## Laws of the iterated logarithm for the local times of recurrent random walks on $$Z^ 2$$ and of Lévy processes and random walks in the domain of attraction of Cauchy random variables.(English)Zbl 0805.60069

The authors [Ann. Probab. 22, No. 2, 626-658 (1994)] established recently first and second order laws of the iterated logarithm for the local times of symmetric Lévy processes in the domain of attraction of a stable law with index $$\alpha\in (1,2]$$. The paper under review is a complement to the latter, and concerns symmetric Lévy processes and random walks in the domain of attraction of a Cauchy variable. Typically, let $$X$$ be a real-valued recurrent symmetric Lévy process which possesses local times. Denote the truncated Green function by $$g(t)= \int^ t_ 0 p_ s(0)ds$$, $$t\geq 0$$, where $$p_ s(\cdot)$$ stands for the continuous version of the semigroup. Under some technical conditions, if $$g$$ is slowly varying at infinity, then a.s. \begin{aligned} \limsup_{t\to\infty} {{L^ 0_ t} \over {g(t/ \log\log g(t))\log \log g(t)}} &=1\\ \text{and} \limsup_{t\to\infty} {{L^ 0_ t- L^ x_ t} \over {g^{1/2} (t/\log \log g(t))\log \log g(t)}} &= \sqrt{2} \sigma(x), \end{aligned} where $$L^ y_ s$$ denotes the local time at level $$y$$ and time $$s$$ and $$\sigma^ 2(x)= \int^ \infty_ 0 (p_ t(0)- p_ t(x))^ 2 dt$$. There is a similar result for symmetric random walks in dimensions 1 and 2.
It was recently observed by Caballero and the reviewer [On the rate of growth of subordinators with slowly varying Laplace exponent (to appear)] that these results can also be extended to the asymmetric case by using a general theorem of B. E. Fristedt and W. E. Pruitt [Z. Wahrscheinlichkeitstheorie Verw. Geb. 18, 167-182 (1971; Zbl 0197.442)].
Reviewer: J.Bertoin (Paris)

### MSC:

 60J55 Local time and additive functionals

### Citations:

Zbl 0207.487; Zbl 0197.442
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