Corpus ID: 118687385

# CLT for the capacity of the range of stable random walks

@article{Cygan2019CLTFT,
title={CLT for the capacity of the range of stable random walks},
author={Wojciech Cygan and Nikola Sandri'c and Stjepan vSebek},
journal={arXiv: Probability},
year={2019}
}
• Published 2019
• Mathematics
• arXiv: Probability
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 $d\ge3\alpha$. 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.
4 Citations
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#### References

SHOWING 1-10 OF 25 REFERENCES
Capacity of the range of random walk on Z
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. TheExpand
Capacity of the range of random walk on $\mathbb{Z}^{4}$
• Mathematics
• The Annals of Probability
• 2019
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 aExpand
Capacity of the range of random walk on $\mathbb{Z}^d$
• Mathematics
• 2016
We study the capacity of the range of a transient simple random walk on Z^d. Our main result is a central limit theorem for the capacity of the range for d ≥ 6. We present a few open questions inExpand
The Range of Stable Random Walks
• Mathematics
• 1991
Limit theorems are proved for the range of d-dimensional random walks in the domain of attraction of a stable process of index P3. The range Rn is the number of distinct sites of Zd visited by theExpand
Deviations for the capacity of the range of a random walk
• Mathematics
• 2018
We obtain estimates for downward deviations for the centered capacity of the range of a random walk on Z d , in dimension d ≥ 5. Our regime of deviations runs from large to moderate. We describe pathExpand
Limit theorems for random walks
• Mathematics
• 2015
We consider a random walk $S_{\tau}$ which is obtained from the simple random walk $S$ by a discrete time version of Bochner's subordination. We prove that under certain conditions on theExpand
On subordinate random walks
In this article subordination of random walks in $R^d$ is considered. We prove that subordination of random walks in the sense of [BSC12] yields the same process as subordination of Levy processesExpand
Intersections of random walks
We study the large deviation behaviour of simple random walks in dimension three or more in this thesis. The first part of the thesis concerns the number of lattice sites visited by the random walk.Expand
Some properties of random walk paths
• Mathematics
• 1973
Abstract For random walk on the d-dimensional integer lattice we consider again the problem of deciding when a set is recurrent, that is visited infinitely often with probability one by the randomExpand
Propriétés d'intersection des marches aléatoires
We study intersection properties of multi-dimensional random walks. LetX andY be two independent random walks with values in ℤd (d≦3), satisfying suitable moment assumptions, and letIn denote theExpand