An original method for computing the Probability Density Function (PDF) of the Line Spectrum Pair (LSP) coefficient vector as a function of the PDF of the AutoRegressive (AR) parameter vector is derived. Being linked by non-linear relations, AR parameters and LSP coefficients cannot both be Gaussian. This strict theoretical result could hide an approximately Gaussian distribution for the LSP parameters. Some distances between the LSP parameter PDF and its asymptotic Gaussian PDF are then studied. In a pattern recognition application, the LSP parameter statistics allow us to study the convexity of LSP point clusters, conditioned to each class: the LSP parameters seem to be attractive for pattern recognition when used with the centroid distance rule.