overline{\zeta_F(0,\lambda]}$ and $B, \tilde{B}$ and $F$ as in section~\ref{spre}.
\end{Proposition}
\begin{proof}
It is quite clear that by taking the L\'evy measure $\Lambda_L=L\Lambda$, process $X_L/\sqrt{L}$ has the spectral representation of Section~\ref{spre} with the pure jump process
$$
G_L(v)=G(L\cdot v)/L,
$$
where $G$ is the corresponding jump process in the spectral representation of a process with the L\'evy measure $\Lambda$.
Consequently, in the spectral measure of $X_L/\sqrt{L}$ given by
\begin{equation}
\zeta_{F,L}(0,\lambda]=\frac{\sqrt{2}}{2}\bigg(B(G_L(F(0,\lambda]))+i\tilde{B}(G_L(F(0,\lambda]))\bigg)
\end{equation}
the subordinator has the form
$$
G_L(F(0,\lambda])=\frac{G(L\cdot F(0,\lambda])}{L\cdot F(0,\lambda]} F(0,\lambda].
$$
For L\'evy subordinators  with finite moment, we have
$$
\lim_{L\rightarrow \infty} \frac{G(L\cdot F(0,\lambda])}{L\cdot F(0,\lambda]} =1.
$$
From this it follows easily that $\zeta_{F,L}$ converges to $\zeta_F$
The conclusion then can be easily obtained for example, by conditioning on $G$ and establishing second order convergence of two Gaussian processes.
\end{proof}
\begin{Remark}
From the above result it follows that in order to argue about the ergodicity for large $L$ ($m$) while requiring that the amplitude distribution does not depend on $L$ ($m$), one inevitably ends up with the process that asymptotically is Gaussian.
\end{Remark}

In what follows, we show that the ergodicity property does not hold for processes of the form (\ref{eq:series_Poisson_2}) or equivalently (\ref{eq:Gamma_series_trunc2}).
Despite this lack of ergodicity, the model can be still valuable for practical applications since we also establish the limiting distributions of the time averages expressed in terms of the amplitude distributions.

We will use  \emph{ergodicity in autocorrelation}, which is a variant of ergodicity of a process $X(t)$ requiring that for a centered process
 \begin{equation*}
     \label{eq:Birkoff}
 \lim_{T\mapsto \infty}\frac{1}{T}\int_0^T X(t)X(t+\tau)~dt=r_X(\tau).
 \end{equation*}

\begin{Proposition}
\label{prop:time_averages}
Consider  the conditioned process $(X(t)\mcond N(L)=m)$  in (\ref{eq:series_Poisson_2}) defined as finite sum of harmonics, i.e.
$$
\tilde{X}(t)\stackrel{d}{=}\sum_{i=1}^m \xi_{i,L}\cos(\lambda_i t+\phi_i)
$$
with amplitudes, frequencies and phases being random and described earlier is Subsection~\ref{fsia}.
Then
    \begin{align}
    \label{eq:sum_1}
    \lim_{T\mapsto \infty}\frac{1}{T}\int_0^{\infty} \tilde{X}(t)~dt&=0,\\
   \label{eq:sum_2}
       \lim_{T\mapsto \infty}\frac{1}{T}\int_0^{\infty} \tilde{X}(t)\tilde{X}(t+\tau)~dt&=\sum_{i=1}^m \frac{\xi_{i,L}^2}{2}\cos(\lambda_i\tau).
\end{align}
 \end{Proposition}
 \begin{proof}
The relation in (\ref{eq:sum_1}) is a direct consequence of the properties of integrals of trigonometric functions and the fact that
 $$
 \lim_{T\mapsto \infty} \frac{\sin(T)}{T}=0
 $$
To prove (\ref{eq:sum_2}) notice that for  $\tau>0$,
 \begin{eqnarray*}
 \label{eq:proof_2}

  &\frac{1}{T}\int_0^{T} \tilde{X}(t)\tilde{X}(t+\tau)~dt=\\
  &\frac{1}{T}\sum_{i,j=1}^m \xi_{i,L}\xi_{j,L} \int_{0}^{T}\cos(\lambda_it+\phi_i)\cos(\lambda_{j}(t+\tau)+\phi_j)~dt=\\
  &\frac{1}{T}\sum_{i,j=1}^m \frac{\xi_{i,L}\xi_{j,L}}{2} \bigg(\int_{0}^{T}\cos\big((\lambda_{i}+\lambda_{j})t+\lambda_j\tau +\phi_i+\phi_j\big)~dt + \\
  &+\int_{0}^{T}\cos\big((\lambda_{i}-\lambda_{j})t -\lambda_j\tau +\phi_i-\phi_j\big)~dt\bigg)

 \end{eqnarray*}
 Notice that the first integral in the last equation  is of the form
 $$
 \int_0^T \cos(at+b)~dt=\frac{1}{a}(\sin(a'T+b')-\sin(a'')),
 $$
 for some constants $a,b, a',b'$ and $a''$, so that divided by $T$ as $T$ increases without bound goes to zero. The same is true for the second integral also, unless $i=j$, giving
 $$
\lim_{T\mapsto \infty}\frac{1}{T}\sum_{i=1}^m \frac{\xi_{i,L}^2}{2}  \int_0^T \cos(\lambda_i\tau)~dt = \sum_{i=1}^m\frac{\xi_{i,L}^2}{2}\cos(\lambda_i\tau).
 $$

 \end{proof}

  Concluding we present  the main result.
  \begin{Theorem}
  The  process ${X}$ defined  in (\ref{process_series}) that in a harmonizable form is also given in Proposition~\ref{prop:spectral} is not ergodic in autocorrelation.
  Moreover, almost surely we have
  \begin{align*}

