A stochastic equation for the law of the random Dirichlet variance

This paper shows some new results concerning the law of the random variance VV of a Dirichlet process PP, expressed as the solution of a stochastic equation involving the squared difference between two independent copies of the mean of PP. An explicit solution of this equation is obtained via the Zolotarev transform of VV. Moreover, we discuss the correspondence between the distribution of the variance and the parameter of the Dirichlet process with given total mass.