arXiv subjects

Alexander Gnedin

Publications and source records attributed to Alexander Gnedin.

At least 19 recordsLinked to original sources

arXiv2026

The Dual-Population Benchmark Model

Motivated by Fisher's and Hill's \cite{Fisher,Hill} ideas of ranking and pivotal quantities, we introduce a planar Poisson process (PPP) model in which the negative component $\Pi_-$, acted upon by a choice operator {\rm C}, supplies a set of past benchmarks that divide the future points of the positive component $\Pi_+$ into ranked categories. The benchmarks act as separators in an ordered paintbox while simultaneously acquiring the dual role of past standards established by the choice operator. The exceptional homogeneity properties of the PPP offer an infinitude of possibilities which absorb many existing combinatorial structures and enrich the toolbox of Bayesian distribution-free inference. A particular benchmark generating mechanism considered here (the exponential race) amounts to the device of splitting into spacings of nonhomogeneous order statistics. On the methodological side, the paper aims to highlight the role of order as important characteristic of an exchangeable structure, complementary to the description in terms of the components size. Moreover, we advocate the viewpoint that the order induced by a latent strength parameter is {\it intrinsically} inherited from the distant-past temporal sampling order, hence the decoupled orders may coexist within the framework of population duality without disturbing size-biasedness of components and the full exchangeability within the sample. This implies that the indistinguishability of components in the nonlinear CRP, sometimes regarded as nonexchangeability, does not in fact destroy exchangeability and should be reconciled with the arrival ordering within the paradigm of ordered structures.

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arXiv2026

Exchangeable Testing Against an Unknown Benchmark

We generate infinite binary exchangeable sequences by sequential comparison of data points against a latent benchmark. Assuming a prior distribution of the benchmark rank $R_0$ within an unobserved group, we set up the Bayesian machinery that determines the posterior distribution of the running rank $R_n$ in purely combinatorial terms. This yields an explicitly computable predictive probability of winning against the benchmark. The normalised running rank converges to a latent strength variable $X$ with polynomial density, possibly Beta-tilted. Some min-max tournaments lead to particularly simple multiplicative formulae for predictive probabilities related to priors that generalise the Topp--Leone distribution; for that class we analyse the asymptotics of the associated fixed-$n$ up-down Markov chains. The limiting diffusion has the classical Wright--Fisher variance but a nonlinear drift expressed explicitly via the prior density of the benchmark. Mixtures of Beta densities are classical objects in the theory of exchangeable sequences. The contribution of the present work is the combinatorial rank-based updating mechanism and the resulting explicit predictive laws for sequential testing against an unknown benchmark.

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arXiv2026

The Memoryless Best-Choice Problem

A random sequence sampled from a known continuous distribution is observed with the objective to choose an item with the overall rank one. A rejected item cannot be recalled and is immediately erased from the memory. Under this memory constraint, the choice problem is not amenable to recursive methods of optimal stopping and becomes a global optimisation task. We focus on a heavy-traffic form of the problem with infinitely many choice opportunities, which we state in terms of a planar Poisson process (PPP). Symmetries of the PPP are used to derive basic structural properties of the optimal stopping rule, including the balance at the boundary equation, and two key integral identities. Throughout, we make throrough comparison to the classic full-information counterpart of the problem, revisiting both discrete- and continuous-time models. The optimal value, stopping rule and other characteristics of the problem are determined analytically and approximated numerically with high precision.

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arXiv2026

Symmetries of Random Partitions

This paper is motivated by a recent result of Pitman and Yakubovich stating that a partially exchangeable, stationary (PES) random partition of $\mathbb{N}$ is exchangeable. This echoes an earlier theorem of Kallenberg on the equivalence of spreadability (contractability) and exchangeability for infinite partitions. We revise the hierarchy of symmetries with a focus on partitions of finite sets $[n]$, and ask about the extent to which these relaxed symmetry types differ from exchangeability, framing the question in terms of the geometry of the polytope of distributions defined by the symmetry constraints. We show that single-orbit exchangeable partitions remain extreme among spreadable and PES distributions, and that $n=5$ is the first case where non-exchangeable extreme partitions occur, causing the associated polytopes to deviate from a simplex structure.

