Generalized Mittag Leffler distributions arising as limits in preferential attachment models

Abstract

For $0<α<1,$ and $θ>-α,$ let $(S^{-α}_{α,θ+r})_{\{r\ge 0\}}$ denote an increasing(decreasing) sequence of variables forming a time inhomogeneous Markov chain whose marginal distributions are equivalent to generalized Mittag Leffler distributions. We exploit the property that such a sequence may be connected with the two parameter $(α,θ)$ family of Poisson Dirichlet distributions. We demonstrate that the sequences serve as limits in certain types of preferential attachment models. As one illustrative application, we describe the explicit joint limiting distribution of scaled degree sequences arising under a class of linear weighted preferential attachment models as treated in Móri (2005), with weight $β>-1.$ When $β=0$ this corresponds to the Barbasi-Albert preferential attachment model. We then construct sequences of nested $(α,θ)$ Chinese restaurant partitions of $[n]$. From this, we identify and analyze relevant quantities that may be thought of as mimics for vectors of degree sequences, or differences in tree lengths. We also describe connections to a wide class of continuous time coalescent processes that can be seen as a variation of stochastic flows of bridges related to generalized Fleming-Viot models. Under a change of measure our results suggest the possibilities for identification of limiting distributions related to consistent families of nested Gibbs partitions of $[n]$ that would otherwise be difficult by methods using moments or Laplace transforms. In this regard, we focus on special simplifications obtained in the case of $α=1/2.$ That is to say, limits derived from a $\mathrm{PD}(1/2|t)$ distribution. Throughout we present some distributional results that are relevant to various settings. We describe nestings across the $α$ parameter in section 6