arXiv subjects

Alexander Marynych

Publications and source records attributed to Alexander Marynych.

At least 55 records · Page 3Linked to original sources

arXiv2019

Random walks in a strongly sparse random environment

The integer points (sites) of the real line are marked by the positions of a standard random walk. We say that the set of marked sites is weakly, moderately or strongly sparse depending on whether the jumps of the standard random walk are supported by a bounded set, have finite or infinite mean, respectively. Focussing on the case of strong sparsity we consider a nearest neighbor random walk on the set of integers having jumps $\pm 1$ with probability $1/2$ at every nonmarked site, whereas a random drift is imposed at every marked site. We prove new distributional limit theorems for the so defined random walk in a strongly sparse random environment, thereby complementing results obtained recently in Buraczewski et al. (2018+) for the case of moderate sparsity and in Matzavinos et al. (2016) for the case of weak sparsity. While the random walk in a strongly sparse random environment exhibits either the diffusive scaling inherent to a simple symmetric random walk or a wide range of subdiffusive scalings, the corresponding limit distributions are non-stable.

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arXiv2019

Self-similar solutions of kinetic-type equations: the boundary case

For a time dependent family of probability measures $(ρ_t)_{t\ge 0}$ we consider a kinetic-type evolution equation $\partial ϕ_t/\partial t + ϕ_t = \widehat{Q} ϕ_t$ where $\widehat{Q}$ is a smoothing transform and $ϕ_t$ is the Fourier--Stieltjes transform of $ρ_t$. Assuming that the initial measure $ρ_0$ belongs to the domain of attraction of a stable law, we describe asymptotic properties of $ρ_t$, as $t\to\infty$. We consider the critical regime when the standard normalization leads to a degenerate limit and find an appropriate scaling ensuring a non-degenerate self-similar limit. Our approach is based on a probabilistic representation of probability measures $(ρ_t)_{t\ge 0}$ that refines the corresponding construction proposed in Bassetti and Ladelli [Ann. Appl. Probab. 22(5): 1928--1961, 2012].

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arXiv2019

Cones generated by random points on half-spheres and convex hulls of Poisson point processes

Let $U_1,U_2,\ldots$ be random points sampled uniformly and independently from the $d$-dimensional upper half-sphere. We show that, as $n\to\infty$, the $f$-vector of the $(d+1)$-dimensional convex cone $C_n$ generated by $U_1,\ldots,U_n$ weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the $f$-vector of $C_n$ and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of $C_n$ can be expressed through the expected $f$-vector. This yields convergence of expected Grassmann angles and conic intrinsic volumes and answers thereby a question of Bárány, Hug, Reitzner and Schneider [Random points in halfspheres, Rand. Struct. Alg., 2017]. Our approach is based on the observation that the random cone $C_n$ weakly converges, after a suitable rescaling, to a random cone whose intersection with the tangent hyperplane of the half-sphere at its north pole is the convex hull of the Poisson point process with power-law intensity function proportional to $\|x\|^{-(d+γ)}$, where $γ=1$. We compute the expected number of facets, the expected intrinsic volumes and the expected $T$-functional of this random convex hull for arbitrary $γ>0$.

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arXiv2018

Stable limit laws for random walk in a sparse random environment I: moderate sparsity

A random walk in a sparse random environment is a model introduced by Matzavinos et al. [Electron. J. Probab. 21, paper no. 72: 2016] as a generalization of both a simple symmetric random walk and a classical random walk in a random environment. A random walk $(X_n)_{n\in \mathbb{N}\cup\{0\}}$ in a sparse random environment $(S_k,λ_k)_{k\in\mathbb{Z}}$ is a nearest neighbor random walk on $\mathbb{Z}$ that jumps to the left or to the right with probability $1/2$ from every point of $\mathbb{Z}\setminus \{\ldots,S_{-1},S_0=0,S_1,\ldots\}$ and jumps to the right (left) with the random probability $λ_{k+1}$ ($1-λ_{k+1}$) from the point $S_k$, $k\in\mathbb{Z}$. Assuming that $(S_k-S_{k-1},λ_k)_{k\in\mathbb{Z}}$ are independent copies of a random vector $(ξ,λ)\in \mathbb{N}\times (0,1)$ and the mean $\mathbb{E}ξ$ is finite (moderate sparsity) we obtain stable limit laws for $X_n$, properly normalized and centered, as $n\to\infty$. While the case $ξ\leq M$ a.s.\ for some deterministic $M>0$ (weak sparsity) was analyzed by Matzavinos et al., the case $\mathbb{E} ξ=\infty$ (strong sparsity) will be analyzed in a forthcoming paper.

