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Alexander Gnedin

Publications and source records attributed to Alexander Gnedin.

70 records · Page 4Linked to original sources
arXiv2006

Asymptotic laws for compositions derived from transformed subordinators

A random composition of $n$ appears when the points of a random closed set $\widetilde{\mathcal{R}}\subset[0,1]$ 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 $\widetilde{\mathcal{R}}=ϕ(S_{\bullet})$, where $(S_t,t\geq0)$ is a subordinator and $ϕ:[0,\infty]\to[0,1]$ is a diffeomorphism. We derive the asymptotics of $K_n$ when the Lévy 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.

math.PR↗
arXiv2006

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. M{ö}hle described the recursion which determines the generalization of the Ewens sampling formula in the situation when the lines of descent are governed by a $Λ$-coalescent, which allows multiple mergers. Here we show that the basic integral representation of transition rates for the $Λ$-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.

math.PR↗
arXiv2006

The boundary of the Eulerian number triangle

The Eulerian triangle is a classical array of combinatorial numbers defined by a linear recursion. The associated boundary problem asks one to find all extreme nonnegative solutions to a dual recursion. Exploiting connections with random permutations and Markov chains we show that the boundary is discrete and explicitly identify its elements.

math.PR↗
arXiv2006

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 Young's 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 Young's 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.

math.CO↗
arXiv2006

Recognising the Last Record of a Sequence

We study the best-choice problem for processes which generalise the process of records from Poisson-paced i.i.d. observations. Under the assumption that the observer knows distribution of the process and the horizon, we determine the optimal stopping policy and for a parametric family of problems also derive an explicit formula for the maximum probability of recognising the last record.

math.PR↗
arXiv2005

Self-similar and Markov composition structures

The bijection between composition structures and random closed subsets of the unit interval implies that the composition structures associated with $S \cap [0,1]$ for a self-similar random set $S\subset {\mathbb R}_+$ are those which are consistent with respect to a simple truncation operation. Using the standard coding of compositions by finite strings of binary digits starting with a 1, the random composition of $n$ is defined by the first $n$ terms of a random binary sequence of infinite length. The locations of 1s in the sequence are the places visited by an increasing time-homogeneous Markov chain on the positive integers if and only if $S = \exp(-W)$ for some stationary regenerative random subset $W$ of the real line. Complementing our study in previous papers, we identify self-similar Markovian composition structures associated with the two-parameter family of partition structures.

math.PR↗
arXiv2005

Rank Independence and Rearrangements of Random Variables

A rearrangement of $n$ independent uniform $[0,1]$ random variables is a sequence of $n$ random variables $Y_1,...,Y_n$ whose vector of order statistics has the same distribution as that for the $n$ uniforms. We consider rearrangements satisfying the strong rank independence condition, that the rank of $Y_k$ among $Y_1,...,Y_k$ is independent of the values of $Y_1,...,Y_{k-1}$, for $k=1,...,n$. Nontrivial examples of such rearrangements are the travellers' processes defined by Gnedin and Krengel. We show that these are the only examples when $n=2$, and when certain restrictive assumptions hold for $n\geq 3$; we also construct a new class of examples of such rearrangements for which the restrictive assumptions do not hold.

math.PR↗
arXiv2004

Exchangeable Gibbs partitions and Stirling triangles

For two collections of nonnegative and suitably normalised weights $\W=(\W_j)$ and $\V=(\V_{n,k})$, a probability distribution on the set of partitions of the set $\{1,...,n\}$ is defined by assigning to a generic partition $\{A_j, j\leq k\}$ the probability $\V_{n,k} \W_{|A_1|}... \W_{|A_k|}$, where $|A_j|$ is the number of elements of $A_j$. We impose constraints on the weights by assuming that the resulting random partitions $Π_n$ of $[n]$ are consistent as $n$ varies, meaning that they define an exchangeable partition of the set of all natural numbers. This implies that the weights $\W$ must be of a very special form depending on a single parameter $α\in [-\infty,1]$. The case $α=1$ is trivial, and for each value of $α\neq 1$ the set of possible $\V$-weights is an infinite-dimensional simplex. We identify the extreme points of the simplex by solving the boundary problem for a generalised Stirling triangle. In particular, we show that the boundary is discrete for $-\infty\leqα<0$ and continuous for $0\leqα<1$. For $α\leq 0$ the extremes correspond to the members of the Ewens-Pitman family of random partitions indexed by $(α,θ)$, while for $0<α<1$ the extremes are obtained by conditioning an $(α,θ)$-partition on the asymptotics of the number of blocks of $Π_n$ as $n$ tends to infinity.

math.PR↗
arXiv2004

Regenerative partition structures

We consider Kingman's partition structures which are regenerative with respect to a general operation of random deletion of some part. Prototypes of this class are the Ewens partition structures which Kingman characterised by regeneration after deletion of a part chosen by size-biased sampling. We associate each regenerative partition structure with a corresponding regenerative composition structure, which (as we showed in a previous paper) can be associated in turn with a regenerative random subset of the positive halfline, that is the closed range of a subordinator. A general regenerative partition structure is thus represented in terms of the Laplace exponent of an associated subordinator. We also analyse deletion properties characteristic of the two-parameter family of partition structures.

math.PR↗
arXiv2004

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 composition to the L{é}vy parameters of a subordinator whose range is the corresponding regenerative set. In particular, the only reversible regenerative composition structures are those associated with the interval partition of $[0,1]$ generated by excursions of a standard Bessel bridge of dimension $2 - 2 α$ for some $α\in [0,1]$.

math.PR↗
arXiv2004

Derangement characters of the full general linear group

We focus on derangement characters of $GL(n,q)$ which depend solely on the dimension of the space of fixed vectors. This family includes Thoma characters which become asymptotically irreducible as $n\to\infty$. We find explicit decomposition of Thoma characters into irreducibles, construct further derangement characters and seek for extremes in the family of derangement characters.

math.RT↗
arXiv2004

Asymptotic laws for nonconservative self-similar fragmentations

We consider a self-similar fragmentation process in which the generic particle of size $x$ is replaced at probability rate $x^α$, by its offspring made of smaller particles, where $α$ is some positive parameter. The total of offspring sizes may be both larger or smaller than $x$ with positive probability. We show that under certain conditions the typical size in the ensemble is of the order $t^{-1/α}$ and that the empirical distribution of sizes converges to a random limit which we characterise in terms of the reproduction law.

math.PR↗
arXiv2003

Bernoulli Sieve

Bernoulli sieve is a recursive construction of a random composition (ordered partition) of integer $n$. This composition can be induced by sampling from a random discrete distribution which has frequencies equal to the sizes of component intervals of a stick-breaking interval partition of $[0,1]$. We exploit Markov property of the composition and its renewal representation to derive asymptotics of the moments and to prove a central limit theorem for the number of parts.

math.PR↗
arXiv2002

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 and add a new family to the list.

math.PR↗
arXiv2002

Best Choice from the Planar Poisson Process

Various best-choice problems related to the planar homogeneous Poisson process in finite or semi-infinite rectangle are studied. The analysis is largely based on properties of the one-dimensional box-area process associated with the sequence of records. We prove a series of distributional identities involving exponential and uniform random variables, and resolve the Petruccelli-Porosinski-Samuels paradox on coincidence of asymptotic values in certain discrete-time optimal stopping problems.

math.PR↗