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arXiv2026

Optimal Stopping for the Uniform Distribution

Many discrete-time optimal stopping problems are known to have more tractable limit forms based on a planar Poisson process. Using this tool we find a solution to the optimal stopping problem for i.i.d. sequence of $n$ discrete uniform random variables, in the asymptotic regime where $n$ and the range of distribution are of the same order. The optimal stopping rule in the Poisson problem is identified, by means of a time change, with known asymptotic solution to Lindley's problem of minimising the expected rank.

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arXiv2025

Maximal Counts in the Stopped Occupancy Problem

We revisit a version of the classic occupancy scheme, where balls are thrown until almost all boxes receive a given number of balls. Special cases are widely known as coupon-collectors and dixie cup problems. We show that as the number of boxes tends to infinity, the distribution of the maximal occupancy count does not converge, but can be approximated by a convolution of two Gumbel distributions, with the approximating distribution having oscillations close to periodic on a logarithmic scale. We pursue two approaches: one relies on lattice point processes obtained by poissonisation of the number of balls and boxes, and the other employs interpolation of the multiset of occupancy counts to a point process on reals. This way we gain considerable insight in known asymptotics obtained previously by mostly analytic tools. Further results concern the moments of maximal occupancy counts and ties for the maximum.

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arXiv2024

Cross Modality of the Extended Binomial Sums

For a family of probability functions (or a probability kernel), cross modality occurs when every likelihood maximum matches a mode of the distribution. This implies existence of simultaneous maxima on the modal ridge of the family. The paper explores the property for extended Bernoulli sums, which are random variables representable as a sum of independent Poisson and any number (finite or infinite) of Bernoulli random variables with variable success probabilities. We show that the cross modality holds for many subfamilies of the class, including power series distributions derived from entire functions with totally positive series expansion. A central role in the study is played by the extended Darroch's rule \cite{Darroch, Pitman}, which originally localised the mode of Poisson-binomial distribution in terms of the mean. We give different proofs and geometric interpretation to the extended rule and point at other modal properties of extended Bernoulli sums, in particular discuss stability of the mode in the context of a transport problem.

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arXiv2023

Infinite Size-Biased Orders

The infinite random size-biased order with arbitrary positive size parameters is introduced in terms of independent exponential random variables. We collect basic properties and constructions of the order, some of which belong to the folklore, and show how the order type (e.g. ${\mathbb Z}_{>0}, {\mathbb Q}$ or any other possible) depends on parameters.

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arXiv2023

Records in the Infinite Occupancy Scheme

We consider the classic infinite occupancy scheme, where balls are thrown in boxes independently, with probability $p_j$ of hitting box $j$. Each time a box receives its first ball we speak of a record and, more generally, call an $r$-record every event when a box receives its $r$th ball. Assuming that the sequence $(p_j)$ is not decaying too fast, we show that after many balls have been thrown, the suitably scaled point process of $r$-record times is approximately Poisson. The joint convergence of $r$-record processes is argued under a condition of regular variation.

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arXiv2023

A Random Graph Growth Model

A growing random graph is constructed by successively sampling without replacement an element from the pool of virtual vertices and edges. At start of the process the pool contains $N$ virtual vertices and no edges. Each time a vertex is sampled and occupied, the edges linking the vertex to previously occupied vertices are added to the pool of virtual elements. We focus on the edge-counting at times when the graph has $n\leq N$ occupied vertices. Two different Poisson limits are identified for $n\asymp N^{1/3}$ and $N-n\asymp 1$. For the bulk of the process, when $n\asymp N$, the scaled number of edges is shown to fluctuate about a deterministic curve, with fluctuations being of the order of $N^{3/2}$ and approximable by a Gaussian bridge.

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arXiv2022

Random Permutations and Queues

Given a growth rule which sequentially constructs random permutations of increasing degree, the stochastic process version of the rencontre problem asks what is the limiting proportion of time that the permutation has no fixed points (singleton cycles). We show that the discrete-time Chinese Restaurant Process (CRP) does not exhibit this limit. We then consider the related embedding of the CRP in continuous time and thereby show that it does have this and other limits of the time averages. By this embedding the cycle structure of the permutation can be represented as a tandem of infinite-server queues. We use this connection to show how results from the queuing theory can be interpreted in terms of the evolution of the cycle counts of permutations.