    \lim_{T\mapsto \infty}\frac{1}{T}\int_0^{\infty} {X}(t)~dt&=0,\\

       \lim_{T\mapsto \infty}\frac{1}{T}\int_0^{\infty} {X}(t){X}(t+\tau)~dt&=\sum_{i=1}^\infty
       \frac{
       \Lambda^{-1}(\Gamma_i) R_i^2
       }{2}\cos(\lambda_i\tau).
\end{align*}
  \end{Theorem}
  \begin{proof}
  This result follows from Proposition~\ref{prop:time_averages}  and convergence of $X_L$  introduced at the end of Subsection~\ref{fsia} to $X$.
  Standard arguments for the change of the order of the two limits, one in $L$ and the other in $T$ are omitted here.
 \end{proof}

\section{Conclusions}
The important model of stationary stochastic signals arising from allowing for random frequencies in harmonics has been thoroughly studied by representing it as a harmonizable process with respect to non-Gaussian measure. It features  distributional properties decoupled from spectral properties, as the former are governed by the distribution of  amplitudes and the latter by the distribution of frequencies. One important property that is obtained from that approach is that the time averages do preserve randomness over the long time horizon. This can be viewed as important property in some applications. On the other hand it follows that if the ergodic property is required from the model for practical reasons, one has to resort to other models that feature arbitrary spectrum and non-Gaussian marginal distributions, such as non-Gaussian moving averages considered in \cite{bib:PodgorskiW}, \cite{bib:BaxevaniPW}. These models   also feature non-Gaussian tail distributions but they require some coupling (dependence) between different harmonics both in amplitudes and frequencies, see \cite{bib:BaxevaniP}.
Despite that the asymptotics of the time averages cannot retrieve such properties of the model as the moments, autocovariance, and sample distributions, the ergodic theorem yields the stochastic form of the limiting values that allows for the model estimation and studies of its sample to sample properties.

\bibliographystyle{apalike}

redpajama_set_name: RedPajamaArXiv


text:
Alfa Laval - New 1500 m3/h reactor for Alfa Laval PureBallast 3 will handle large ballast water flows with even greater efficiency
2018-09-05 Product news
New 1500 m3/h reactor for Alfa Laval PureBallast 3 will handle large ballast water flows with even greater efficiency
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Alfa Laval is a leading global provider of specialized products and engineering solutions based on its key technologies of heat transfer, separation and fluid handling.
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www.alfalaval.com
All Marine
Anders Lindmark
General Manager, Business Centre PureBallast
anders.lindmark@alfalaval.com
AL PureBallast Large Reactor PR September 2018.pdf 2019-04-19 258 kB
Large reactor.zip 2018-08-31 1610 kB
New Alfa Laval PureCool, developed with engine designer WinGD, enables up to 50% methane slip reduction
Turkish Shipowners Ballast Water Treatment Group chooses Alfa Laval PureBallast 3 for its vessels
Alfa Laval's groundbreaking HCO filter technology has been approved by MAN Diesel & Turbo
Alfa Laval's 100th anniversary year will close with new innovations at Marintec

redpajama_set_name: RedPajamaCommonCrawl


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text:
Diversidoris crocea is a species of colourful sea slug, a dorid nudibranch, a shell-less marine gastropod mollusk in the family Chromodorididae.

Distribution
This species is found in the tropical western Pacific, including: Indonesia, Philippines, Solomon Islands, Guam, and south to Queensland, Australia.

Description
This nudibranch is usually less than  in length. It has a yellow body with a pale mantle margin that has two semi-permanent mantle folds about mid-body. The rhinophores and branchia (gills) are also yellow. However, identifying individual yellow sea slugs within the genus Diversidoris can be challenging because yellow forms which mimic their food, yellow sponges, exist in many related species.

Ecology
This species is often found on the yellow sponge Darwinella, which appears to be its preferred food source.

