A functional central limit theorem (Brownian motion limit in Skorohod space) is proved for the capacity and cardinality of the range of stable random walks when d/alpha is greater than 5/2 and 3/2, respectively.
CLT for the capacity of the range of stable random walks
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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.
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Functional CLT for the range of stable random walks
A functional central limit theorem (Brownian motion limit in Skorohod space) is proved for the capacity and cardinality of the range of stable random walks when d/alpha is greater than 5/2 and 3/2, respectively.