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Zbl 1223.60021
Nándori, Péter
Recurrence properties of a special type of heavy-tailed random walk. (English)
[J] J. Stat. Phys. 142, No. 2, 342-355 (2011). ISSN 0022-4715; ISSN 1572-9613/e

Consider the heavy tail distribution $$p_k:=\frac{1}{2\zeta(3)}\,k^{-3},\qquad k\in\Bbb Z\setminus\{0\},$$ 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.
[Achim Klenke (Mainz)]

MSC 2000:: *60F05 Weak limit theorems
60G50 Sums of independent random variables

Keywords: random walk; local limit theorem; recurrence; heavy tail

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