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Alexander Gnedin

Publications and source records attributed to Alexander Gnedin.

At least 55 records · Page 3Linked to original sources
arXiv2011

Lambda-coalescents with dust component

We consider the lambda-coalescent processes with positive frequency of singleton clusters. The class in focus covers, for instance, the beta$(a,b)$-coalescents with $a>1$. We show that some large-sample properties of these processes can be derived by coupling the coalescent with an increasing L{é}vy process (subordinator), and by exploiting parallels with the theory of regenerative composition structures. In particular, we discuss the limit distributions of the absorption time and the number of collisions.

math.PR↗
arXiv2010

$q$-exchangeability via quasi-invariance

For positive $q\neq1$, 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 $\mathbb{R}$ plays a significant role for the $q$-analogs. An explicit construction of ergodic $q$-exchangeable measures involves random shuffling of $\mathbb{N}=\{1,2,...\}$ 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.

math.PR↗
arXiv2010

The Bernoulli sieve: an overview

The Bernoulli sieve is a version of the classical balls-in-boxes occupancy scheme, in which random frequencies of infinitely many boxes are produced by a multiplicative random walk, also known as the residual allocation model or stick-breaking. We give an overview of the limit theorems concerning the number of boxes occupied by some balls out of the first $n$ balls thrown, and present some new results concerning the number of empty boxes within the occupancy range.

math.PR↗
arXiv2010

A Species Sampling Model with Finitely many Types

A two-parameter family of exchangeable partitions with a simple updating rule is introduced. The partition is identified with a randomized version of a standard symmetric Dirichlet species-sampling model with finitely many types. A power-like distribution for the number of types is derived.

math.PR↗
arXiv2010

Limit theorems for the number of occupied boxes in the Bernoulli sieve

The Bernoulli sieve is a version of the classical `balls-in-boxes' occupancy scheme, in which random frequencies of infinitely many boxes are produced by a multiplicative renewal process, also known as the residual allocation model or stick-breaking. We focus on the number $K_n$ of boxes occupied by at least one of $n$ balls, as $n\to\infty$. A variety of limiting distributions for $K_n$ is derived from the properties of associated perturbed random walks. Refining the approach based on the standard renewal theory we remove a moment constraint to cover the cases left open in previous studies.

math.PR↗
arXiv2009

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 $n$ individuals is chosen uniformly at random, independently of the random partition $π_n$ of these individuals into various types, and all individuals of the same type as the chosen individual are deleted, then for each $r > 0$, given that $r$ individuals remain, these individuals are partitioned according to $π_r'$ for some sequence of random partitions $(π_r')$ that does not depend on $n$. An analogous result characterizes the associated 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 neutral-to-the right process.

math.PR↗
arXiv2009

Boundaries from inhomogeneous Bernoulli trials

The boundary problem is considered for inhomogeneous increasing random walks on the square lattice ${\mathbb Z}_+^2$ with weighted edges. Explicit solutions are given for some instances related to the classical and generalized number triangles.

math.PR↗
arXiv2009

A q-analogue of de Finetti's theorem

A q-analogue of de Finetti's theorem is obtained in terms of a boundary problem for the q-Pascal graph. For q a power of prime this leads to a characterisation of random spaces over the Galois field F_q that are invariant under the natural action of the infinite group of invertible matrices with coefficients from F_q.

math.PR↗
arXiv2008

On asymptotics of exchangeable coalescents with multiple collisions

We study the number of collisions $X_n$ of an exchangeable coalescent with multiple collisions ($Λ$-coalescent) which starts with $n$ particles and is driven by rates determined by a finite characteristic measure $ν({\rm d}x)=x^{-2}Λ({\rm d}x)$. Via a coupling technique we derive limiting laws of $X_n$, using previous results on regenerative compositions derived from stick-breaking partitions of the unit interval. The possible limiting laws of $X_n$ include normal, stable with index $1\leα<2$ and Mittag-Leffler distributions. The results apply, in particular, to the case when $ν$ is a beta$(a-2,b)$ distribution with parameters $a>2$ and $b>0$. The approach taken allows to derive asymptotics of three other functionals of the coalescent, the absorption time, the length of an external branch chosen at random from the $n$ external branches, and the number of collision events that occur before the randomly selected external branch coalesces with one of its neighbours.

math.PR↗
arXiv2008

Moments of convex distribution functions and completely alternating sequences

We solve the moment problem for convex distribution functions on $[0,1]$ in terms of completely alternating sequences. This complements a recent solution of this problem by Diaconis and Freedman, and relates this work to the Lévy-Khintchine formula for the Laplace transform of a subordinator, and to regenerative composition structures.

math.PR↗
arXiv2008

Small parts in the Bernoulli sieve

Sampling from a random discrete distribution induced by a `stick-breaking' process is considered. Under a moment condition, it is shown that the asymptotics of the sequence of occupancy numbers, and of the small-parts counts (singletons, doubletons, etc) can be read off from a limiting model involving a unit Poisson point process and a self-similar renewal process on the halfline.

math.PR↗
arXiv2007

Corners and Records of the Poisson Process in Quadrant

The scale-invariant spacings lemma due to Arratia, Barbour and Tavar{é} establishes the distributional identity of a self-similar Poisson process and the set of spacings between the points of this process. In this note we connect this result with properties of a certain set of extreme points of the unit Poisson process in the positive quadrant.

math.PR↗
arXiv2007

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 samples from populations with infinitely many species. We present moments of this random variable, discuss asymptotic relations among them and with related random variables, and draw connections with regular variation, which appears in various manifestations.

math.PR↗
arXiv2007

On the number of collisions in $Λ$-coalescents

We examine the total number of collisions $C_n$ in the $Λ$-coalescent process which starts with $n$ particles. A linear growth and a stable limit law for $C_n$ are shown under the assumption of a power-like behaviour of the measure $Λ$ near 0 with exponent $0<α<1$.

math.PR↗
arXiv2006

Coherent random permutations with record statistics

Random permutations with distribution conditionally uniform given the set of record values can be generated in a unified way, coherently for all values of $n$. Our central example is a two-parameter family of random permutations that are conditionally uniform given the counts of upper and lower records. This family interpolates between two versions of Ewens' distribution. We discuss characterisations of the conditionally uniform permutations, their asymptotic properties, constructions and relations to random partitions.

math.PR↗
arXiv2006

Constrained exchangeable partitions

For a class of random partitions of an infinite set a de Finetti-type representation is derived, and in one special case a central limit theorem for the number of blocks is shown.

math.PR↗
arXiv2006

Poisson representation of a Ewens fragmentation process

A simple explicit construction is provided of a partition-valued fragmentation process whose distribution on partitions of $[n]=\{1,...,n\}$ at time $θ\ge 0$ is governed by the Ewens sampling formula with parameter $θ$. These partition-valued processes are exchangeable and consistent, as $n$ varies. They can be derived by uniform sampling from a corresponding mass fragmentation process defined by cutting a unit interval at the points of a Poisson process with intensity $θx^{-1} \diff x$ on ${\mathbb R}_+$, arranged to be intensifying as $θ$ increases.

math.PR↗