References

External links

Chromodorididae
Gastropods described in 1986

redpajama_set_name: RedPajamaWikipedia


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Arctic Beringia
NEW YORK, NY | March 26, 2018
The Climate is Changing – New Report Says So Should Wildlife Conservation Strategies
A new report released today by WCS shows real world examples of how conservationists in the U.S. have successfully changed their conservation strategies to adapt to climate change.
Original Arctic Explorer George Schaller's Powerful Statement on Drilling in Arctic National Wildlife Refuge
December 19, 2017 -- The following statement is by WCS Senior Conservation Scientist George Schaller on allowing drilling in the Arctic National Wildlife Refuge, which is part of the tax bill expected to pass Congress and be signed into law this week. Schaller was part of the original scientific expedition in 1956 that led to the Refuge's creation:
WCS Urges Senate Committee to Reject Arctic Refuge Drilling Legislation
WASHINGTON (Nov. 13, 2017) – WCS today released a letter sent to the Senate Committee on Energy and Natural Resources, Chair Lisa Murkowski (R-AK) and Ranking Member Maria Cantwell (D-WA), urging a rejection of pending legislation to open portions of the Arctic National Wildlife Refuge to oil and gas development.
WCS: Senate's failure to remove Arctic Refuge drilling from budget endangers one of America's most pristine places
WASHINGTON (October 19, 2017) – Today, an amendment was offered to the Senate's budget resolution that would have stopped the effort to open the Arctic National Wildlife Refuge to oil and gas drilling as a means to fund other priorities, including tax cuts. That amendment failed, and the budget subsequently passed, signifying one step closer to irreparable damage to one of America's last remaining pristine places.
Congressional Efforts to Open up Arctic National Wildlife Refuge Would Despoil One of America's Most Pristine Places
WASHINGTON (September 29, 2017) – Allowing oil and gas drilling in the Arctic National Wildlife Refuge would be an unnecessary and harmful despoiling of a place of great natural value to wildlife, to the people who rely on this land, and for the American public – both now and in the future.
WCS: NPR-A Must Preserve the Current Balance of Wildlife Conservation and Energy Development
Fairbanks, Alaska (June 2, 2017) – WCS released a statement in response to the Department of the Interior's announcement that it will reopen a 2013 comprehensive management plan on the future of the National Petroleum Reserve-Alaska.
WCS Announces Request for Proposals for Nature-Based Climate Change Adaptation Projects
February 27, 2017 – Through its Climate Adaptation Fund, Wildlife Conservation Society (WCS) today solicited proposals from nonprofit conservation organizations to explore and implement new methods for helping wildlife adapt to rapidly-shifting environmental conditions brought about by climate change.
WCS Applauds New International Agreement to Protect Polar Regions
January 19, 2017 – WCS (Wildlife Conservation Society) commends the International Maritime Organization and partners for the development and implementation of the "Polar Code," a binding international agreement that will help ensure the safety of mariners operating in polar waters, and the protection of both the Arctic and Antarctic's marine environment from the risks of vessel activities.
WOLVERINES! WCS Arctic Team Collaring Elusive Predators on Alaska's North Slope
April 22, 2016 – A WCS-led research team, working in cooperation with the Alaska Department of Fish and Game has successfully placed satellite collars on two wolverines as part of a study to see how these elusive predators are faring on Alaska's North Slope.
Siberian Muskoxen Study Continues
Dr. Joel Berger of WCS and the University of Montana taking part in a landmark U.S.–Russian bilateral conservation expedition on Wrangel Island off the coast of northeastern Siberia Berger and his Russian colleagues braving wind-chills down to –70°F to gather information on the impacts of climate change on the Arctic's largest land mammal—the muskoxen(April 9, 2014) — Dr. Joel Berger, Senior Scientist with the Wildlife Conservation Society's North America Program and John J. Craighead Chair in ...
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Archil Sikharulidze
Working Experience
Category Archives: Foreign Policy
Azerbaijan, Commentary, Domestic Affairs, EU, Foreign Policy, Georgia, Global Politics, RIA Novosti, Russia, South Caucasus, Ukraine
"Yanukovich regreted about it". Eastern Partnership countries took stock
May 16, 2019 @achosikha	Leave a comment
Representatives of the so-called Eastern Partnership countries gathered in Brussels to celebrate 10th anniversary of the initiative. RIA Novosti journalist, Galia Ibragimova with support of local experts summarizes main achievements and failures.
Read the article in Russian here.
Eastern Partnershipeuropean unionforeign policygeorgiaGeorgia-EU relationsgeorgian-russian relationsrussiaUkraine
Domestic Affairs, Foreign Policy, Georgia, GeorgiaToday, Global Politics, NATO, Opinion, Russia, South Caucasus, Ukraine, US
NATO's "Small Talk"
March 28, 2019 @achosikha	2 Comments
On March 26, NATO Secretary-General Jens Stoltenberg arrived in Tbilisi and met high officials, including PM Giorgi Bakhtadze and President Salome Zurabishvili. During his "blitzkrieg" visit, Stoltenberg firmly re-iterated the Alliance's official standpoint once again: Georgia will become a member of the military bloc and Russia has no rights or power to oppose it. Yet, according to mainstream media outlets, this time Secretary-General added "very soon," wording that offered more concrete timing and raised hopes that the so-called MAP (Membership Action Plan) will finally be granted to the small Caucasus country.
Unarguable, such high-level visits are extremely important to maintaining Georgia's aspirations for NATO membership and, generally, to keep spirits high on the ground. But Tbilisi has been hearing such promises of commitment on a regular basis for the last decade, with visits from NATO officials having become something of an annual tradition. We can easily argue that Jens Stoltenberg's trip to Tbilisi was another example of NATO "small talk" without real outcomes for MAP issuance, but with more serious consequences for local society and its perception of the real ongoing processes in global politics.
NATO & Democracy
NATO has little to do with democracy. The Alliance is a military bloc based on ideology and with concrete geopolitical interests; in this case, democracy is just an additional advantage. These assumptions must be made clear when it comes issuing membership. It is obvious that, according to Western democratic standards, neither Georgia nor Ukraine are actually ready to satisfy such strong requirements; to say nothing of the 2008 period when Tbilisi and Kiev hoped to get MAP and had strong support from the US government. The promise given by the Bucharest Summit Declaration to open doors for these two states in the future was clearly the result of geopolitical calculations that dealt with strengthening the Eastern flank of the bloc and bringing NATO forces closer to the Russian border.