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arXiv2018

On perpetuities with gamma-like tails

An infinite convergent sum of independent and identically distributed random variables discounted by a multiplicative random walk is called perpetuity, because of a possible actuarial application. We give three disjoint groups of sufficient conditions which ensure that the distribution right tail of a perpetuity $\mathbb{P}\{X>x\}$ is asymptotic to $ax^ce^{-bx}$ as $x\to\infty$ for some $a,b>0$ and $c\in\mathbb{R}$. Our results complement those of Denisov and Zwart [J. Appl. Probab. 44 (2007), 1031--1046]. As an auxiliary tool we provide criteria for the finiteness of the one-sided exponential moments of perpetuities. Several examples are given in which the distributions of perpetuities are explicitly identified.

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arXiv2018

Limit theorems for the least common multiple of a random set of integers

Let $L_{n}$ be the least common multiple of a random set of integers obtained from $\{1,\ldots,n\}$ by retaining each element with probability $θ\in (0,1)$ independently of the others. We prove that the process $(\log L_{\lfloor nt\rfloor})_{t\in [0,1]}$, after centering and normalization, converges weakly to a certain Gaussian process that is not Brownian motion. Further results include a strong law of large numbers for $\log L_{n}$ as well as Poisson limit theorems in regimes when $θ$ depends on $n$ in an appropriate way.

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arXiv2017

General Edgeworth expansions with applications to profiles of random trees

We prove an asymptotic Edgeworth expansion for the profiles of certain random trees including binary search trees, random recursive trees and plane-oriented random trees, as the size of the tree goes to infinity. All these models can be seen as special cases of the one-split branching random walk for which we also provide an Edgeworth expansion. These expansions lead to new results on mode, width and occupation numbers of the trees, settling several open problems raised in Devroye and Hwang [Ann. Appl. Probab. 16(2): 886--918, 2006], Fuchs, Hwang and Neininger [Algorithmica, 46 (3--4): 367--407, 2006], and Drmota and Hwang [Adv. in Appl. Probab., 37 (2): 321--341, 2005]. The aforementioned results are special cases and corollaries of a general theorem: an Edgeworth expansion for an arbitrary sequence of random or deterministic functions $\mathbb L_n:\mathbb Z\to\mathbb R$ which converges in the mod-$ϕ$-sense. Applications to Stirling numbers of the first kind will be given in a separate paper.

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arXiv2017

The collision spectrum of $Λ$-coalescents

$Λ$-coalescents model the evolution of a coalescing system in which any number of blocks randomly sampled from the whole may merge into a larger block. For the coalescent restricted to initially $n$ singletons we study the collision spectrum $(X_{n,k}:2\le k\le n)$, where $X_{n,k}$ counts, throughout the history of the process, the number of collisions involving exactly $k$ blocks. Our focus is on the large $n$ asymptotics of the joint distribution of the $X_{n,k}$'s, as well as on functional limits for the bulk of the spectrum for simple coalescents. Similarly to the previous studies of the total number of collisions, the asymptotics of the collision spectrum largely depends on the behaviour of the measure $Λ$ in the vicinity of $0$. In particular, for beta$(a,b)$-coalescents different types of limit distributions occur depending on whether $0 2$.

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arXiv2017

A functional limit theorem for random processes with immigration in the case of heavy tails

Let $(X_k,ξ_k)_{k\in \mathbb {N}}$ be a sequence of independent copies of a pair $(X,ξ)$ where $X$ is a random process with paths in the Skorokhod space $D[0,\infty)$ and $ξ$ is a positive random variable. The random process with immigration $(Y(u))_{u\in \mathbb {R}}$ is defined as the a.s. finite sum $Y(u)=\sum_{k\geq0}X_{k+1}(u- ξ_1-\cdots-ξ_k)1\mkern-4.5mu\mathrm{l}_{\{ξ_1+\cdots+ξ_k\leq u\}}$. We obtain a functional limit theorem for the process $(Y(ut))_{u\geq 0}$, as $t\to\infty$, when the law of $ξ$ belongs to the domain of attraction of an $α$-stable law with $α\in(0,1)$, and the process $X$ oscillates moderately around its mean $\mathbb{E}[X(t)]$. In this situation the process $(Y(ut))_{u\geq0}$, when scaled appropriately, converges weakly in the Skorokhod space $D(0,\infty)$ to a fractionally integrated inverse stable subordinator.

