arXiv subjects

Alexander Marynych

Publications and source records attributed to Alexander Marynych.

At least 19 recordsLinked to original sources

arXiv2026

Peelings and Wrappings of Families of Convex Sets with Applications to Strongly Convex Sets Generated by Random Samples

We introduce and study peeling and wrapping operations for families of compact convex sets. The two peeling procedures considered in the paper are the $m$-point peeling, obtained by intersecting the convex hulls remaining after all possible deletions of $m$ members of the family, and the recursive convex hull peeling, obtained by repeatedly removing the contributing sets, that is, those members whose deletion strictly changes the convex hull. Using polarity, we also introduce the dual wrapping operations for intersections of convex sets. The deterministic part of the paper develops the geometric framework needed for these constructions. In particular, we study contributing sets under general position assumptions, explain the role of compactness of convex hulls of subfamilies, and prove continuity results for both peeling procedures with respect to a suitable vague convergence of locally finite point measures on the space of compact convex sets. The probabilistic part applies this framework to $K$-hulls generated by random samples from a convex body $K$. Assuming that $K$ is strictly convex and regular, we prove that the m-point and recursive peelings of the polar bodies associated with the random $K$-hulls converge in distribution to the corresponding peelings of the limiting Poisson object. By polarity, this also yields distributional convergence of the associated wrapping operations for the rescaled random sets themselves.

math.MG↗

arXiv2026

Set-indexed and multiple sums in high dimensions

We consider multiple and set-indexed sums of random vectors taking values in Euclidean space of growing dimension. It is shown that, when viewed as finite metric spaces, the sets of values of such sums converge in probability. The limit is identified as a generalisation of the Wiener spiral, which appears as the high-dimensional limit of single-index sums.

math.PR↗

arXiv2026

Intrinsic volumes of $\ell_p$-balls and a continuum of Maxwell--Poincar\'e--Borel laws for their curvature measures

For $p>1$, we derive explicit formulas for the intrinsic volumes $V_0(\mathbb B_p^n),\dots,V_{n-1}(\mathbb B_p^n)$ of the $n$-dimensional $\ell_p$-balls $$ \mathbb B_p^n = \{x\in\mathbb R^n:\ |x_1|^p+\ldots+|x_n|^p\le 1\} $$ and, more generally, of their coordinate-weighted analogues. The formula is given in terms of a one-dimensional integral involving the special function $$ \mathcal F_p(t;\nu) = \int_{\mathbb R}|u|^\nu e^{-|u|^p-t|u|^{2p-2}}\,du. $$ Previously known formulas for the intrinsic volumes of ellipsoids, weighted crosspolytopes, and rectangular boxes arise as special or limiting cases. We also obtain asymptotic formulas for $V_{j(n)}(\mathbb B_p^n)$ in the high-dimensional regime $n\to\infty$, where the index $j(n)$ is allowed to depend on $n$. We further investigate the curvature measures of $\mathbb B_p^n$. These are finite measures $$ \Phi_0(\mathbb B_p^n,\cdot),\dots,\Phi_{n-1}(\mathbb B_p^n,\cdot) $$ on $\partial\mathbb B_p^n$ that localize the intrinsic volumes. We prove a Maxwell--Poincar\'{e}--Borel type limit theorem: if $X_n$ is a random boundary point of $\mathbb B_p^n$ distributed according to the normalized curvature measure $\Phi_{j(n)}(\mathbb B_p^n,\cdot)/V_{j(n)}(\mathbb B_p^n)$, where $j(n)/n\to\alpha\in[0,1]$ as $n\to\infty$, then for every fixed $r\in\mathbb N$, the joint distribution of the first $r$ coordinates of $n^{1/p}X_n$ converges weakly to the product measure $\nu_{p,\alpha}^{\otimes r}$. Here $\nu_{p,\alpha}$ is an explicit probability measure on $\mathbb R$ depending on $p>1$ and $\alpha\in[0,1]$. The main tool underlying these results is an explicit characterization of the curvature measures of coordinate-weighted $\ell_p$-balls, and in particular an explicit formula for their mixed moments.

math.PR↗

arXiv2025

On the distribution patterns of zeros for random polynomials with regularly varying coefficients