So far, using lack of democracy in Georgia as an explanatory variable for MAP refusal is an outdated trick to avoid naming the real reasons; seemingly, the only group of people who might still trust such arguments are regular citizens.
Russia, Rights & the Black Sea Region
Despite loud claims that Russia has no rights and no powers to oppose Georgia's integration, it actually has. Moscow's geopolitical interests stopped Tbilisi from getting MAP in 2008 and it is the only opinion that matters when it comes to the decision-making process. Putting aside political rhetoric, Russia does not need special rights to counter NATO's policy on the Eastern flank. Every single political actor by default has the right to protect its own political, economic and social standpoints. Thus, the Kremlin has the right to defend its national interests without being granted "permission" from the West or any other actor. What actually matters is whether Russia also has the economic and military strength to back up any protest. Even though Moscow is no match for Washington in general, the state currently possesses enough accumulated power to counter the West, especially near its borders and geopolitically important life-space. Furthermore, the Russian policy of containing NATO enlargement is perceived as crucial by the majority of society; this is not constructed by Putin's regime but a factual given. As such, the Kremlin will do its best not to let Tbilisi or Kiev be granted Alliance membership.
Whether officially noted or not, the highest and most influential political actors in the West are well-aware of Russia's NATO fears; additionally, Georgia's strategic partners understand that, historically, Moscow has always been keen to fight such threats by all means necessary. That is why they are not in a hurry to give a green light for the next stage of the military bloc enlargement. Even worse, NATO representatives are unsure whether bringing the organization to the Black Sea region will actually lead to more stability and prosperity or, by intervening in the so-called Russian "zone of interest," instead lead to a direct or indirect military clash that will destabilize the whole post-Soviet space. Today, the West, and Western European states in particular, are not ready to be involved in such dramatic processes.
Negative effects of the NATO "small talk"
While looking at NATO's annual "small talk" in Georgia, we need to raise another issue that deals with negative consequences. First, such harsh and provocative statements by high-rank officials create and maintain false expectations and perceptions of global political processes. Georgian society is being kept in a virtual reality where Russia allegedly has no rights and powers while Tbilisi's NATO membership issue has been stretched over a decade due to unrealistic claims of a "lack of democracy." Secondly, NATO has been transformed into a fixed idea, an obsession that clouds the judgement of local elites and regular citizens alike, while human resources are shifted from improving domestic challenges to the desperate effort to get that MAP. People have really started to believe that once in the military Alliance, democratic institutions will appear by default and the Russian threat will simply vanish. It's a bitter pill that at the end of the day may lead to dramatic nihilism and even to a foreign policy shift, as happened with Turkey and its European integration dreams. Ad notam, that is why all main elitist INGOs and NGOs in Georgia tried hard to put the issue of foreign policy orientation into the new constitution. This is the third possible negative outcome of a short-sighted NATO policy.
By and large, NATO Secretary-General Jens Stoltenberg's visit to Tbilisi is important for the country to feel the Alliance's support. But it could have been even more positive if the organization's high-officials had held back from giving false promises and "small talk" about granting MAP to Georgia; especially using the argument that Russia has nothing to do with NATO enlargement. These assumptions mislead local elites as well as society. Georgia has achieved a respectable enough level of democracy to appeal and to start slowly becoming an inherent part of the military bloc. But this aspiration will not be satisfied unless the Kremlin stops opposing it. Moscow does not need permission to defend its national interests; moreover, it has enough strength to do so. At the same time, NATO's annual "small talk" has significant negative outcomes for Georgia. People are being kept in a bubble of falsehood where the basic laws of global politics are abandoned; secondly, the issue itself has become so politicized that it clouds the judgement of decision-makers on the ground who put more effort into becoming a NATO member than into building democratic institutions. And finally, the absence of real outcomes of Georgian-NATO cooperation may push for more nihilism and even to a foreign policy shift.
The article was originally published by GeorgiaToday. It is available here.
georgiaMAPNATOrussiaSecuritysouth caucasusUkraineUSA
Domestic Affairs, Foreign Policy, Georgia, GeorgiaToday, Global Politics, Opinion, Russia, South Caucasus, US
Georgia's North Macedonian Dream
On March 12, the North Atlantic Treaty Organization celebrated the 20th anniversary of the membership of the Czech Republic, Poland and Hungary. Prominent American politician and former Secretary of State Madeleine Albright re-iterated NATO's open-door policy and assured Georgia and Ukraine that they will become Alliance members once the two countries are ready. Earlier, on June 12, 2018, Greece and the Republic of Macedonia signed the so-called Prespa Agreement according to which the latter will change its name to the Republic of North Macedonia while Greece will finally agree to support its long-standing aspirations of acquiring the NATO Membership Action Plan (MAP). Currently, the Republic of North Macedonia is on its way to joining the military bloc. The latest happenings raised new hopes among the so-called pro-NATO groups in Georgia that their state is next on the list.
Meanwhile, there are serious doubts as to whether the offering of NATO membership to the Republic of North Macedonia is a real precedent for Georgia and Ukraine. These Eastern European states face far more serious challenges attached to global politics, geopolitics and confrontation between the West and the East. At present, it is unlikely the North Macedonian experience can be readily applied.
Names Matter
The name dispute is a long-standing confrontation with old roots. Macedonia is a historical region associated with Greece and Alexander the Great. Nowadays, ancient Macedonia approximately corresponds with the modern Greek region of Macedonia and has little to do with the Republic of North Macedonia. Athens blamed Skopje of trying to assign symbols and figures that were initially perceived as part of Greek culture. Furthermore, there were fears of irredentism. Millions of Greeks identify themselves as Macedonians, and officials in Athens had concerns this might lead to some concepts of so-called United Macedonia threatening the territorial integrity of the state.