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arXiv2016

A leader-election procedure using records

The study of the number of collisions in a Poisson-Dirichlet coalescent leads to the analysis of the following version of a stochastic leader-elec\-tion algorithm. Consider an infinite family of persons, labeled by $1,2,3,\ldots$, who generate iid random numbers from an arbitrary continuous distribution. Those persons who have generated a record value, that is, a value larger than the values of all previous persons, stay in the game, all others must leave. The remaining persons are relabeled by $1,2,3,\ldots$ maintaining their order in the first round, and the election procedure is repeated independently from the past and indefinitely. We prove limit theorems for a number of relevant functionals for this procedure, notably the number of rounds $T(M)$ until all persons among $1,\ldots,M$, except the first one, have left (as $M\to\infty$). For example, we show that the sequence $(T(M)-\log^{*}M)_{M\in\mathbb{N}}$, where $\log^{*}$ denotes the iterated logarithm, is tight, and study its weak subsequential limits. We further provide an appropriate and apparently new kind of normalization (based on tetrations) such that the original labels of persons who stay in the game until round $n$ converge (as $n\to\infty$) to some random non-Poissonian point process and study its properties. The results are applied to study subsequential distributional limits for the number of collisions in the Poisson-Dirichlet coalescent.

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arXiv2016

Leader election using random walks

In the classical leader election procedure all players toss coins independently and those who get tails leave the game, while those who get heads move to the next round where the procedure is repeated. We investigate a generalizion of this procedure in which the labels (positions) of the players who remain in the game are determined using an integer-valued random walk. We study the asymptotics of some relevant quantities for this model such as: the positions of the persons who remained after $n$ rounds; the total number of rounds until all the persons among $1,2,\ldots,M$ leave the game; and the number of players among $1,2,\ldots,M$ who survived the first $n$ rounds. Our results lead to some interesting connection with Galton-Watson branching processes and with the solutions of certain stochastic-fixed point equations arising in the context of the stability of point processes under thinning. We describe the set of solutions to these equations and thus provide a characterization of one-dimensional point processes that are stable with respect to thinning by integer-valued random walks.

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arXiv2016

Mode and Edgeworth expansion for the Ewens distribution and the Stirling numbers

We provide asymptotic expansions for the Stirling numbers of the first kind and, more generally, the Ewens (or Karamata-Stirling) distribution. Based on these expansions, we obtain some new results on the asymptotic properties of the mode and the maximum of the Stirling numbers and the Ewens distribution. For arbitrary $θ>0$ and for all sufficiently large $n\in\mathbb N$, the unique maximum of the Ewens probability mass function $$ \mathbb L_n(k) = \frac{θ^k}{θ(θ+1)\ldots(θ+n-1)} \genfrac{[}{]}{0pt}{}{n}{k}, \quad k=1,\ldots,n, $$ is attained at $k= \left\lfloor θ\log n + \frac{θΓ'(θ)}{Γ(θ)} - \frac 12\right\rfloor$ or $k=\left\lceil θ\log n + \frac{θΓ'(θ)}{Γ(θ)} + \frac 12\right\rceil$. We prove that the mode is $$ k=\left\lfloor θ\log n - \frac{θΓ'(θ)}{Γ(θ)}\right\rfloor $$ for a set of $n$'s of asymptotic density $1$, yet this formula is not true for infinitely many $n$'s.

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arXiv2016

Local universality for real roots of random trigonometric polynomials

Consider a random trigonometric polynomial $X_n: \mathbb R \to \mathbb R$ of the form $$ X_n(t) = \sum_{k=1}^n \left( ξ_k \sin (kt) + η_k \cos (kt)\right), $$ where $(ξ_1,η_1),(ξ_2,η_2),\ldots$ are independent identically distributed bivariate real random vectors with zero mean and unit covariance matrix. Let $(s_n)_{n\in\mathbb N}$ be any sequence of real numbers. We prove that as $n\to\infty$, the number of real zeros of $X_n$ in the interval $[s_n+a/n, s_n+ b/n]$ converges in distribution to the number of zeros in the interval $[a,b]$ of a stationary, zero-mean Gaussian process with correlation function $(\sin t)/t$. We also establish similar local universality results for the centered random vectors $(ξ_k,η_k)$ having an arbitrary covariance matrix or belonging to the domain of attraction of a two-dimensional $α$-stable law.

