On financial applications of the two-parameter Poisson-Dirichlet distribution

Capital distribution curve is defined as log-log plot of normalized stock capitalizations ranked in descending order. The curve displays remarkable stability over periods of time. Theory of exchangeable distributions on set partitions, developed for purposes of mathematical genetics and recently applied in non-parametric Bayesian statistics, provides probabilistic-combinatorial approach for analysis and modeling of the capital distribution curve. Framework of the two-parameter Poisson-Dirichlet distribution contains rich set of methods and tools, such as consistent family of distributions on partitions, limiting distribution in infinite ranked simplex and associated diffusion process, induced by Markov chain of special form. The purpose of this note is to provide an informal introduction to the framework of exchangeable distributions on partitions in financial context. In particular, it is shown that averaged samples from the Poisson-Dirichlet distribution provide approximation to the capital distribution curves in equity markets. This suggests that the two-parameter model can be employed for modeling evolution of ranked market weights fluctuating in stochastic equilibrium.