Alexander Gnedin

Regenerative Composition Structures

A new class of random composition structures (the ordered analog of Kingman’s partition structures) is defined by a regenerative description of component sizes. Each regenerative composition structure is represented by a process of random sampling of points from an exponential distribution on the positive halfline, and separating the points into clusters by an independent regenerative random set. Examples are composition structures derived from residual allocation models, including one associated with the Ewens sampling formula, and composition structures derived from the zero set of a Brownian motion or Bessel process. We provide characterisation results and formulas relating the distribution of the regenerative

Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws ∗

This paper collects facts about the number of occupied boxes in the classical balls-in-boxes occupancy scheme with infinitely many positive frequencies: equivalently, about the number of species represented in sam- ples from populations with infinitely many species. We present moments of this random variable, discuss asymptotic relations among them and with re- lated random variables, and draw connections with regular variation, which appears in various manifestations. AMS 2000 subject classifications: Primary 60F05, 60F15; secondary

Asymptotic laws for compositions derived from transformed subordinators

A random composition of n appears when the points of a random closed set R ⊂ [0,1] are used to separate into blocks n points sampled from the uniform distribution. We study the number of parts K n of this composition and other related functionals under the assumption that R = Φ(S • ), where (S t , t ≥ 0) is a subordinator and Φ:[0,∞] → [0, 1] is a diffeomorphism. We derive the asymptotics of K n when the Levy measure of the subordinator is regularly varying at 0 with positive index. Specializing to the case of exponential function Φ(x) = 1 - e -x , we establish a connection between the asymptotics of K n and the exponential functional of the subordinator.

Exchangeable partitions derived from Markovian coalescents

Kingman derived the Ewens sampling formula for random partitions describing the genetic variation in a neutral mutation model defined by a Poisson process of mutations along lines of descent governed by a simple coalescent process, and observed that similar methods could be applied to more complex models. Mohle described the recursion which determines the general- ization of the Ewens sampling formula in the situation when the lines of descent are governed by a �-coalescent, which allows multiple mergers. Here we show that the basic integral representa- tion of transition rates for the �-coalescent is forced by sampling consistency under more general assumptions on the coalescent process. Exploiting an analogy with the theory of regenerative partition structures, we provide various characterizations of the associated partition structures in terms of discrete-time Markov chains.

q-EXCHANGEABILITY VIA QUASI-INVARIANCE

For positive q is not 1, the q-exchangeability of an infinite random word is introduced as quasi-invariance under permutations of letters, with a special cocycle which accounts for inversions in the word. This framework allows us to extend the q-analog of de Finetti’s theorem for binary sequences—see Gnedin and Olshanski [Electron. J. Combin. 16 (2009) R78]—to general real-valued sequences. In contrast to the classical case of exchangeability (q = 1), the order on ℝ plays a significant role for the q-analogs. An explicit construction of ergodic q-exchangeable measures involves random shuffling of ℕ = {1, 2, …} by iteration of the geometric choice. Connections are established with transient Markov chains on q-Pascal pyramids and invariant random flags over the Galois fields.

Coherent permutations with descent statistic and the boundary problem for the graph of zigzag diagrams

The graph of zigzag diagrams is a close relative of Youngs lattice. The boundary problem for this graph amounts to describing coherent random permutations with descent-set statistic, and is also related to certain positive characters on the algebra of quasi-symmetric functions. We establish connections to some further relatives of Youngs lattice and solve the boundary problem by reducing it to the classification of spreadable total orders on integers, as recently obtained by Jacka and Warren.

ON A BEST-CHOICE PROBLEM WITH DEPENDENT CRITERIA

We study the problem of maximizing the probability of stopping at an object which is best in at least one of a given set of criteria, using only stopping rules based on the knowledge of whether the current object is relatively best in each of the criteria. The asymptotic results for the case of independent criteria are shown to hold in certain cases where the componentwise maxima are, pairwise, either asymptotically independent or asymptotically full dependent. An example of the former is a random sample from a bivariate correlated normal distribution; thus our results settle a question posed recently by T. S. Ferguson.

Characterizations of exchangeable partitions and random discrete distributions by deletion properties

We prove a long-standing conjecture which characterises the Ewens-Pitman two- parameter family of exchangeable random partitions, plus a short list of limit and exceptional cases, by the following property: for each n = 2, 3,..., if one of n indi- viduals is chosen uniformly at random, independently of the random partitionn of these individuals into various types, and all individuals of the same type as the cho- sen individual are deleted, then for each r > 0, given that r individuals remain, these individuals are partitioned according to � 0 for some sequence of random partitions (� 0 ) which does not depend on n. An analogous result characterizes the associ- ated Poisson-Dirichlet family of random discrete distributions by an independence property related to random deletion of a frequency chosen by a size-biased pick. We also survey the regenerative properties of members of the two-parameter family, and settle a question regarding the explicit arrangement of intervals with lengths given by the terms of the Poisson-Dirichlet random sequence into the interval partition induced by the range of a homogeneous neutral-to-the right process.

Records from a multivariate normal sample

Given a sequence X1, X2... of iid random vectors, call Xa a record if there is a record simultaneously in all coordinates at index n. We study the asymptotic behaviour of the probability that Xa is a record under the assumption that the underlying distribution is normal. AMS classification: primary 60G70

Three Sampling Formulas

Sampling formulas describe probability laws of exchangeable combinatorial structures like partitions and compositions. We give a brief account of two known parametric families of sampling formulas for compositions and add a new family to the list.