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\title{ \textbf{ Un titlu remarcabil de unic}}
\author{ Frentescu Stefan}
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\abstract
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\section*{Section 1}
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\section*{Section 2}
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\section*{Section 3}
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\hspace{-5.3mm}\textbf{DOUBLE EXPONENTIAL \\
DISTRIBUTIONS}

\hspace{5mm}Let be $\mathrm{a_t}$ be a serios of independent identically \hspace{5mm} double-exponentially \hspace{5mm} (Laplace) distributed random variables, i.e. with probability density function(\textbf{PDF}) given by $$\displaystyle{ f(a) = \frac{\lambda}{2}e^{-\lambda |a|} , \lambda > 0 \hspace{5mm} \forall a}$$

Let be the observed stationary time series $ \mathrm{\{X_t\}} $ be generated by the \textbf{ARMA} scheme
$$\Phi ( B ) X_t = \Theta ( B ) a_t $$ where $$ \Theta ( B ) = (1 - \phi_{_1} B - ... - \phi_q B^q ) $$
$$ \Phi ( B )  = ( 1 - \phi_{_1} B - \phi_{_2} B^2 - ... - \phi_p B^p ) $$ and B is the backward shift operator so that $$\mathrm{ B^k X_t = X_{t-k}.} $$

\hspace{5mm} Since the series $\mathrm{\{X_t\}} $ is assumed to be stationary, all the roots of \hspace{2mm} lie outside the unit circle, and we can write the moving average
$$ X_{_t} = \Phi^{-1} ( B )  \Theta ( B ) a_t = \Psi (B ) \sum\nolimits_{j = 0}^{m} \mathit{\Psi_j a_{t-j}} $$
where the coefficients \boldsymbol{$ \Psi ( B )  = 1 + \psi_1 B + ...    $} can be found by equating coefficients in
$$ \Phi( B )  \Psi ( B ) = \Theta ( B ). $$
Let
\begin{equation} Z_{_n} = \sum_{j = 0}^n \mathit{\Psi_{_j} a_{_{t-j}}} \end{equation}
and assume $\scriptstyle{\Psi_y \neq \Psi_j} $ for i $\neq$ j.

\hspace{5mm} The \textbf{PDF} of $ \mathrm{Z_n} $ is given by
\large{$$ \displaystyle{ f_n (z) = \frac{\lambda}{2} \sum\nolimits_{j=0}^{n} \alpha_{_j}^{ ( n )} \left| \mathit{\Psi}_{_j} \right|^{-1}  \mathrm{exp} ( - \lambda \left| \frac{z}{\mathit{\Psi}_{_j}} \right|)}$$} \normalsize{\hspace{-1.5mm}where $\mathrm{\alpha_{j}^{(n)} }$ are functions of \hspace{1mm} $\mathrm{\scriptstyle{\{\Psi_i\}}}$
the given by
} $$ \alpha_{_j}^{(n)} = \prod_{i=0,i \neq j} (1 - \left|\mathit{\frac{\Psi_{_i}}{\Psi_{_j}}} \right| )^{-1}, \hspace{2mm} \mathit{j} = 1,2,...,n $$

\hspace{5mm}Now
\begin{equation} \displaystyle\ f(x) = \lim_{_{n \rightarrow \infty}} f_n(x)  $$
and obtain the following expression for the marginal \textbf{PDF} of $\mathrm{X_t}$
$$ f(x) = \frac{\lambda}{2} \sum\nolimits_{j=0}^{\infty} \alpha_{_j} \left| \mathit{\Psi_{_j}} \right|^{-1 } \mathrm{exp} (-\lambda \left| \mathit{\frac{x}{\Psi_{_j}}} \right| )   \
\end{equation}
and $$ \alpha_{_j} = \prod_{_{i=0,i \neq j}}^{\infty} {(1 - \left| \mathit{\frac{\Psi_{_i}}{\Psi_{_j}}} \right| ) }^{-1} $$

\hspace{5mm} These results follows as special case of \textit{Box's} [3], where he derives the distribution of any linear combination of independent $ \chi^2 $ variables with even degree of freedom by nothing that each $\mathrm{a_i}$ may be written as a constant times as difference \hspace{3mm} between \hspace{3mm} two independent \hspace{3mm} $\chi^2$ variables. \textit{Preda} [8] generalized the above fact to mixed double-exponentially.

\subsection*{\center{1. STATISTICAL MODEL}}
\subsubsection*{\center{Autoregressive model}}

\hspace{16.4mm}Let be the time series Z and Y be represented by autoregressive models of order p
$$ Z_t = \mu + \sum\nolimits_{i=1}^p \phi $$

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