arXiv subjects

Andrii Ilienko

Publications and source records attributed to Andrii Ilienko.

11 recordsLinked to original sources

arXiv2026

Random valuations

A valuation is a finitely additive function on the family of compact convex sets in $\mathbb{R}^d$. We study non-negative infinitely divisible random valuations, with particular emphasis on monotone, $σ$-continuous models with independent increments along nested families. After separating the deterministic part, we show that the Lévy measure of such a valuation is generated by pairs $(F,r)$, where $F$ is a non-empty closed convex set and $r>0$, with each pair contributing $r\mathbf{1}_{F\cap K=\emptyset}$. This yields a Poisson representation and an equivalent formulation through a pure-jump completely random measure on the space of closed convex sets. For stationary valuations, we derive a cylinder-Grassmannian representation of the Lévy measure. In the stationary isotropic case, we obtain a McMullen-type decomposition, at the level of one-dimensional distributions, into independent components stable under dilation of the argument.

math.PR↗

arXiv2026

Explosion and non-explosion in pure birth Crump--Mode--Jagers branching processes

In this short note, we provide an explicit sufficient condition for non-explosion of Crump--Mode--Jagers branching processes with pure birth reproduction. It shows that the standard sufficient condition for explosion, namely the convergence of the series of reciprocals of the birth rates, is -- at least for rate sequences without excessive oscillations -- remarkably close to being necessary. At the same time, it is not necessary in full generality: we construct a counterexample which also yields a general preferential attachment tree without fitness with an infinite path and no vertices of infinite degree, thereby answering an open question previously raised in the literature.

math.PR↗

arXiv2026

Pareto points in growing dimensions

We consider $n$ independent random points uniformly distributed in the $d_n$-dimensional unit cube and study Pareto points, that is, points that do not coordinatewise dominate any other point. We identify the critical growth rate of $d_n$ at which a phase transition occurs: below this threshold, the number of non-Pareto points diverges in probability, whereas above it there are asymptotically no such points. At criticality, the number of non-Pareto points converges in distribution to a Poisson random variable. We further describe their asymptotic spatial distribution in terms of convergence of random point measures. We also investigate points that dominate exactly $r$ other points and establish analogous phase transitions. For $r=1$, the critical dimension is the same as for non-Pareto points, whereas for every fixed $r\geq 2$ it is different, but, surprisingly, common to all such $r$.

math.PR↗

arXiv2026

Integer-valued valuations

We obtain a complete characterization of planar monotone $σ$-continuous valuations taking integer values, without assuming invariance under any group of transformations. We further investigate the consequences of dropping monotonicity or $σ$-continuity and give a full classification of line valuations. We also introduce a construction of the product for valuations of this type.

math.MG↗

arXiv2025

Scaling limit for small blocks in the Chinese restaurant process

The Chinese restaurant process is a basic sequential construction of consistent random partitions. We consider random point measures describing the composition of small blocks in such partitions and show that their scaling limit is given by the projective limit of certain inhomogeneous Poisson measures on cones of increasing dimension. This result makes it possible to derive classical and functional limit theorems in the Skorokhod topology for various characteristics of the Chinese restaurant process.

math.PR↗

arXiv2024

Strong limit theorems for empirical halfspace depth trimmed regions

We study empirical variants of the halfspace (Tukey) depth of a probability measure $μ$, which are obtained by replacing $μ$ with the corresponding weighted empirical measure. We prove analogues of the Marcinkiewicz--Zygmund strong law of large numbers and of the law of the iterated logarithm in terms of set inclusions and for the Hausdorff distance between the theoretical and empirical variants of depth trimmed regions. In the special case of $μ$ being the uniform distribution on a convex body $K$, the depth trimmed regions are convex floating bodies of $K$, and we obtain strong limit theorems for their empirical estimators.

math.PR↗

arXiv2023

Short cycles of random permutations with cycle weights: point processes approach

We study the asymptotic behavior of short cycles of random permutations with cycle weights. More specifically, on a specially constructed metric space whose elements encode all possible cycles, we consider a point process containing all information on cycles of a given random permutation on $\{1,\ldots,n\}$. The main result of the paper is the distributional convergence with respect to the vague topology of the above processes towards a Poisson point process as $n\to\infty$ for a wide range of cycle weights. As an application, we give several limit theorems for various statistics of cycles.

math.PR↗

arXiv2020

Limit theorems in the extended coupon collector's problem

We consider an extended variant of the classical coupon collector's problem with infinite number of collections. An arriving coupon is placed in the $r^{th}$ collection, $r\ge0$, if $r$ is the smallest index such that the corresponding collection still does not have a coupon of this type. We derive distributional limit theorems for the number of empty spots in different collections at the time when the $0^{th}$ collection was completed, as well as after some delay. We also obtain limiting distributions for completion times of different collections. All main results are given in an ultimate infinite-dimensional form in the sense of distributional convergence in $\mathbb R^\infty$. The main tool in the proofs is convergence of specially constructed point processes.

math.PR↗

arXiv2019

Convergence of point processes associated with coupon collector's and Dixie cup problems

We prove that, in the coupon collector's problem, the point processes given by the times of $r$-th arrivals for coupons of each type, centered and normalized in a proper way, converge toward a non-homogeneous Poisson point process. This result is then used to derive some generalizations and infinite-dimensional extensions of classical limit theorems on the topic.

math.PR↗

arXiv2017

Limit theorems for multidimensional renewal sets

Consider multiple sums $S_n$ on the $d$-dimensional integer grid,which are generated by i.i.d.\ random variables with a positive expectation. We prove the strong law of large numbers, the law of the iterated logarithm and the distributional limit theorem for random sets ${\mathcal M}_t$ that appear as inversion of the multiple sums, that is, as the set of all arguments $x\in{\mathbb R}_+^d$ such that the interpolated multiple sum $S_x$ exceeds $t$. The moment conditions are identical to those imposed in the almost sure limit theorems for multiple sums. The results are expressed in terms of set inclusions and using distances between sets.

math.PR↗

arXiv2013

Continuous counterparts of Poisson and binomial distributions and their properties

On the basis of integral representations of Poisson and binomial distribution functions via complete and incomplete Euler Γ- and B-functions, we introduce and discuss continuous counterparts of the Poisson and binomial distributions. The former turns out to be closely related to classical Volterra functions as well. Under usual conditions, we also prove that the sequence of continuous binomial distributions converges weakly to the continuous Poisson one. At the end, we discuss a relationship between the continuous Poisson distribution and the Γ-process.

math.PR↗