arXiv subjects

Alexander Marynych

Publications and source records attributed to Alexander Marynych.

At least 37 records · Page 2Linked to original sources

arXiv2023

Random Walks in the High-Dimensional Limit II: The Crinkled Subordinator

A crinkled subordinator is an $\ell^2$-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 $d,n\to\infty$.

math.PR↗

arXiv2023

Random Walks in the High-Dimensional Limit I: The Wiener Spiral

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.

math.PR↗

arXiv2023

Mod-$φ$ convergence of Stirling distributions and limit theorems for zeros of their generating functions

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 $n$.

math.PR↗

arXiv2022

Limit theorems for random Dirichlet series

We prove a functional limit theorem in a space of analytic functions for the random Dirichlet series $D(α;z)=\sum_{n\geq 2}(\log n)^α(η_n+{\rm i} θ_n)/n^z$, properly scaled and normalized, where $(η_n,θ_n)_{n\in\mathbb{N}}$ is a sequence of independent copies of a centered $\mathbb{R}^2$-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 $\mathbb{P}\{θ=0\}=1$, we also prove a law of the iterated logarithm for $D(α;z)$, properly normalized, as $z\to (1/2)+$.

math.PR↗

arXiv2022

Limit theorems for discounted convergent perpetuities II

Let $(ξ_1, η_1)$, $(ξ_2, η_2),\ldots$ be independent identically distributed $\mathbb{R}^2$-valued random vectors. Assuming that $ξ_1$ has zero mean and finite variance and imposing three distinct groups of assumptions on the distribution of $η_1$ we prove three functional limit theorems for the logarithm of convergent discounted perpetuities $\sum_{k\geq 0}e^{ξ_1+\ldots+ξ_k-ak}η_{k+1}$ as $a\to 0+$. Also, we prove a law of the iterated logarithm which corresponds to one of the aforementioned functional limit theorems. The present paper continues a line of research initiated in the paper Iksanov, Nikitin and Samoillenko (2022), which focused on limit theorems for a different type of convergent discounted perpetuities.

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arXiv2022

Lah distribution: Stirling numbers, records on compositions, and convex hulls of high-dimensional random walks

Let $ξ_1,ξ_2,\ldots$ be a sequence of independent copies of a random vector in $\mathbb R^d$ having an absolutely continuous distribution. Consider a random walk $S_i:=ξ_1+\cdots+ξ_i$, and let $C_{n,d}:=\text{conv}(0,S_1,S_2,\ldots,S_n)$ be the convex hull of the first $n+1$ points it has visited. The polytope $C_{n,d}$ is called $k$-neighborly if for every indices $0\leq i_0 <\cdots < i_k\leq n$ the convex hull of the $k+1$ points $S_{i_0},\ldots, S_{i_k}$ is a $k$-dimensional face of $C_{n,d}$. We study the probability that $C_{n,d}$ is $k$-neighborly in various high-dimensional asymptotic regimes, i.e. when $n$, $d$, and possibly also $k$ diverge to $\infty$. There is an explicit formula for the expected number of $k$-dimensional faces of $C_{n,d}$ which involves Stirling numbers of both kinds. Motivated by this formula, we introduce a distribution, called the Lah distribution, and study its properties. In particular, we provide a combinatorial interpretation of the Lah distribution in terms of random compositions and records, and explicitly compute its factorial moments. Limit theorems which we prove for the Lah distribution imply neighborliness properties of $C_{n,d}$. This yields a new class of random polytopes exhibiting phase transitions parallel to those discovered by Vershik and Sporyshev, Donoho and Tanner for random projections of regular simplices and crosspolytopes.

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arXiv2022

Asymptotics of arithmetic functions of GCD and LCM of random integers in hyperbolic regions

We prove limit theorems for the greatest common divisor and the least common multiple of random integers. While the case of integers uniformly distributed on a hypercube with growing size is classical, we look at the uniform distribution on sublevel sets of multivariate symmetric polynomials, which we call hyperbolic regions. Along the way of deriving our main results, we obtain some asymptotic estimates for the number of integer points in these hyperbolic domains, when their size goes to infinity.

math.NT↗

arXiv2022

Stable fluctuations of iterated perturbed random walks in intermediate generations of a general branching process tree

Consider a general branching process, a.k.a. Crump-Mode-Jagers process, generated by a perturbed random walk $η_1$, $ξ_1+η_2$, $ξ_1+ξ_2+η_3,\ldots$. Here, $(ξ_1,η_1)$, $(ξ_2, η_2),\ldots$ are independent identically distributed random vectors with arbitrarily dependent positive components. Denote by $N_j(t)$ the number of the $j$th generation individuals with birth times $\leq t$. Assume that $j=j(t)\to\infty$ and $j(t)=o(t^a)$ as $t\to\infty$ for some explicitly given $a>0$ (to be specified in the paper). The corresponding $j$th generation belongs to the set of intermediate generations. We provide sufficient conditions under which finite-dimensional distributions of the process $(N_{\lfloor j(t)u\rfloor}(t))_{u>0}$, properly normalized and centered, converge weakly to those of an integral functional of a stable Lévy process with finite mean.