So far, the case of the Republic of North Macedonia is unique and inapplicable to Georgia and Ukraine due to its comparative simplicity. The only actor that opposed Macedonia joining NATO was Alliance member Greece. No global politics was involved. Furthermore, there was only one demand that Athens had for Skopje – to change its constitutional name. Generally speaking, the whole dispute surrounded the name of the state and the main challenge was to persuade local elites to negotiate and make a deal using a "win-win" approach. Former confronting sides belonged to so-called small actors that are more vulnerable to external pressure and influence.
The Georgia Case
Whether you perceive NATO as a military bloc, democracy promoter, hybrid organization or something else, it is obvious that modern Ukraine is not ready to join the club. This lack of readiness can be seen on every level of the state build-up. On the opposite side, there is Georgia, which has by all means managed to satisfy NATO requirements; Tbilisi does not need to be ready for Alliance membership because it already is. Thus, we need to abandon this frequently-used-by-various-Western-high officials argument and be clear about the real challenges, recognizing that they are much more complicated than Athens and Skopje.
Perhaps the biggest need is to acknowledge separatism and irredentism as actually-happened facts. Georgia has two separatist regions supported by its northern neighbor, the Russian Federation. Additionally, at least some actors in Tbilisi realize that, besides the Russian factor, there are also significant issues to deal with on the ground, especially when it comes to conflict in the Abkhazia region. An even more complicated situation is seen in Ukraine, where the Crimean Peninsula was integrated by the Kremlin into the state while self-proclaimed Donbas and Luhansk on the east receive military and financial support.
We also need to consider Moscow and its geopolitical agenda. Unlike Greece and Macedonia, Russia is a "great power". Of course, it is no match for the USSR, US and/or China, but it still possesses enough accumulated military power, backed by a vast territory and natural resources, to promote national interests and counter any other global actor. The Kremlin perceives both Georgia and Ukraine as zones of influence, "living space" and "security belt" and so the integration of these two states into NATO is a direct threat to its fundamental interests. And there is no point presenting the Alliance as a Russia-friendly military bloc: the Kremlin will perceive it as a threat while NATO prospers and Russia is not a member.
Unfortunately for Tbilisi and Kiev, their separatist regions have become battlefields between the united West and the East, meaning Russia, China and all those actors that aspire to shake the existing global political order, with the US on top. This fact makes it even more complicated to solve disputes on the ground due to their transformation from local (especially in Georgia's case) to global: global actors are involved, and their goals go far beyond simple conflict resolution.
The reasons outlined above are real challenges that Georgia and Ukraine experience on their way to NATO membership. And while on the ground, Alliance representatives may speak about an open-door policy, but factually these "doors" will be opened only verbally.
By and large, the so-called Macedonian precedent has nothing to do with NATO aspirations for Tbilisi and Kiev. The name dispute between Greece and the Republic of Macedonia was comparatively simple, while Georgia and Ukraine face extremely serious challenges, including separatism, irredentism, the Kremlin's geopolitical agenda and, of course, war for the future of global politics.
Thus, Georgia will need to wait a while on the list. But it is highly important for these two countries to continue state building, whether with MAP or not; officials must finally understand and accept that the ongoing build-up is not for NATO but for the prosperity of the people on the ground.
The article was initially published by GeorgiaToday. It is available here.
georgiagreecemacedoniaNATONATO enlargementrussian federation
Armenia, Commentary, Foreign Policy, Georgia, South Caucasus, Sputnik-Georgia
"Together we can be better" – messages that Zurabishvili brought to Armenia
March 14, 2019 @achosikha	Leave a comment
President of Georgia, Salome Zurabishvili has finished her official visit to Armenia. She brought some important messages to Yerevan. Georgian political analyst Archil Sikharulidze argues that Georgia has no real South Caucasian foreign policy.
Read more in Russian here.
armeniageorgiaGeorgian-armenian relationsinternational relationssalome zurabishvilisouth caucasus
Azerbaijan, Commentary, Domestic Affairs, Foreign Policy, Georgia, Media.Az, Russia, South Caucasus
Archil Sikharulidze: Azerbaijan is a guarantor of Georgia's energy security
February 28, 2019 @achosikha	Leave a comment
President of Georgia, Salome Zurabishvili has visited Baku with official meetings. Georgian political analyst Archil Sikharulidze argues that key issue for discussion will be to keep Azerbaijan as a guarantor of Georgia's energy security.
azerbaijanenergy securityforeign policygeorgiaGeorgian-Azerbaijanian relationssalome zurabishviliSecuritysouth caucasus
Azerbaijan, Commentary, Foreign Policy, Georgia, South Caucasus, Trends News Media
Archil Sikharulidze: Visit of Georgian president to Baku will further strengthen interstate relations
President of Georgia, Salome Zurabishvili has visited Baku, Republic of Azerbaijan. She had meetings with local high officials. Georgian political analyst Archil Sikharulidze argues that this visit will further strengthen interstate relations.
azerbaijanforeign policygeorgiaGeorgian-Azerbaijanian relationsRussiansalome zurabishviliSecuritysouth caucasus
Abkhazia, Domestic Affairs, EU, Eurasia, Foreign Policy, Georgia, Global Politics, Opinion, Religion, Russia, Ukraine
God Engaged: Orthodox Christianity in the "Big Game"
January 17, 2019 @achosikha	Leave a comment
On January 5, Ecumenical Patriarch Bartholomew, the head of the global Orthodox Church and the first among equals, signed a decree "Tomos" in Istanbul granting the Ukrainian Church autonomy from the Russian Orthodox Church (alternatively, the Moscow Patriarchate, abb. ROC) and recognizing its autocephaly (independence). The event was attended by the head of the Ukrainian government, President Petro Poroshenko and other high officials. Earlier, Bartholomew called on leaders of other autocephalous churches, most importantly in Serbia, Bulgaria, Romania and Georgia, to support his initiative. The decision to remove the Ukrainian Church from the subordination of the ROC which had existed as such for centuries was highly criticized by Patriarch Kirill of Moscow, the head of the Moscow Patriarchate, as a rude violation of religious laws and an obvious political step leading to religious strife and a division in the global Orthodox community.
