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Recurrence properties of a special type of heavy-tailed random walk. (English) Zbl 1223.60021

Consider the heavy tail distribution
\[ p_k:=\frac{1}{2\zeta(3)}\,k^{-3},\qquad k\in\mathbb Z\setminus\{0\}, \]
on the integer lattice and let \(Q=(Q_n)_{n=0,1,2,\dots}\) be the corresponding integer-valued random walk. Furthermore, let \(S=(S_n)_{n=0,1,2,\dots}\) be the random walk on the two-dimensional integer lattice with step distribution
\[ p'_{(0,k)}=p'_{(k,0)}=\tfrac12p_k. \]
The author derives local limit theorems for \(Q\) and \(S\) as well as the asymptotics for the time of first return to the origin, and the number of visits to the origin in the first \(n\) steps.

MSC:

60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
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