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arXiv2021

Facial structure of strongly convex sets generated by random samples

The $K$-hull of a compact set $A\subset\mathbb{R}^d$, where $K\subset \mathbb{R}^d$ is a fixed compact convex body, is the intersection of all translates of $K$ that contain $A$. A set is called $K$-strongly convex if it coincides with its $K$-hull. We propose a general approach to the analysis of facial structure of $K$-strongly convex sets, similar to the well developed theory for polytopes, by introducing the notion of $k$-dimensional faces, for all $k=0,\dots,d-1$. We then apply our theory in the case when $A=Ξ_n$ is a sample of $n$ points picked uniformly at random from $K$. We show that in this case the set of $x\in\mathbb{R}^d$ such that $x+K$ contains the sample $Ξ_n$, upon multiplying by $n$, converges in distribution to the zero cell of a certain Poisson hyperplane tessellation. From this results we deduce convergence in distribution of the corresponding $f$-vector of the $K$-hull of $Ξ_n$ to a certain limiting random vector, without any normalisation, and also the convergence of all moments of the $f$-vector.

math.MG↗

arXiv2020

Moderate parts in regenerative compositions: the case of regular variation

A regenerative random composition of integer $n$ is constructed by allocating $n$ standard exponential points over a countable number of intervals, comprising the complement of the closed range of a subordinator $S$. Assuming that the Lévy measure of $S$ is infinite and regularly varying at zero of index $-α$, $α\in(0,\,1)$, we find an explicit threshold $r=r(n)$, such that the number $K_{n,\,r(n)}$ of blocks of size $r(n)$ converges in distribution without any normalization to a mixed Poisson distribution. The sequence $(r(n))$ turns out to be regularly varying with index $α/(α+1)$ and the mixing distribution is that of the exponential functional of $S$. The result is derived as a consequence of a general Poisson limit theorem for an infinite occupancy scheme with power-like decay of the frequencies. We also discuss asymptotic behavior of $K_{n,\,w(n)}$ in cases when $w(n)$ diverges but grows slower than $r(n)$. Our findings complement previously known strong laws of large numbers for $K_{n,\,r}$ in case of a fixed $r\in\mathbb{N}$. As a key tool we employ new Abelian theorems for Laplace--Stiletjes transforms of regularly varying functions with the indexes of regular variation diverging to infinity.

math.PR↗

arXiv2020

Renewal theory for iterated perturbed random walks on a general branching process tree: intermediate generations

Let $(ξ_k,η_k)_{k\in\mathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $(T_k)_{k\in\mathbb{N}}$ defined by $T_k:=ξ_1+\cdots+ξ_{k-1}+η_k$ for $k\in\mathbb{N}$. Further, by an iterated perturbed random walk is meant the sequence of point processes defining the birth times of individuals in subsequent generations of a general branching process provided that the birth times of the first generation individuals are given by a perturbed random walk. For $j\in\mathbb{N}$ and $t\geq 0$, denote by $N_j(t)$ the number of the $j$th generation individuals with birth times $\leq t$. In this article we prove counterparts of the classical renewal-theoretic results (the elementary renewal theorem, Blackwell's theorem and the key renewal theorem) for $N_j(t)$ under the assumption that $j=j(t)\to\infty$ and $j(t)=o(t^{2/3})$ as $t\to\infty$. According to our terminology, such generations form a subset of the set of intermediate generations.

math.PR↗

arXiv2020

On intermediate levels of nested occupancy scheme in random environment generated by stick-breaking II

A nested occupancy scheme in random environment is a generalization of the classical Karlin infinite balls-in-boxes occupancy scheme in random environment (with random probabilities). Unlike the Karlin scheme in which the collection of boxes is unique, there is a nested hierarchy of boxes, and the hitting probabilities of boxes are defined in terms of iterated fragmentation of a unit mass. In the present paper we assume that the random fragmentation law is given by stick-breaking in which case the infinite occupancy scheme defined by the first level boxes is known as the Bernoulli sieve. Assuming that $n$ balls have been thrown, denote by $K_n(j)$ the number of occupied boxes in the $j$th level and call the level $j$ intermediate if $j=j_n\to\infty$ and $j_n=o(\log n)$ as $n\to\infty$. We prove a multidimensional central limit theorem for the vector $(K_n(\lfloor j_n u_1\rfloor),\ldots, K_n(\lfloor j_n u_\ell\rfloor)$, properly normalized and centered, as $n\to\infty$, where $j_n\to\infty$ and $j_n=o((\log n)^{1/2})$. The present paper continues the line of investigation initiated in Buraczewski, Dovgay and Iksanov [Electron. J. Probab. 25: paper no. 123, 2020] in which the occupancy of intermediate levels $j_n\to\infty$, $j_n=o((\log n)^{1/3})$ was analyzed.

math.PR↗

arXiv2020

How long is the convex minorant of a one-dimensional random walk?