And while supporters of Bartholomew's historic move both in Ukraine and abroad, especially in Georgia, react according to how they perceive the victory over the corrupt Russian Orthodox Church controlled by Putin's regime, it is obvious that there is much more than just the right of Ukrainian people for religious independence at stake here. The Orthodox Christian god has been officially engaged by all sides in the so-called "big game" now: political and ideological rivalry where religion is used for concrete goals. The absolute majority of the so-called pro-Western and pro-Ukrainian analysts and experts fiercely defend Bartholomew's step as fair and logical, bringing forth arguments about historical justice and more. At the same time, there are those who are not so positive about it. This pessimism comes from a few extremely puzzling moments that are mainly, frequently intentionally, overlooked and neglected.
Orthodox Christianity and Politics
Unlike other Christian denominations, Orthodox Christianity was and still is extremely involved in domestic and foreign politics. Political actors actively attract religious institutions into various state, political and ideological activities to gain electoral support, legitimacy and to justify their activities, while local churches are keen to use this "dependency" to influence political decision-making and, sometimes, even pursue a harsh policy of proselytism. This mutual interdependence and interconnection make it impossible to build a stable secular state without interference from religious institutions and persons in state affairs.
And so, here is perhaps the biggest question that puzzles experts and analysts: was Bartholomew's initiative purely a religious one? It is pretty questionable that Ecumenical Patriarch would have dared and/or decided to push the topic of an autocephalous Ukrainian Church in the midst of the fight between Kiev and Moscow and, in general, the West and the East, without any serious political back-up; especially if we take into consideration that Bartholomew, in theory, holds sway over more than 300 million Orthodox Christians around the world, from which the majority are Russian and when the biggest Orthodox Christian Church is the Moscow Patriarchate.
The second important question regards the strong bond between political elites and local religious institutions that have a place in almost all Orthodox Christian countries. We can easily outline at least three exceptional cases: Russian Federation, Ukraine and Georgia.
The Moscow Patriarchate seems to be a continuation of the Russian state (ad notam, historically given condition) and is being actively used by the Kremlin to pursue its domestic and foreign policies through support and justification from the religious institution and its representatives. At the same time, the State acknowledges the Moscow Patriarchate's support and backs its frequently questionable domestic (the law on protecting religious feelings) and foreign (dominance over global Orthodox Christian community) initiatives. The same could be seen in Ukraine. Until it received "Tomos," there were three Orthodox Church branches: the Ukrainian Orthodox Church or Kiev Patriarchate (independent), the Ukrainian Autocephalous Orthodox Church (independent) and the Ukrainian Orthodox Church (abb. UOC, subordinated to the Moscow Patriarchate). After accepting autocephaly, the first two independent churches formed the Orthodox Church of Ukraine (abb. OCU) that, by initial design and primary mission, was to counter and oppose the UOC that was and still is the leading Orthodox institution in the country. So far, the OCU, a priori, is being considered by all sides and, most importantly, by Poroshenko's government, as a political tool against the Kremlin; and we may argue that the Ukrainian state will do its best to put the newly formed local autocephalous church at the service of political elites. It is also highly questionable whether the OCU can really counter the UOC without political, ideological, administrative and financial support from the government and affiliated individuals.
Where the Moscow Patriarchate and the Orthodox Church of Ukraine are being used by political elites, the Georgian Orthodox Church (abb. GOC) is intelligently manipulating the Georgian elites; where in the Russian Federation and Ukraine we are witnessing the existence of so-called "pocket churches" that lie as a huge burden both on governments and societies, Georgian society is easily "pocketed" by the Georgian Orthodox Church. As still the most trusted and influential institution in the country, the GOC interferes in almost every single aspect of everyday life in Georgia. This frequently harmful practice restrains the country from transforming into the Western-type secular state it claims it wants to be.
Orthodox Christianity at a Crossroads
The establishment of the Ukrainian autocephalous Orthodox Church not only officially involved Orthodox Christian god in global politics, but also led to historic strife; and if decision of the Moscow Patriarchate to cut ties with the Ecumenical Patriarchate of Constantinople was somehow a logical retaliation, other autocephalous churches suddenly appeared to be at a crossroads, with only two options – follow Bartholomew or support the ROC. We may argue that no one would want to be involved into this dispute; furthermore, for some autocephalous churches, such as those of Serbia and Georgia, this decision is highly sensitive and is directly attached to important political issues.
It is no secret that Serbia holds strong political, cultural and religious ties with the Russian Federation. Furthermore, Serbian officials openly re-iterate the importance of these ties and are still extremely thankful to the Kremlin for the support given during the breakup of Yugoslavia. Additionally, Moscow was in the vanguard of those states that did not support the separation of Kosovo from Serbia, and Russia is still the main and most important ally of Belgrade in its attempts to bring the partially recognized state under its jurisdiction. It seems the Serbian Orthodox Church will definitely think twice before backing Bartholomew's decision and by default directly oppose the Moscow Patriarchate.
This situation is even more challenging for the Georgian Orthodox Church. Despite fierce support for the Orthodox Church of Ukraine from local pro-Western forces and parts of Georgian society, which perceive these events as the restoration of historical justice, it is obvious the situation is much more complicated. Georgia has two separatist regions (Abkhazia and the so-called South Ossetia) which were recognized as independent states by the Russian Federation, Venezuela, Nicaragua, Nauru and Syria. Despite supporting Abkhazian "sovereignty," the Russian Orthodox Church does not officially recognize the Abkhazian Orthodox Church (abb. AOC) and it is, de jure, still subordinated to the Georgian Orthodox Church. Representatives of the GOC are afraid that by recognizing the establishment of the Ukrainian autocephalous Church, they may directly push the Moscow Patriarchate to recognize the AOC. Some may argue that the ROC has de facto already done so and representatives of the Russian Orthodox Church are working secretly on the ground, and yet there is a huge difference between official recognition and masked activities. Finally, Georgia has been surrounded by non-Christian nations for centuries with the Moscow Patriarchate its only neighboring religious ally; thus, this historical memory is also present.
Useless Independence?
