  Stochastic feedback systems give rise to a variety of notions of stability.
The conditions for the stability of the median, mean, and variance stability
conditions differ. These conditions can be stated explicitly for scalar
discrete-time systems with (almost) arbitrary distributions of the stochastic
feedback gain. The state variable in such systems evolves towards a
heavy-tailed distribution and exhibits some non-intuitive characteristics. For
example, one can use stochastic feedback to stabilise unstable systems where
one does not know the sign of the unstable pole or the sign of the system gain.
A more dramatic example is an investment scheme which simultaneously yields
unbounded expected profit and almost certain bankruptcy to every investor.
