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\begin{titlepage}
 \begin{center}



  \vspace*{3cm}
 {\bf \huge Seminar Bewertungsmethoden in der Personenversicherungsmathematik}
 \vspace*{1cm} \\
 {\Large spezielles Thema }
 \vspace*{2cm} \\
 {\large Name}
 \vspace*{4cm} \\






 Mathematisches Institut \\
 der \\
Universität zu Köln
 \vspace*{1cm} \\
 Sommersemester 2010\\
 Betreuung: Prof. Schmidli, J. Eisenberg
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\pagenumbering{roman} % Seitennummerierung durch roemische Ziffern
\tableofcontents % Inhaltsverzeichnis
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\section{Zentraler Grenzwertsatz} 

% Abschnitt

\bf{Dies soll lediglich ein Beispiel fürs Texen sein und steht daher inhaltlich nicht mit dem seminar in Verbindung. Das Literaturverzeichnis ist allerdings schon für das spezielle Seminar angefangen.}

%Satz
\begin{satz}[Zentraler Grenzwertsatz]
 $X_n:(\Omega,{\cal A},\mathbb{P}) \to \mathbb{R}$ seien i.i.d. Zufallsvariablen mit
 $0<Var(X_1)<\infty$. Dann gilt:
 \begin{equation} \label{grenzwert}
    \frac{\sum_{i=1}^n (X_i-E(X_i))}{\sqrt{n Var(X_1)}} \to N(0,1)
 \end{equation}
 
%Für Formeln ohne Nummern benutze \begin{equation*} \end{equation*}.
in Verteilung. 
\\\eqref{grenzwert} heißt standardisierte Summenvariable und es gilt $E()=0$ und $Var()=1$.
\end{satz}
\begin{proof}
.....
\end{proof}

%Bemerkung
\begin{bem}
 Gilt $X_n \sim N(a,\sigma^2)$, so folgt, dass die standardisierte Summenvariable in \eqref{grenzwert}
 $N(0,1)$-verteilt ist.
\end{bem}


\newpage
\section{Der zentrale Grenzwertsatz von \\Lindeberg-Feller} 

% Abschnitt

Im folgenden bezeichne $(X_{n,j})_{j \leq n}$ ein Dreiecksschema von zeilenweise unabhängigen
Zufallsvariablen. Es gelte:
\begin{eqnarray}
 && E(X_{n,j})=0 \forall j,n; \\
 && \mbox{Setze } \sigma_{nj}^2:=E(X_{n,j}^2); s_n^2:=\sum_{i=1}^n \sigma_{n,j}^2; \\
 && \mbox{Lindeberg-Bedingung: } \forall \varepsilon >0 \\
 && L_n(\varepsilon):=\frac{1}{s_n^2}\sum_{j=1}^n \int_{\{|X_{n,j}\geq \varepsilon s_n\}} X_{n,j}^2 \ dP \to 0 \nonumber \\
 && S_n:=\frac{1}{s_n} \sum_{j=1}^n X_{n,j}.
\end{eqnarray}

%Satz
\begin{satz}[ZGS von Lindeberg-Feller]
 Unter den obigen Voraussetzungen gilt
 \begin{equation}
    S_n \to N(0,1)
 \end{equation}
in Verteilung.
\end{satz}
%
\begin{proof}

\end{proof}

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\bibliographystyle{plain}
\begin{thebibliography}{99}
\addcontentsline{toc}{section}{Literaturverzeichnis}
\footnotesize

\bibitem{MS} M\o ller, T. und Steffensen, M. (2007). \textit{Market-Valuation Methods in Life and Pension Insurance}. Cambridge University Press, New York.
										 
\end{thebibliography}

\end{document}