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arXiv2016

Renewal shot noise processes in the case of slowly varying tails

We investigate weak convergence of renewal shot noise processes in the case of slowly varying tails of the inter-shot times. We show that these processes, after an appropriate non-linear scaling, converge in the sense of finite-dimensional distributions to an inverse extremal process.

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arXiv2016

Moment convergence of first-passage times in renewal theory

Let $ξ_1, ξ_2, \ldots$ be independent copies of a positive random variable $ξ$, $S_0 = 0$, and $S_k = ξ_1+\ldots+ξ_k$, $k \in \mathbb{N}$. Define $N(t) = \inf\{k \in \mathbb{N}: S_k>t\}$ for $t\geq 0$. The process $(N(t))_{t\geq 0}$ is the first-passage time process associated with $(S_k)_{k\geq 0}$. It is known that if the law of $ξ$ belongs to the domain of attraction of a stable law or $\mathbb{P}(ξ>t)$ varies slowly at $\infty$, then $N(t)$, suitably shifted and scaled, converges in distribution as $t \to \infty$ to a random variable $W$ with a stable law or a Mittag-Leffler law. We investigate whether there is convergence of the power and exponential moments to the corresponding moments of $W$. Further, the analogous problem for first-passage times of subordinators is considered.

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arXiv2016

Weak convergence of renewal shot noise processes in the case of slowly varying normalization

We investigate weak convergence of finite-dimensional distributions of a renewal shot noise process $(Y(t))_{t\geq 0}$ with deterministic response function $h$ and the shots occurring at the times $0 = S_0 < S_1 < S_2<\ldots$, where $(S_n)$ is a random walk with i.i.d.\ jumps. There has been an outbreak of recent activity around this topic. We are interested in one out of few cases which remained open: $h$ is regularly varying at $\infty$ of index $-1/2$ and the integral of $h^2$ is infinite. Assuming that $S_1$ has a moment of order $r>2$ we use a strong approximation argument to show that the random fluctuations of $Y(s)$ occur on the scale $s=t+g(t,u)$ for $u\in [0,1]$, as $t\to\infty$, and, on the level of finite-dimensional distributions, are well approximated by the sum of a Brownian motion and a Gaussian process with independent values (the two processes being independent). The scaling function $g$ above depends on the slowly varying factor of $h$. If, for instance, $\lim_{t\to\infty}t^{1/2}h(t)\in (0,\infty)$, then $g(t,u)=t^u$.

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arXiv2016

Fractionally integrated inverse stable subordinators

A fractionally integrated inverse stable subordinator (FIISS) is the convolution of a power function and an inverse stable subordinator. We show that the FIISS is a scaling limit in the Skorokhod space of a renewal shot noise process with heavy-tailed, infinite mean `inter-shot' distribution and regularly varying response function. We prove local Hölder continuity of FIISS and a law of iterated logarithm for both small and large times.

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arXiv2016

Functional limit theorems for the number of occupied boxes in the Bernoulli sieve

The Bernoulli sieve is the infinite Karlin "balls-in-boxes" scheme with random probabilities of stick-breaking type. Assuming that the number of placed balls equals $n$, we prove several functional limit theorems (FLTs) in the Skorohod space $D[0,1]$ endowed with the $J_{1}$- or $M_{1}$-topology for the number $K_{n}^{*}(t)$ of boxes containing at most $[n^{t}]$ balls, $t\in[0,1]$, and the random distribution function $K_{n}^{*}(t)/K_{n}^{*}(1)$, as $n\to\infty$. The limit processes for $K_{n}^{*}(t)$ are of the form $(X(1)-X((1-t)-))_{t\in[0,1]}$, where $X$ is either a Brownian motion, a spectrally negative stable Lévy process, or an inverse stable subordinator. The small values probabilities for the stick-breaking factor determine which of the alternatives occurs. If the logarithm of this factor is integrable, the limit process for $K_{n}^{*}(t)/K_{n}^{*}(1)$ is a Lévy bridge. Our approach relies upon two novel ingredients and particularly enables us to dispense with a Poissonization-de-Poissonization step which has been an essential component in all the previous studies of $K_{n}^{*}(1)$. First, for any Karlin occupancy scheme with deterministic probabilities $(p_{k})_{k\ge 1}$, we obtain an approximation, uniformly in $t\in[0,1]$, of the number of boxes with at most $[n^{t}]$ balls by a counting function defined in terms of $(p_{k})_{k\ge 1}$. Second, we prove several FLTs for the number of visits to the interval $[0,nt]$ by a perturbed random walk, as $n\to\infty$.

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