On generalisations of the log-Normal distribution by means of a new product definition in the Kapteyn process

We discuss the modification of the Kapteyn multiplicative process using the qq-product of Borges [E.P. Borges, A possible deformed algebra and calculus inspired in nonextensive thermostatistics, Physica A 340 (2004) 95]. Depending on the value of the index qq a generalisation of the log-Normal distribution is yielded. Namely, the distribution increases the tail for small (when q<1q<1) or large (when q>1q>1) values of the variable upon analysis. The usual log-Normal distribution is retrieved when q=1q=1, which corresponds to the traditional Kapteyn multiplicative process. The main statistical features of this distribution as well as related random number generators and tables of quantiles of the Kolmogorov-Smirnov distance are presented. Finally, we illustrate the validity of this scenario by describing a set of variables of biological and financial origin.