Derivation of nonstationary Lotka-type distributions without divergence of moments

Using the “Success breeds success” (SBS) theory to explain the inverse power distributions of scientific productivity (Lotka-type distributions), several theoretical distributions have been proposed by Simon and other scientists. However, most of them assume a stationary distribution, so that the distributions they produce cannot explain the change in the shape of the distribution according to the change in the total number of items, and have the problem that moments above a certain order diverge. The purpose of this study is to propose a model that leads to a nonstationary Lotka-type distribution without divergence of moments and to derive a probability density function for the distribution of the number of items contained in a source according to this model. Four probability density functions were obtained by making different assumptions about the new source emergence probability α, one of which explains relatively well the distributions obtained empirically. This model allows for the calculation of changes in the distribution shape and moments with changes in the total number of items. A detailed analysis of the results obtained by Yule (1925), the only existing study that considered the nonstationarity part of the distribution, showed that his distribution was almost identical to that obtained by the author.