This paper investigates asymptotic distribution of complex zeros of random polynomials $P_n(z):=\sum_{k=0}^{n}b(k)\xi_k z^k$, as $n\to\infty$, where $b$ is a regularly varying function at infinity with index $\alpha\in \mathbb{R}$ and $(\xi_k)_{k\geq 0}$ is a sequence of independent copies of a complex-valued random variable $\xi$. The limiting distribution of zeros both inside and outside the unit disk is determined assuming $\mathbb{E}[\log^{+}|\xi|]<\infty$. Under the additional assumptions $\mathbb{E}[\xi]=0$ and $\mathbb{E}[|\xi|^2]<\infty$, local universality results for zeros near the boundary of the unit disk are established. Notably, it is shown that the point process of zeros undergoes a transition from liquid-like to crystalline phases as $\alpha$ crosses the critical value $\alpha_c = -1/2$ from right to left. In the liquid phase ($\alpha > \alpha_c$), the limiting point process of zeros is universal. In the crystalline phase, it is universal if and only if $\alpha = \alpha_c$ and $\sum_k b^2(k) = +\infty$ (the weak crystalline phase), and non-universal when $\sum_k b^2(k) < +\infty$ (the strong crystalline phase). The zeros of the so-called random self-inversive polynomials on the unit circle exhibit a similar phase transition.

math.PR↗

arXiv2025

Zeros and exponential profiles of polynomials II: Examples

In [Jalowy, Kabluchko, Marynych, arXiv:2504.11593v1, 2025], the authors discuss a user-friendly approach to determine the limiting empirical zero distribution of a sequence of real-rooted polynomials, as the degree goes to $\infty$. In this note, we aim to apply it to a vast range of examples of polynomials providing a unifying source for limiting empirical zero distributions. We cover Touchard, Fubini, Eulerian, Narayana and little $q$-Laguerre polynomials as well as hypergeometric polynomials including the classical Hermite, Laguerre and Jacobi polynomials. We construct polynomials whose empirical zero distributions converge to the free multiplicative normal and Poisson distributions. Furthermore, we study polynomials generated by some differential operators. As one inverse result, we derive coefficient asymptotics of the characteristic polynomial of random covariance matrices.

math.CA↗

arXiv2025

On local large deviations for decoupled random walks

A decoupled standard random walk is a sequence of independent random variables $(\hat{S}_n)_{n \geq 1}$ such that, for each $n \geq 1$, the distribution of $\hat{S}_n$ is the same as that of $S_n = \xi_1 + \ldots + \xi_n$, where $(\xi_k)_{k \geq 1}$ are independent copies of a nonnegative random variable $\xi$. We consider the counting process $(\hat{N}(t))_{t\geq 0}$ defined as the number of terms $\hat{S}_n$ in the sequence $(\hat{S}_n)_{n \geq 1}$ that lie within the interval $[0, t]$. Under various assumptions on the tail distribution of $\xi$, we derive logarithmic asymptotics for the local large deviation probabilities $\mathbb{P}\{\hat{N}(t) = \lfloor b \, \mathbb{E}[\hat{N}(t)] \rfloor\}$ as $t \to \infty$ for a fixed constant $b > 0$. These results are then applied to obtain a logarithmic local large deviations asymptotic for the counting process associated with the infinite Ginibre ensemble and, more generally, for determinantal point processes with the Mittag-Leffler kernel.

math.PR↗

arXiv2025

Zeros and exponential profiles of polynomials I: Limit distributions, finite free convolutions and repeated differentiation

Given a sequence of polynomials $(P_n)_{n \in \mathbb{N}}$ with only nonpositive zeros, the aim of this article is to present a user-friendly approach for determining the limiting zero distribution of $P_n$ as $\mathrm{deg}\, P_n \to \infty$. The method is based on establishing an equivalence between the existence of a limiting empirical zero distribution $\mu$ and the existence of an exponential profile $g$ associated with the coefficients of the polynomials $(P_n)_{n \in \mathbb{N}}$. The exponential profile $g$, which can be roughly described by $[z^k]P_n(z) \approx \exp(n g(k/n))$, offers a direct route to computing the Cauchy transform $G$ of $\mu$: the functions $t \mapsto tG(t)$ and $\alpha \mapsto \exp(-g'(\alpha))$ are mutual inverses. This relationship, in various forms, has previously appeared in the literature, most notably in the paper [Van Assche, Fano and Ortolani, SIAM J. Math. Anal., 1987]. As a first contribution, we present a self-contained probabilistic proof of this equivalence by representing the polynomials as generating functions of sums of independent Bernoulli random variables. This probabilistic framework naturally lends itself to tools from large deviation theory, such as the exponential change of measure. The resulting theorems generalize and unify a range of previously known results, which were traditionally established through analytic or combinatorial methods. Secondly, using the profile-based approach, we investigate how the exponential profile and the limiting zero distribution behave under certain operations on polynomials, including finite free convolutions, Hadamard products, and repeated differentiation. In particular, our approach yields new proofs of the convergence results `$\boxplus_n \to \boxplus$' and `$\boxtimes_n \to \boxtimes$', extending them to cases where the distributions are not necessarily compactly supported.

