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ON A CLASS OF CHARACTERIZATION PROBLEMS

FOR RANDOM CONVEX COMBINATIONS

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LUDWIG BARINGHAUS
AND RUDOLF GRÜBEL

*Institut für Mathematische Stochastik, Universität Hannover,*

Postfach 6009, 30060 Hannover, Germany
(Received October 20, 1995; revised November 8, 1996)

**Abstract.**
We consider stochastic equations of
the form *X* =_{d} *W*_{1}*X*+*W*_{2}*X'*, where (*W*_{1},*W*_{2}), *X* and
*X'* are independent, `=_{d}' denotes equality in
distribution, *EW*_{1}+*EW*_{2} = 1 and *X* =_{d} *X'*. We discuss
existence, uniqueness and stability of the solutions, using
contraction arguments and an approach based on moments. The
case of {0,1}-valued *W*_{1} and constant *W*_{2} leads to
a characterization of exponential distributions.

*Key words and phrases*:
Stochastic difference
equations, exponential distributions, characterization
problems, contractions.

**Source**
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