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  \textbf{\textsc{STANFORD UNIVERSITY}}\\[5pt]
  \textbf{\textsc{DEPARTMENT OF STATISTICS}}\\[5pt]
  \Large{\textbf\textsc{{DEPARTMENTAL SEMINAR}}}
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\begin{center}
  4:15 p.m., Tuesday, August 1, 2006\\
  Sequoia Hall Room 200\\
  (Cookies at 3:45 in 1st Floor Lounge)
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  \textsl{Gerold Alsmeyer} \\
University of M\"unster\\
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  \textbf{ A Stochastic Fixed Point Equation For Weighted Minima 
and Maxima}
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\noindent
This is joint work with  Uwe R\"osler (Kiel):
Given a finite or infinite sequence $T=(T_{j})_{j\in J}$ of positive numbers
(so $J=\{1,...,n\}$ or $J={\mathbb N}$), we will consider the stochastic fixed
point equation
$$ W\ \ {\buildrel d\over =}\ \ \inf_{j\in J}T_{j}W_{j},\eqno(1) $$
for i.i.d.\ real-valued random variables $W,W_{1},W_{2},...$, where $\ {\buildrel d\over =}\ $
means equality in law. The task is to find all solutions to (1). This 
requires to
distinguish between various cases as to the constants $T_{j}$. The 
most interesting
ones are
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\begin{itemize} 
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\item{(i)} $|J|<\infty$ and $\inf_{j\in J}T_{j}>1$\\
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and
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\item{(ii)} $J={\mathbb N}$, $\inf_{j\in J}T_{j}>1$ and $\lim_{j\to\infty}T_{j}=\infty$.
\end{itemize}
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For these all solutions to (1) are found by using arguments from 
harmonic analysis on
trees and Choquet theory.

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