Without doubt, the Ukrainian people deserve the right to have religious institutions that reflect their attitudes and aspirations. Nevertheless, it is questionable whether the reception of autocephaly was and is something that Ukraine really needed. Will it solve the country's challenges and lead to the unification of society? Unlikely.
As said above, the Poroshenko's government sees autocephaly as a pre-election campaign project that should, in theory, raise support among the local electorate and help its leader to keep his presidential post. This is despite the unpleasant fact that Poroshenko failed to lead the state into a better future. Particularly, we may argue that Ukraine was always and still is in dramatic need of independence from its own oligarchs and other questionable figures rather than autocephaly from the Moscow Patriarchate. The reasoning that the inability of every single pro-Western revolutionary movement and leader to handle the corruption and other anti-state practices lies in the dependence of the Ukrainian Church from the Moscow Patriarchate is highly dubious. The reception of autocephaly will not lead to a dismantling of the corrupt political system that exists in Ukraine, as that is just how Poroshenko wants it to be.
Secondly, frequently both international and domestic observers overlook the issue of social divisions in Ukraine. Approximately half of the eastern population are ethnic Russians, and these will most likely not transfer from the Moscow Patriarchate to the Ukrainian Orthodox Church, especially taking when taking into consideration the political context. Furthermore, even among Ukrainians there are plenty who are devoted to the Ukrainian Orthodox Church subordinated to the ROC. Thus, the establishment of an autocephalous church may further deepen the social divisions that have developed due to various questionable decisions made over time by the political elite, such as the controversial language bill restricting and downgrading use of the Russian language in Ukraine.
In conclusion, the establishment of the Ukrainian autocephalous Church can be perceived as an official engagement of the Orthodox Christian god in global politics, where all involved actors try to use religious institutions and affiliated individuals for their own political and ideological interests while local churches try to grab their own benefits. Other autocephalous churches, especially in Serbia and Georgia, now find themselves at a crossroads that may lead to some extremely interesting political developments in the future. If the Georgian Orthodox Church backs Bartholomew's initiative, we may witness the appearance of another partially recognized Church, the Abkhazian, which will result in the further deterioration of Georgian-Russian relations. At the same time, the Ukrainian autocephalous Church may be not the answer that the majority of Ukrainians hoped for, as it is uncertain how it will help the country to finally escape its corrupt circle or build the modern prosperous and secular state that they seek.
domestic affairsforeign policygeorgiageorgian orthodox churchKyiv Patriarchateorthodox christianityRussian orthodox churchRussian patriarchateUkraine
Azerbaijan, Commentary, Foreign Policy, Georgia, Global Politics, South Caucasus, Trends News Media, Turkey
Archil Sikharulidze on Military Agreement between Turkey, Azerbaijan and Georgia
November 21, 2018 @achosikha	Leave a comment
Turkey, Azerbaijan and Georgia signed military agreement that will strengthen cooperation between miitary institutions of these countries and, so far, ensure more safety in the South Caucasus region.
For more read in Russian here. The original comment was published by Trend News Agency and is available here.
azerbaijangeorgiamilitary agreementregional securitysouth caucasusturkey
Debate, Democracy, Domestic Affairs, EU, Eurasia, Foreign Policy, Georgia, Global Politics, Russia, South Caucasus, The New Past, US
"The Decline of Europe" a Hundred Years Later (Debate), Journal «The New Past», №2, 2018
July 1, 2018 @achosikha	Leave a comment
Questions for Discussion are following:
Is there a "decline of Europe" as a conceptual model today? What are its characteristics?
Can we talk about the specifics in the perception of the "decline of Europe" from the inside and outside? And if so, what are the differences in conceptual approaches to this problem?
Can we assess the migration crisis, regionalism, identity crisis and solidarity in Europe as an "evidence of the decline of Europe"?
Are the integration projects of the Eurasian Union, the New Silk Road (One Belt — One Road) a symptom of the extinction of European civilization in the post-war format familiar to us from the 1940s?
Did Oswald Spengler's forecasts come true?
Disputants:
Archil Sikharulidze, co-founder, Center of System Political Research (CSPR), editor of the Georgian Journal of System Politics (GJSP), Georgia;
Thomas Kruessmann, Prof. Dr. Dr. h.c., LL.M. (King's College), School of Politics and International Relations of University of Kent (Great Britain), co-ordinator of the Jean Monnet Network "Developing European Studies in the Caucasus" with the University of Tartu / Johan Skytte Institute of Political Studies and Senior Expert in the EU Technical Assistance Project "Strengthening Teaching and Research Capacity at ADA University" in Baku (Azerbaijan);
Rustem Davletgildeev, Doctor of Law, Academic coordinator of the Jean Monnet Centre of Excellence in European Studies VOICES+, Vice-dean of the Faculty of Law on International Activity, Kazan Federal University, Russia;
Bryan Christiansen, Chief Executive Officer, Global Research Society, LLC, USA;
Goran Ilik, PhD, Faculty of Law, "St. Kliment Ohridski" University — Bitola, Republic of Macedonia;
Aijarkyn Kojobekova, PhD in Philosophy, Associate professor, Senior Adviser on Research, UNUM, Training, Research and Analytics Company, Kyrgyzstan.
«Закат Европы» Сто Лет Спустя/"The Decline of Europe" a Hundred Years Later
eugeopoliticsgeorgiamigration crisisone beltsilk roadsolidarity crisisthe decline of europe
Abkhazia, Domestic Affairs, Foreign Policy, Georgia, Global Politics, MediaMetrics Radio, Opinion, RIAC, Russia, South Caucasus, South Ossetia
Conflict Management in Georgia: Georgian vs. Russian Narratives
On August 13, 2017 a prominent Russian scientist on post-Soviet issues, Sergey Markedonov participated in a radio show, Geopolitical Cuisine, with Igor Shatrov aired by MediaMetrics Radio, dealing with the August 2008 War. During the broadcast both, Shatrov and Markedonov raised some very interesting and significant issues that were presented in scopes of Russian political agenda. We will try to balance the analysis and offer a Georgian narrative.
Three topics can be identified: a) a debate on the possible transformation of Georgia's political system into a federal or confederal model; b) allegations of treaty violations; and c) assumptions of unchanged Georgian society and unanimity regarding the August 2008 War in general.
Federalism and Confederation
Since the beginning of the conflicts, Russian diplomats actively promoted an idea of 'Georgian Federation' or a possible confederation that could have included Georgia, Abkhazia and the so-called South Ossetia. This approach was especially prevalent in the late 90s and early 2000. Even nowadays, Russian experts and scholars argue that Tbilisi should have chosen these options to deal with separatism. during the radio broadcast, Igor Shatrov argued that Georgia was historically a multi-national union and its government should have reflected this background in the country's political architecture. Additionally, some comparisons were made between Russia and Georgia.
