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.

For p>1, we derive explicit formulas for the intrinsic volumes V_0(𝔹_p^n),…,V_n-1(𝔹_p^n) of the n-dimensional ℓ_p-balls 𝔹_p^n = {x∈ℝ^n: |x_1|^p+…+|x_n|^p≤ 1} and, more generally, of their coordinate-weighted analogues. The formula is given in terms of a one-dimensional integral involving the special function ℱ_p(t;ν) = ∫_ℝ|u|^νe^-|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)(𝔹_p^n) in the high-dimensional regime n→∞, where the index j(n) is allowed to depend on n. We further investigate the curvature measures of 𝔹_p^n. These are finite measures Φ_0(𝔹_p^n,·),…,Φ_n-1(𝔹_p^n,·) on ∂𝔹_p^n that localize the intrinsic volumes. We prove a Maxwell–Poincaré–Borel type limit theorem: if X_n is a random boundary point of 𝔹_p^n distributed according to the normalized curvature measure Φ_j(n)(𝔹_p^n,·)/V_j(n)(𝔹_p^n), where j(n)/n∈[0,1] as n→∞, then for every fixed r∈ℕ, the joint distribution of the first r coordinates of n^1/pX_n converges weakly to the product measure ν_p,α^⊗ r. Here ν_p,α is an explicit probability measure on ℝ depending on p>1 and α∈[0,1]. The main tool underlying these results is an explicit characterization of the curvature measures of coordinate-weighted ℓ_p-balls, and in particular an explicit formula for their mixed moments.

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.

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.

For a right-continuous nondecreasing and unbounded function V of at most exponential growth, which vanishes on the negative half-line, we investigate the asymptotic behavior of the Lebesgue-Stieltjes convolution powers V & lowast;( j ) ( t ) as both j and t tend to infinity. We obtain a comprehensive asymptotic formula for V & lowast;( 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. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The standard closed convex 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, ℍ)-hull, and which is obtained from the above construction by replacing a half-space with some other closed convex subset K of the Euclidean space, and a group of rigid motions by a subset ℍ of the group of invertible affine transformations. The main focus is on the analysis of (K, ℍ)-convex hulls of random samples from K.

Given a sequence of polynomials (P_n)_n ∈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 deg P_n →∞. The method is based on establishing an equivalence between the existence of a limiting empirical zero distribution μ and the existence of an exponential profile g associated with the coefficients of the polynomials (P_n)_n ∈ℕ. The exponential profile g, which can be roughly described by [z^k]P_n(z) ≈exp(n g(k/n)), offers a direct route to computing the Cauchy transform G of μ: the functions t ↦ tG(t) and α↦exp(-g'(α)) 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 `⊞_n →⊞' and `⊠_n →⊠', extending them to cases where the distributions are not necessarily compactly supported.

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 f is a Besicovitch almost periodic function and V V is a random variable uniformly distributed on [ − 1 , 1 ] [-1,1] , then the random variables f ( L ⋅ V ) f(L\cdot V) converge in distribution, as L → ∞ L\to \infty , to a proper non-degenerate random variable. We prove a functional extension of this result for the random processes ( f ( L ⋅ V + t ) ) t ∈ R (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. Further we investigate the properties of the limiting stationary process and demonstrate applications in analytic number theory by extending the one-dimensional results of Akbary, Ng and Shahabi (2014) and earlier works.

A decoupled standard random walk is a sequence of independent random variables (Ŝ_n)_n ≥ 1 such that, for each n ≥ 1, the distribution of Ŝ_n is the same as that of S_n = ξ_1 + … + ξ_n, where (ξ_k)_k ≥ 1 are independent copies of a nonnegative random variable ξ. We consider the counting process (N̂(t))_t≥ 0 defined as the number of terms Ŝ_n in the sequence (Ŝ_n)_n ≥ 1 that lie within the interval [0, t]. Under various assumptions on the tail distribution of ξ, we derive logarithmic asymptotics for the local large deviation probabilities ℙ{N̂(t) = ⌊ b 𝔼[N̂(t)] ⌋} as t →∞ 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.

We study vantage-point trees constructed using an independent sample from the uniform distribution on a fixed convex body K in (ℝ^d,‖·‖ ) , where ‖·‖ is an arbitrary norm on 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 (ℝ^d,‖·‖ ) . The limiting object is a ball polyhedron, that is, an a.s. finite intersection of closed balls in (ℝ^d,‖·‖ ) of possibly different radii. As a consequence, we derive a limit theorem for the length of the leftmost path of a vantage-point tree.

