Convergence in distribution of the multidimensional Kohonen algorithm

Here we consider the Kohonen algorithm with a constant learning rate as a Markov process evolving in a topological space.It is shown that the process is an irreducible and aperiodic T-chain, regardless of the dimension of both data space and network and the special shape of the neighborhood function.Moreover the validity of Deoblin's condition is proved. These imply the convergence in distribution of the process to a nite invariant measure with a uniform geometric rate. In addition we show the process is positive Harris recurrent, which enables us to use statistical devices to measure its centrality and variability as the time goes to innnity.