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

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