## 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

### Keywords:

random walk; local limit theorem; recurrence; heavy tail
Full Text:

### References:

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