math.PR↗

arXiv2025

Almost periodic stochastic processes with applications to analytic number theory

A classical fact of the theory of almost periodic functions is the existence of their asymptotic distributions. In probabilistic terms, this means that if $f$ is a Besicovitch almost periodic function and $V$ is a random variable uniformly distributed on $[-1,1]$, then the random variables $f(L\cdot V)$ converge in distribution, as $L\to\infty$, to a proper non-degenerate random variable. We prove a functional extension of this result for the random processes $(f(L\cdot V+t))_{t\in\mathbb{R}}$ in the space of Besicovitch almost periodic functions, and also in the sense of weak convergence of finite-dimensional distributions. We further investigate the properties of the limiting stationary process and demonstrate applications in analytic number theory by extending the one-dimensional results of [Limiting distributions of the classical error terms of prime number theory, Quart. J. Math. 65 (2014), 743--780] and earlier works.

math.PR↗

arXiv2024

Generalised convexity with respect to families of affine maps

The standard convex closed hull of a set is defined as the intersection of all images, under the action of a group of rigid motions, of a half-space containing the given set. In this paper we propose a generalisation of this classical notion, that we call a $(K,\mathbb{H})$-hull, and which is obtained from the above construction by replacing a half-space with some other convex closed subset $K$ of the Euclidean space, and a group of rigid motions by a subset $\mathbb{H}$ of the group of invertible affine transformations. The main focus is put on the analysis of $(K,\mathbb{H})$-convex hulls of random samples from $K$.

math.MG↗

arXiv2024

Convolution powers of unbounded measures on the positive half-line

For a right-continuous nondecreasing and unbounded function $V$ of at most exponential growth, which vanishes on the negative halfline, we investigate the asymptotic behavior of the Lebesgue-Stieltjes convolution powers $V^{\ast(j)}(t)$ as both $j$ and $t$ tend to infinity. We obtain a comprehensive asymptotic formula for $V^{\ast(j)}(t)$, which is valid across different regimes of simultaneous growth of $j$ and $t$. Our main technical tool is an exponential change of measure, which is a standard technique in the large deviations theory. Various applications of our result are given.

math.PR↗

arXiv2024

Multinomial random combinatorial structures and $r$-versions of Stirling, Eulerian and Lah numbers

We introduce multinomial and $r$-variants of several classic objects of combinatorial probability, such as the random recursive and Hoppe trees, random set partitions and compositions, the Chinese restaurant process, Feller's coupling, and some others. Just as various classic combinatorial numbers - like Stirling, Eulerian and Lah numbers - emerge as essential ingredients defining the distributions of the mentioned processes, the so-called $r$-versions of these numbers appear in exact distributional formulas for the multinomial and $r$-counterparts. This approach allows us to offer a concise probabilistic interpretation for various identities involving $r$-versions of these combinatorial numbers, which were either unavailable or meaningful only for specific values of the parameter $r$. We analyze the derived distributions for fixed-size structures and establish distributional limit theorems as the size tends to infinity. Utilizing the aforementioned generalized Stirling numbers of both kinds, we define and analyze $(r,s)$-Lah distributions, which have arisen in the existing literature on combinatorial probability in various contexts.

math.PR↗

arXiv2023

Set-valued recursions arising from vantage-point trees

We study vantage-point trees constructed using an independent sample from the uniform distribution on a fixed convex body $K$ in $(\mathbb{R}^d,\|\cdot\|)$, where $\|\cdot\|$ is an arbitrary norm on $\mathbb{R}^d$. We prove that a sequence of sets, associated with the left boundary of a vantage-point tree, forms a recurrent Harris chain on the space of convex bodies in $(\mathbb{R}^d,\|\cdot\|)$. The limiting object is a ball polyhedron, that is, an a.s.~finite intersection of closed balls in $(\mathbb{R}^d,\|\cdot\|)$ of possibly different radii. As a consequence, we derive a limit theorem for the length of the leftmost path of a vantage-point tree.

