arXiv subjects

Stjepan Šebek

Publications and source records attributed to Stjepan Šebek.

26 records · Page 2Linked to original sources

arXiv2021

Packing density of combinatorial settlement planning models

Recently, a combinatorial settlement planning model was introduced. The idea underlying the model is that the houses are randomly being built on a rectangular tract of land according to the specified rule until the maximal configuration is reached, that is, no further houses can be built while still following that rule. Once the building of the settlement is done, the main question is what percentage of the tract of land on which the settlement was built has been used, i.e. what is the building density of the maximal configuration that was reached. In this article, with the aid of simulations, we find an estimate for the average building density of maximal configurations and we study what happens with this average when the size of a tract of land grows to infinity.

math.CO↗

arXiv2021

Combinatorial settlement planning

In this article, we consider a combinatorial settlement model on a rectangular grid where at least one side (east, south or west) of each house must be exposed to sunlight without obstructions. We are interested in maximal configurations, where no additional houses can be added. For a fixed $m\times n$ grid we explicitly calculate the lowest number of houses, and give close to optimal bounds on the highest number of houses that a maximal configuration can have. Additionally, we provide an integer programming formulation of the problem and solve it explicitly for small values of $m$ and $n$.

math.CO↗

arXiv2021

CLT for the capacity of the range of stable random walks

In this article, we establish a central limit theorem for the capacity of the range process for a class of $d$-dimensional symmetric $α$-stable random walks with the index satisfying $d > 5α/2$. Our approach is based on controlling the limit behavior of the variance of the capacity of the range process which then allows us to apply the Lindeberg-Feller theorem.

math.PR↗

arXiv2021

On asymptotic fairness in voting with greedy sampling

The basic idea of voting protocols is that nodes query a sample of other nodes and adjust their own opinion throughout several rounds based on the proportion of the sampled opinions. In the classic model, it is assumed that all nodes have the same weight. We study voting protocols for heterogeneous weights with respect to fairness. A voting protocol is fair if the influence on the eventual outcome of a given participant is linear in its weight. Previous work used sampling with replacement to construct a fair voting scheme. However, it was shown that using greedy sampling, i.e., sampling with replacement until a given number of distinct elements is chosen, turns out to be more robust and performant. In this paper, we study fairness of voting protocols with greedy sampling and propose a voting scheme that is asymptotically fair for a broad class of weight distributions. We complement our theoretical findings with numerical results and present several open questions and conjectures.

math.PR↗

arXiv2020

Limit theorems for a stable sausage

In this article, we study fluctuations of the volume of a stable sausage defined via a $d$-dimensional rotationally invariant $α$-stable process. As the main results, we establish a functional central limit theorem (in the case when $d/α>3 /2$) with a standard one-dimensional Brownian motion in the limit, and Khintchine's and Chung's laws of the iterated logarithm (in the case when $d/α>9 /5$).

math.PR↗

arXiv2020

Functional CLT for the range of stable random walks

In this note, we establish a functional central limit theorem for the capacity of the range for a class of $α$-stable random walks on the integer lattice $\mathbb{Z}^d$ with $d > 5α/2$. Using similar methods, we also prove an analogous result for the cardinality of the range when $d > 3α/ 2$.

math.PR↗

arXiv2020

Transition probability estimates for subordinate random walks

Let $S_n$ be the simple random walk on the integer lattice $\mathbb{Z}^d$. For a Bernstein function $ϕ$ we consider a random walk $S^ϕ_n$ which is subordinated to $S_n$. Under a certain assumption on the behaviour of $ϕ$ at zero we establish global estimates for the transition probabilities of the random walk $S^ϕ_n$. The main tools that we apply are the parabolic Harnack inequality and appropriate bounds for the transition kernel of the corresponding continuous time random walk.

math.PR↗

arXiv2017

Harnack inequality for subordinate random walks

In this paper, we consider a large class of subordinate random walks $X$ on integer lattice $\mathbb{Z}^d$ via subordinators with Laplace exponents which are complete Bernstein functions satisfying a certain lower scaling condition at zero. We establish estimates for one-step transition probabilities, the Green function and the Green function of a ball, and prove the Harnack inequality for non-negative harmonic functions.

math.PR↗