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arXiv2022

The Last-Success Stopping Problem with Random Observation Times

Suppose $N$ independent Bernoulli trials are observed sequentially at random times of a mixed binomial process. The task is to maximise, by using a nonanticipating stopping strategy, the probability of stopping at the last success. We focus on the version of the problem where the $k^\text{th}$ trial is a success with probability $p_k=\theta/(\theta+k-1)$ and the prior distribution of $N$ is negative binomial with shape parameter $\nu$. Exploring properties of the Gaussian hypergeometric function, we find that the myopic stopping strategy is optimal if and only if $\nu\geq\theta$. We derive formulas to assess the winning probability and discuss limit forms of the problem for large $N$.

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arXiv2021

Running minimum in the best-choice problem

We consider the best-choice problem for independent (not necessarily iid) observations $X_1, \cdots, X_n$ with the aim of selecting the sample minimum. We show that in this full generality the monotone case of optimal stopping holds and the stopping domain may be defined by the sequence of monotone thresholds. In the iid case we get the universal lower bounds for the success probability. We cast the general problem with independent observations as a variational first-passage problem for the running minimum process which simplifies obtaining the formula for success probability. We illustrate this approach by revisiting the full-information game (where $X_j$'s are iid uniform-$[0,1]$), in particular deriving new representations for the success probability and its limit by $n \rightarrow \infty$. Two explicitly solvable models with discrete $X_j$'s are presented: in the first the distribution is uniform on $\{j,\cdots,n\}$, and in the second the distribution is uniform on $\{1,\cdots, n\}$. These examples are chosen to contrast two situations where the ties vanish or persist in the large-$n$ Poisson limit.

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arXiv2021

Trapping the Ultimate Success

We introduce a betting game, where the gambler aims to guess the last success epoch from past observed data. The player may bet on the event that no further successes occur, or choose a `trap' which is any span of future times. In the latter case winning is achieved if the last success turns out to be the only one falling in the trap. The game is closely related to the sequential decision problem of maximising the probability of stopping on the last success in a finite sequence of trials. We use this connection to analyse the problem of stopping at the last record for trials paced by a Polya-Lundberg process with log-series distribution of the total number of trials.

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arXiv2021

How to beat the 1/e-strategy of best choice (the random arrivals problem)

In the best choice problem with random arrivals, an unknown number $n$ of rankable items arrive at times sampled from the uniform distribution. As is well known, a real-time player can ensure stopping at the overall best item with probability at least $1/e$, by waiting until time $1/e$ then selecting the first relatively best item to appear (if any). This paper discusses the issue of dominance in a wide class of stopping strategies of best choice, and argues that in fact the player faces a trade-off between success probabilities for various values of $n$. We argue that the $1/e$-strategy is not a unique minimax strategy and that it can be improved in various ways.

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arXiv2020

On sequential selection and a first passage problem for the Poisson process

This note is motivated by connections between the online and offline problems of selecting a possibly long subsequence from a Poisson-paced sequence of uniform marks under either a monotonicity or a sum constraint. The offline problem with the sum constraint amounts to counting the Poisson arrivals before their total exceeds a certain level. A precise asymptotics for the mean count is obtained by coupling with a nonlinear pure birth process.

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arXiv2020

Diffusion Approximations in the Online Increasing Subsequence Problem

The online increasing subsequence problem is a stochastic optimisation task with the objective to maximise the expected length of subsequence chosen from a random series by means of a nonanticipating decision strategy. We study the structure of optimal and near-optimal subsequences in a standardised planar Poisson framework. Following a long-standing suggestion by Bruss and Delbaen (Stoch. Proc. Appl. 114, 2004), we prove a joint functional limit theorem for the transversal fluctuations about the diagonal of the running maximum and the length processes. The limit is identified explicitly with a Gaussian time-inhomogeneous diffusion. In particular, the running maximum converges to a Brownian bridge, and the length process has another explicit non-Markovian limit.

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arXiv2019

Self-intersection local times of random fields in stochastic flows

In this article we study transformations of Gaussian field by stochastic flow on the plane. A stochastic flow is a solution to the equation with interaction whose coefficients depend on the occupation measure of the field. We consider nonsmooth Gaussian field, which has self-intersection local times of any multiplicity. In the article we prove the existence of self-intersection local times for the transformed field and study its asymptotics.

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