A Passage to the Poisson-dirichlet through the Bessel Square Processes

This principal result in this article is that every Poisson-Dirichlet distribution PD(0, θ) is an asymptotically invariant distribution for a growing collection of independent Bessel square processes of dimension zero divided by their total sum, under the condition that the sum total of their initial values grows to infinity in probability. Implications in several areas of Probability theory have been discussed, including Brownian local time, Fernholz & Karatzas’s Volatility Stabilized Market models of Mathematical Finance, Watterson’s Infinitely Many Neutral Alleles model in Statistical Genetics, branching Bessel diffusions, and the Poisson-Dirichlet cascades. A key step involves generalization of a polar decomposition result involving squared Bessel processes that was observed by Warren & Yor in their study of the Brownian burglar.