While these arguments may look attractive they do not show the whole picture. The topic of possible formation of federal or confederal union has two challenges.
The first of these challenges are the associated, security issues. As prominent Georgian expert, Mamuka Areshidze argues, Georgia might think about the confederal model but it is crucial to ensure that respective regulations are set in place to avoid unrecognized regions getting away from the metropolis with international approval. Moreover, there is a strong belief among various local actors that Kremlin's final aspiration is to dominate over Georgia as a whole rather than its secessionist regions. So far, the federal or confederal models can be used by Moscow to undermine their neighbor's sovereignty and influence Tbilisi via a pro-Russian agenda in Sukhumi and Tskhinvali.
Secondly, Georgia was never a multi-national kingdom or state as, for example, Canada or even Russia itself was and still is. So far, a comparison between Russia's federal political model and Tbilisi is probably a stretch. Undoubtedly, the state has a history of cohabitation of various ethnical groups. Even though Georgia is a home for approximately ten ethnic groups, they represent only 14% of the whole population. Moreover, while the case of Abkhazia is complicated and Mikhail Saakashvili, former president of Georgia (2004–2013), even offered to recognize Abkhazian as a second state language, the situation in so called South Ossetia is another case. Ossetians that are residents of Tskhinvali region settled there only in XIX–XX centuries and they arguably never reflected a multi-national nature of Georgia. Lastly, the comparison between Moscow and Tbilisi is a nonsensical due simply to the historical background.
Treaty Violations
Representatives of Georgia's separatist regions and the Kremlin frequently blame Tbilisi for attempts to undermine the so-called Sochi Agreement, a ceasefire agreement that marked the end of Georgian-Abkhazian and Georgian-Ossetian conflicts. This document was signed in Sochi on June 24, 1992 between Georgia and Russia, the ceasefire with Abkhazia was marked on July 27, 1993.
And while this is factually accurate, we should bear in mind that political realities have changed and they must be reflected; the principle that Kremlin is promoting globally. For example, Vladimir Putin frequently argues that the current political order does not represent the redistribution of powers in global politics. Particularly, it is outdated due to the fact that America is still seen as an undisputed political force while the highly increasing role of such countries as Russian Federation, China, India and Brazil is simply overseen. Furthermore, Moscow has already re-negotiated more or less every significant document that was signed during Boris Yeltsin's governance when the country was politically and economically non-competitive. The same story is applicable to Georgia. The Sochi Agreement was negotiated and signed when Tbilisi was unable not only to handle the separatist regions but even such simple and fundamental tasks such as budget management. However, since then, Tbilisi has made a giant step toward political, economic and social stability and prosperity. Hence, it is logical and is totally suitable to the framework offered by Moscow that Georgia was and is determined to re-settle some aspects of the agreement and balance it according to national interests.
All in all, Russian scholars and experts should decide whether their 'reality-match' approach to legal documents is universal, generalizable, or unique, meaning that it is only applicable to big countries, and small states must get used to doing what they can.
2008 August War
One more issue was raised during the broadcasting. Igor Shatrov argued that after the events of the 90s, the residents of the so-called South Ossetia thought that Georgia had changed as a result of democratic reforms. But, the August 2008 War led to a dramatic disillusionment when they realized that Georgian society remained the same.
This assumption is most likely inaccurate due to two facts: a) Georgian society tends to openly recognize its misdeeds; and b) the latest military confrontation was probably more avoidable than that of the 90s.
Of course, Georgia is far away from the Western style democracy but local society is slowly changing and clearly moving forward. As opposite to residents of the separatist regions Georgian society came to a state of condition in which it is capable of re-assessing political decisions and recognizing its share of blame in the conflicts. It has just realized that in order to clap you need two hands; a harsh reality that people in Sukhumi and Tskhinvali are unwilling to take into account.
Secondly, the August 2008 War was probably more predictable and avoidable than those of the 90s. It is rarely mentioned that after political unrests in November 2007 and a questionable outcome of early presidential elections of January 2008, Georgian opposition openly warned about possible military campaign that Saakashvili's government could have conducted to consolidate political power and increase electoral support. The same statement was made by former ambassador of Georgia to Russian Federation, Erosi Kitsmarishvili. in his memoirs, he stated that Saakashvili had aspiration to solve the Abkhazian issue with a military blitzkrieg (something, that Saakashvili and members of his government called a complete lie). Unfortunately, Matthew Bryza, former Deputy Assistant Secretary of State for European and Eurasian Affairs, assured the George W. Bush administration that these allegations were unfounded. Consequently, the state's strategic partners that were capable of restraining Saakashvili and making some serious preventive steps were surprised when military confrontation unfolded (it should be mentioned that some Georgian and Russian experts argue that Bush's administration was aware of Saakashvili's plans and even supported him). At this point, there was no unanimity between the government and Georgian society that, we may argue, was in place during 90s.
By and large, Russian experts and scholars frequently blame the West for not taking into account Russia's national interests. They argue that both, US and EU, must realize that Kremlin has its own political agenda and that Vladimir Putin's government will follow and protect it by all means. Unfortunately, the same people rarely apply the same standards to Georgia and its national interests. And as soon as this is done, Georgian and Russian officials will undoubtedly find a common ground for mutually beneficial political agreements.
The article/blog was initially published by RIAC. It is available here.
'reality-match' approach2008 August Warabkhaziaarchil sikharulidzeAugust 2008 WarFederalism and Confederationgeorgian-russian relationsIgor ShatrovMediaMetrics RadioriacSergeuy MarkedonovSochi Agreementsouth ossetiasukhumitskhinvali
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