Convergence in distribution of the one-dimensional Kohonen algorithms when the stimuli are not uniform. (English) Zbl 0792.60066
Summary: We show that the one-dimensional self-organizing Kohonen algorithm (with zero or two neighbours and constant step \(\varepsilon)\) 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 \(\nu^ \varepsilon\) (support, absolute continuity, etc.) are established as well as its asymptotic behaviour as \(\varepsilon \downarrow 0\) and its robustness with respect to \(\mu\).
MSC:
60J20 | Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) |
60F99 | Limit theorems in probability theory |