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%%    15th Workshop on Stochastic Models and Their Applications
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Bernoulli University, Germany}
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\def\absTitle{Limit Theorems and their Applications}
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\begin{abstract}
We review some basic limit theorems to analyze the stochastic behavior of random phenomena in engineering and finance. Our key application is monitoring of data to detect changes. Historically, this problem has been studied for industrial production processes. However, streams of data which may be affected by changes appear in many disciplines, e.g., finance.
\end{abstract}


\section{Introduction}

Please provide your (extended) abstract using this template. You may use sections, include figures, provide references, etc. However, please respect the following rules.
\begin{compactitem}
  \item Don't modify the preamble inside {\tt $\backslash$ifabstractstandalone}. It will be  ignored when processing your file.
  \item Avoid own macro definitions to minimize conflicts when assembling files.
  \item Please provide citations, labels, references, etc., as illustrated below. \textbf{Do not use other macros like cref} as
  we redefine the standard macros with unique prefixes
\end{compactitem}
Submit your final \LaTeX{} file (\textbf{one} tex together with PNG or PDF images) in \textbf{one zip archive} via the workshop website.

\section{The method}

Change-point detection aims at identifying the first time instant $n$ where the distributional characteristic changes, e.g., the mean. According to \cite{Darkhovski1994}, any change-point problem can be reduced to a change-in-mean problem. For an introduction see \cite{Montgomery2005}.

Suppose $ Y_1, Y_2, \ldots $ are i.i.d. with $ \sigma^2 = \mbox{Var}( Y_1 ) \in (0,\infty)$. Let $ \mu = E(Y_1) $ and
\begin{eqnarray}
  T_n   &=&  n^{-1} \sum\nolimits_{i=1}^n Y_i, \nonumber \\
  T_n^* &=& \sqrt{n} (T_n - \mu)/\sigma,
\label{F1}
\end{eqnarray}

for $ n \in \N $. The central limit theorem (CLT) asserts that
$T_n^* \stackrel{d}{\to} N(0,1)$,
as $ n \to \infty $. For extensions to martingale difference arrays we refer to \cite[somewhere]{Durrett2005}.

The Shewhart individuals control chart is given by $  S = \inf\{ n \in \N : Y_n > c \} $ where $ c $ is a constant called {\it control limit}. One may also study the sequence,
\begin{equation}
  U_N = \inf\{ 1 \le n \le N: T_n^* > c' \}, \qquad N \in \N[+], c' \in \R,
\label{F2}
\end{equation}
with $ T_n^* $ as defined in (\ref{F1}). Its asymptotic distribution is related to Brownian motion.


\section{Example}


The method was used to analyze real CD measurements. Assuming i.i.d. normal data the chart is depicted in Figure~\ref{fig:wafer}.
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\begin{figure}[htb]
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\caption{Monitoring of CD measurements.}
\label{fig:wafer}
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If you want to include a graphic, include a valid PDF file within your zip file or use tikz/pgf etc. (PSTricks is \textbf{not} supported).
For tables, see Table~\ref{tab:mytable}.


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\begin{table}[h]
\begin{center} \footnotesize
\begin{tabular}{l|ccc}
Method &  $ \lambda $ &  ARL \\ \hline
 A     &  $0.1$     &    $ 400 $    \\
 B     &  $0.2$     &    $ 446 $    \\ \hline
\end{tabular}
\caption{Simulated average run lengths.}
\label{tab:mytable}
\end{center} \normalsize
\end{table}
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\begin{thebibliography}{99}

\bibitem{Darkhovski1994}
Darkhovski B. D. (1994). Nonparametric methods in change-point problems: A general approach and some concrete algorithms, {\em Change-Point Problems: Papers from the AMS-IMS-SIAM Summer Research Conference held at Mt. Holyoke College}, Carlstein, E., M\"uller, H. G., Siegmund, D., eds., pp. 99-107, IMS Lecture Notes - Monograph Series, Vol., 23, Institute of Mathematical Statistics: Hayward, CA.

\bibitem{Montgomery2005} Montgomery, D. C. (2005). {\em Introduction to Statistical Quality Control}, 5th ed., Wiley: New York.

\bibitem{Durrett2005}
Durrett, R. (2005). {\em Probability: Theory and Examples}, 3rd ed., Duxbury Advanced Series, Thomson Learning: Australia.

\end{thebibliography}

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