Author
Stjepan Šebek
Other affiliations: Graz University of Technology
Bio: Stjepan Šebek is an academic researcher from University of Zagreb. The author has contributed to research in topics: Random walk & Integer lattice. The author has an hindex of 4, co-authored 16 publications receiving 29 citations. Previous affiliations of Stjepan Šebek include Graz University of Technology.
Papers
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TL;DR: In this article, the capacity of the range process for a class of symmetric $d$-dimensional stable random walks with the index satisfying 5\alpha /2 was established based on controlling the limit behavior of the variance of the capacity.
Abstract: In this article, we establish a central limit theorem for the capacity of the range process for a class of $d$-dimensional symmetric $\alpha$-stable random walks with the index satisfying $d > 5\alpha /2$. 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.
8 citations
TL;DR: In this paper, a functional central limit theorem for the capacity of range for a class of stable random walks on the integer lattice was established, where the cardinality of the range was shown to be polynomial in the number of random walks.
Abstract: In this note, we establish a functional central limit theorem for the capacity of the range for a class of $$\alpha $$
-stable random walks on the integer lattice $$\mathbb {Z}^d$$
with $$ d> 5\alpha /2$$
. Using similar methods, we also prove an analogous result for the cardinality of the range when $$d > 3\alpha / 2$$
.
7 citations
TL;DR: In this article, a large class of subordinator random walks X on the integer lattice Z with Laplace exponents were considered and the Harnack inequality for nonnegative harmonic functions was established.
Abstract: In this paper, we consider a large class of subordinate random walks X on the integer lattice $$\mathbb {Z}^d$$
via subordinators with Laplace exponents which are complete Bernstein functions satisfying some mild scaling conditions 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 nonnegative harmonic functions.
5 citations
TL;DR: In this article, the capacity of α-stable random walks with the index satisfying d/α>5/2 was established for a class of d-dimensional symmetric random walks.
Abstract: 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/α>5/2. Our appro...
4 citations
Posted Content•
TL;DR: In this article, the capacity and cardinality of the range for a class of stable random walks on the integer lattice was shown to be invariant to the number of nodes and the cardinality.
Abstract: In this article, we establish an almost sure invariance principle for the capacity and cardinality of the range for a class of $\alpha$-stable random walks on the integer lattice $\mathbb{Z}^d$ with $d > 5\alpha/2$ and $d>3\alpha/2$, respectively. As a direct consequence, we conclude Khintchine's and Chung's laws of the iterated logarithm for both processes.
4 citations
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Book•
01 Jan 2013
TL;DR: In this paper, the authors consider the distributional properties of Levy processes and propose a potential theory for Levy processes, which is based on the Wiener-Hopf factorization.
Abstract: Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.
1,957 citations
01 Jan 2016
TL;DR: The principles of random walk is universally compatible with any devices to read and is available in the book collection an online access to it is set as public so you can get it instantly.
Abstract: Thank you for reading principles of random walk. Maybe you have knowledge that, people have search hundreds times for their chosen novels like this principles of random walk, but end up in infectious downloads. Rather than enjoying a good book with a cup of coffee in the afternoon, instead they are facing with some malicious virus inside their desktop computer. principles of random walk is available in our book collection an online access to it is set as public so you can get it instantly. Our digital library hosts in multiple locations, allowing you to get the most less latency time to download any of our books like this one. Merely said, the principles of random walk is universally compatible with any devices to read.
241 citations
01 Jan 2001
TL;DR: In this article, it was shown that the optimal strategy to realize the above large deviation is for W a 1 (ct) and W a 2(ct) to form a Swiss cheese, where the two Wiener sausages cover part of the space, leaving random holes whose sizes are of order 1 and whose density varies on scale t 1=d according to a certain optimal prole.
Abstract: For a > 0, let W a 1 (t) and W a 2 (t) be the a-neighbourhoods of two independent standard Brownian motions in R d starting at 0 and observed until time t. We prove that, for d 3 and c > 0, lim t!1 1 t (d 2)=d logP jW a 1 (ct) \W a 2 (ct)j t = I a d (c) and derive a variational representation for the rate constant I a d (c). Here, a is the Newtonian capacity of the ball with radius a. We show that the optimal strategy to realise the above large deviation is for W a 1 (ct) and W a 2 (ct) to \form a Swiss cheese": the two Wiener sausages cover part of the space, leaving random holes whose sizes are of order 1 and whose density varies on scale t 1=d according to a certain optimal prole. We study in detail the function c 7! I a d (c). It turns out that I a d (c) =
27 citations
Posted Content•
TL;DR: In this paper, the scaling limit of the capacity of the range of a simple random walk on the integer lattice in dimension four was studied and a strong law of large numbers and a central limit theorem with a non-gaussian limit were established.
Abstract: We study the scaling limit of the capacity of the range of a simple random walk on the integer lattice in dimension four. We establish a strong law of large numbers and a central limit theorem with a non-gaussian limit. The asymptotic behaviour is analogous to that found by Le Gall in '86 for the volume of the range in dimension two.
8 citations
TL;DR: In this paper, a functional central limit theorem for the capacity of range for a class of stable random walks on the integer lattice was established, where the cardinality of the range was shown to be polynomial in the number of random walks.
Abstract: In this note, we establish a functional central limit theorem for the capacity of the range for a class of $$\alpha $$
-stable random walks on the integer lattice $$\mathbb {Z}^d$$
with $$ d> 5\alpha /2$$
. Using similar methods, we also prove an analogous result for the cardinality of the range when $$d > 3\alpha / 2$$
.
7 citations