CONVERGENCE IN DISTRIBUTION OF THE ONE-DIMENSIONAL KOHONEN ALGORITHMSWHEN THE STIMULI ARE NOT UNIFORM

We show that the one-dimensional self-organizing Kohonen algorithm (wi th zero or two neighbours and constant step epsilon) is a Doeblin recu rrent Markov chain provided that the stimuli distribution mu is lower bounded by the Lebesgue measure on some open set. Some properties of t he invariant probability measure nu(epsilon) (support, absolute contin uity, etc.) are established as well as its asymptotic behaviour as eps ilon down 0 and its robustness with respect to mu.