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Capacity of the range in dimension 5
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abstract
We prove a Central limit theorem for the capacity of the range of a symmetric random walk on $\mathbb Z^5$, under only a moment condition on the step distribution. The result is analogous to the central limit theorem for the size of the range in dimension three, obtained by Jain and Pruitt in 1971. In particular an atypical logarithmic correction appears in the scaling of the variance. The proof is based on new asymptotic estimates, which hold in any dimension $d\ge 5$, for the probability that the ranges of two independent random walks intersect. The latter are then used for computing covariances of some intersection events, at the leading order.
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Cited by 1 Pith paper
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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.
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