This paper investigates asymptotic distribution of complex zeros of random polynomials P_n(z):=∑_k=0^nb(k)ξ_k z^k, as n→∞, where b is a regularly varying function at infinity with index α∈ℝ and (ξ_k)_k≥ 0 is a sequence of independent copies of a complex-valued random variable ξ. The limiting distribution of zeros both inside and outside the unit disk is determined assuming 𝔼[log^+|ξ|]<∞. Under the additional assumptions 𝔼[ξ]=0 and 𝔼[|ξ|^2]<∞, 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 α crosses the critical value α_c = -1/2 from right to left. In the liquid phase (α> α_c), the limiting point process of zeros is universal. In the crystalline phase, it is universal if and only if α= α_c and ∑_k b^2(k) = +∞ (the weak crystalline phase), and non-universal when ∑_k b^2(k) < +∞ (the strong crystalline phase). The zeros of the so-called random self-inversive polynomials on the unit circle exhibit a similar phase transition.

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 ∞. 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.

Let $({\xi _{1}},{\eta _{1}})$, $({\xi _{2}},{\eta _{2}}),\dots $ be independent identically distributed ${\mathbb{N}^{2}}$-valued random vectors with arbitrarily dependent components. The sequence ${({\Theta _{k}})_{k\in \mathbb{N}}}$ defined by ${\Theta _{k}}={\Pi _{k-1}}\cdot {\eta _{k}}$, where ${\Pi _{0}}=1$ and ${\Pi _{k}}={\xi _{1}}\cdot \dots \cdot {\xi _{k}}$ for $k\in \mathbb{N}$, is called a multiplicative perturbed random walk. Arithmetic properties of the random sets $\{{\Pi _{1}},{\Pi _{2}},\dots ,{\Pi _{k}}\}\subset \mathbb{N}$ and $\{{\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{k}}\}\subset \mathbb{N}$, $k\in \mathbb{N}$, are studied. In particular, distributional limit theorems for their prime counts and for the least common multiple are derived.

We study mod-φ convergence of several probability distributions on the set of positive integers that involve Stirling numbers of both kinds and, as a consequence, derive various limit theorems for these distributions. We also derive closely related limit theorems for the distribution of zeros of the corresponding generating functions. For example, we identify the asymptotic distribution of zeros for the generating polynomial of the number of occupied boxes when n balls are allocated equiprobably and independently among θ boxes in the regime when θ grows linearly with n.

We study divisibility properties of a set {f1(Un(s)),…,fm(Un(s))}, where f1,…,fm are polynomials in s variables over Z and Un(s) is a point picked uniformly at random from the set {1,…,n}s. We show that, as n→∞, the GCD and the suitably normalized LCM of this set converge in distribution to a.s. finite random variables under mild assumptions on f1,…,fm. Our approach is based on the known fact that the uniform distribution on {1,…,n} converges to the Haar measure on the ring Zˆ of profinite integers, combined with the Lang–Weil bounds and tools from probability theory.

We prove limit theorems for random walks with $n$ steps in the $d$-dimensional Euclidean space as both $n$ and $d$ tend to infinity. One of our results states that the path of such a random walk, viewed as a compact subset of the infinite-dimensional Hilbert space $\ell^2$, converges in probability in the Hausdorff distance up to isometry and also in the Gromov-Hausdorff sense to the Wiener spiral, as $d,n\to\infty$. Another group of results describes various possible limit distributions for the squared distance between the random walker at time $n$ and the origin.

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.

A crinkled subordinator is an ℓ2-valued random process which can be thought of as a version of the usual one-dimensional subordinator with each out of countably many jumps being in a direction orthogonal to the directions of all other jumps. We show that the path of a d-dimensional random walk with n independent identically distributed steps with heavy-tailed distribution of the radial components and asymptotically orthogonal angular components converges in distribution in the Hausdorff distance up to isometry and also in the Gromov–Hausdorff sense, if viewed as a random metric space, to the closed range of a crinkled subordinator, as d,n→∞.

We prove a functional limit theorem in a space of analytic functions for the random Dirichlet series D(α;z)=∑n≥2(logn)α(ηn+iθn)/nz, properly scaled and normalized, where (ηn,θn)n∈N is a sequence of independent copies of a centered R2-valued random vector (η,θ) with a finite second moment and α>−1/2 is a fixed real parameter. As a consequence, we show that the point processes of complex and real zeros of D(α;z) converge vaguely, thereby obtaining a universality result. In the real case, that is, when P{θ=0}=1, we also prove a law of the iterated logarithm for D(α;z), properly normalized, as z→(1/2)+.