We prove distributional limit theorems for the length of the largest convex minorant of a one-dimensional random walk with independent identically distributed increments. Depending on the increment law, there are several regimes with different limit distributions for this length. Among other tools, a representation of the convex minorant of a random walk in terms of uniform random permutations is utilized.

math.PR↗

arXiv2020

A fundamental problem of hypothesis testing with finite inventory in e-commerce

In this paper, we draw attention to a problem that is often overlooked or ignored by companies practicing hypothesis testing (A/B testing) in online environments. We show that conducting experiments on limited inventory that is shared between variants in the experiment can lead to high false positive rates since the core assumption of independence between the groups is violated. We provide a detailed analysis of the problem in a simplified setting whose parameters are informed by realistic scenarios. The setting we consider is a $2$-dimensional random walk in a semi-infinite strip. It is rich enough to take a finite inventory into account, but is at the same time simple enough to allow for a closed form of the false-positive probability. We prove that high false-positive rates can occur, and develop tools that are suitable to help design adequate tests in follow-up work. Our results also show that high false-negative rates may occur. The proofs rely on a functional limit theorem for the $2$-dimensional random walk in a semi-infinite strip.

math.PR↗

arXiv2020

A Brownian weak limit for the least common multiple of a random m-tuple of integers

Let $B_n(m)$ be a set picked uniformly at random among all $m$-elements subsets of $\{1,2,\ldots,n\}$. We provide a pathwise construction of the collection $(B_n(m))_{1\leq m\leq n}$ and prove that the logarithm of the least common multiple of the integers in $(B_n(\lfloor mt\rfloor))_{t\geq 0}$, properly centered and normalized, converges to a Brownian motion when both $m,n$ tend to infinity. Our approach consists of two steps. First, we show that the aforementioned result is a consequence of a multidimensional central limit theorem for the logarithm of the least common multiple of $m$ independent random variables having uniform distribution on $\{1,2,\ldots,n\}$. Second, we offer a novel approximation of the least common multiple of a random sample by the product of the elements of the sample with neglected multiplicities in their prime decompositions.

math.PR↗

arXiv2020

The laws of iterated and triple logarithms for extreme values of regenerative processes

We analyze almost sure asymptotic behavior of extreme values of a regenerative process. We show that under certain conditions a properly centered and normalized running maximum of a regenerative process satisfies a law of the iterated logarithm for the $\limsup$ and a law of the triple logarithm for the $\liminf$. This complements a previously known result of Glasserman and Kou [Ann. Appl. Probab. 5(2) (1995), 424--445]. We apply our results to several queuing systems and a birth and death process.

math.PR↗

arXiv2020

Sieving random iterative function systems

It is known that backward iterations of independent copies of a contractive random Lipschitz function converge almost surely under mild assumptions. By a sieving (or thinning) procedure based on adding to the functions time and space components, it is possible to construct a scale invariant stochastic process. We study its distribution and paths properties. In particular, we show that it is càdlàg and has finite total variation. We also provide examples and analyse various properties of particular sieved iterative function systems including perpetuities and infinite Bernoulli convolutions, iterations of maximum, and random continued fractions.

math.PR↗

arXiv2019

On the least common multiple of several random integers

Let $L_n(k)$ denote the least common multiple of $k$ independent random integers uniformly chosen in $\{1,2,\ldots ,n\}$. In this note, using a purely probabilistic approach, we derive a criterion for the convergence in distribution as $n\to\infty$ of $\frac{f(L_n(k))}{n^{rk}}$ for a wide class of multiplicative arithmetic functions~$f$ with polynomial growth $r>-1$. Furthermore, we identify the limit as an infinite product of independent random variables indexed by prime numbers. Along the way, we compute the generating function of a trimmed sum of independent geometric laws, occurring in the above infinite product. This generating function is rational; we relate it to the generating function of a certain max-type Diophantine equation, of which we solve a generalized version. Our results extend theorems by Erdős and Wintner (1939), Fernández and Fernández (2013) and Hilberdink and Tóth (2016).

math.PR↗