math.PR↗

arXiv2023

Divisibility properties of polynomial expressions of random integers

We study divisibility properties of a set $\{f_1(\mathbf{U}_n^{(s)}),\ldots,f_m(\mathbf{U}_n^{(s)})\}$, where $f_1,\ldots,f_m$ are polynomials in $s$ variables over $\mathbb{Z}$ and $\mathbf{U}_n^{(s)}$ is a point picked uniformly at random from the set $\{1,\ldots,n\}^s$, $s\in\mathbb{N}$. We show that the ${\rm GCD}$ and the suitably normalized ${\rm LCM}$ of this set converge in distribution to a.s.\ finite random variables under mild assumptions on $f_1,\ldots, f_m$. Our approach is based on the notion of integer adeles and a known fact that the uniform distribution on $\{1,\ldots, n\}$ converges to the Haar measure on the ring of integer adeles combined with the Lang-Weil bounds.

math.NT↗

arXiv2023

Arithmetic properties of multiplicative integer-valued perturbed random walks

Let $(ξ_1, η_1)$, $(ξ_2, η_2),\ldots$ be independent identically distributed $\mathbb{N}^2$-valued random vectors with arbitrarily dependent components. The sequence $(Θ_k)_{k\in\mathbb{N}}$ defined by $Θ_k=Π_{k-1}\cdotη_k$, where $Π_0=1$ and $Π_k=ξ_1\cdot\ldots\cdot ξ_{k}$ for $k\in\mathbb{N}$, is called a multiplicative perturbed random walk. We study arithmetic properties of the random sets $\{Π_1,Π_2,\ldots, Π_k\}\subset \mathbb{N}$ and $\{Θ_1,Θ_2,\ldots, Θ_k\}\subset \mathbb{N}$, $k\in\mathbb{N}$. In particular, we derive distributional limit theorems for their prime counts and for the least common multiple.

math.PR↗

arXiv2023

Solutions of kinetic-type equations with perturbed collisions

We study a class of kinetic-type differential equations $\partial ϕ_t/\partial t+ϕ_t=\widehat{\mathcal{Q}}ϕ_t$, where $\widehat{\mathcal{Q}}$ is an inhomogeneous smoothing transform and, for every $t\geq 0$, $ϕ_t$ is the Fourier--Stieltjes transform of a probability measure. We show that under mild assumptions on $\widehat{\mathcal{Q}}$ the above differential equation possesses a unique solution and represent this solution as the characteristic function of a certain stochastic process associated with the continuous time branching random walk pertaining to $\widehat{\mathcal{Q}}$. Establishing limit theorems for this process allows us to describe asymptotic properties of the solution, as $t\to\infty$.

math.PR↗

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.

math.PR↗

arXiv2023

When does the chaos in the Curie-Weiss model stop to propagate?

We investigate increasing propagation of chaos for the mean-field Ising model of ferromagnetism (also known as the Curie-Weiss model) with $N$ spins at inverse temperature $β>0$ and subject to an external magnetic field of strength $h\in\mathbb{R}$. Using a different proof technique than in [Ben Arous, Zeitouni; 1999] we confirm the well-known propagation of chaos phenomenon: If $k=k(N)=o(N)$ as $N\to\infty$, then the $k$'th marginal distribution of the Gibbs measure converges to a product measure at $β<1$ or $h \neq 0$ and to a mixture of two product measures, if $β>1$ and $h =0$. More importantly, we also show that if $k(N)/N\to α\in (0,1]$, this property is lost and we identify a non-zero limit of the total variation distance between the number of positive spins among any $k$-tuple and the corresponding binomial distribution.

math.PR↗

arXiv2023

Critical branching processes in a sparse random environment

We introduce a branching process in a sparse random environment as an intermediate model between a Galton--Watson process and a branching process in a random environment. In the critical case we investigate the survival probability and prove Yaglom-type limit theorems, that is, limit theorems for the size of population conditioned on the survival event.

math.PR↗