# Regenerative composition structures: the case of slow variation

Stochastic Processes Seminar Series

#### by A.D. Barbour

Institution: Universitat Zurich
Date: Thu 1st September 2005
Time: 1:15 PM
Location: Russell Love Theatre, Richard Berry Building, University of Melbourne

Abstract: A composition is an ordered partition of an integer~$n$. A regenerative composition structure is a family of probabilitydistributions on compositions, which satisfies certain conditionsappropriate to discrete sampling models; the most celebrated
example is the Ewens sampling formula. Here, we are interested in the asymptotics of the number of parts~$K_n$ in the composition for large~$n$. These asymptotics depend on the properties of the L\'evy measure~$\nu$ of an associated subordinator. If~$\nu$ is finite, both mean and variance of~$K_n$ are logarithmic, and~$K_n$ satisfies a central limit theorem.
If~$\nu$ is infinite, and \nu[x,\infty)$is regularly varying with positive exponent~$\a$as$x\to0$, then$K_n/\{n^\a L(n)\}$converges in distribution to a non-gaussian limit, for some slowly varying~$L$. Between these two very different extremes, only the case when~$\nu(dx)$is extremely close to the `gamma' measure$x^{-1}\theta e^{-\theta x}\,dx$had previously been solved; in this case, there is a normal approximation, with$E(K_n) \asymp \log^2n$and${\rm Var\,}(K_n) \asymp \log^3n$. In this talk, we describe the behaviour of~$K_n$for almost any~$\nu$which is infinite but slowly varying at~$0\$, explaining the transition
between the extremes in terms of the behaviour over time of an associated
point process.

Joint work with A.~V.~Gnedin