  We consider a class of discrete time Markov chains with state space [0,1] and
the following dynamics. At each time step, first the direction of the next
transition is chosen at random with probability depending on the current
location. Then the length of the jump is chosen independently as a random
proportion of the distance to the respective end point of the unit interval,
the distributions of the proportions being fixed for each of the two
directions. Chains of that kind were subjects of a number of studies and are of
interest for some applications. Under simple broad conditions, we establish the
ergodicity of such Markov chains and then derive closed form expressions for
the stationary densities of the chains when the proportions are beta
distributed with the first parameter equal to 1. Examples demonstrating the
range of stationary distributions for processes described by this model are
given, and an application to a robot coverage algorithm is discussed.
