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Zbl 0805.60069
Marcus, Michael B.; Rosen, Jay
Laws of the iterated logarithm for the local times of recurrent random walks on $Z\sp 2$ and of Lévy processes and random walks in the domain of attraction of Cauchy random variables. (English)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 30, No.3, 467-499 (1994). ISSN 0246-0203

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 $g(t)= \int\sp t\sb 0 p\sb s(0)ds$, $t\geq 0$, where $p\sb s(\cdot)$ stands for the continuous version of the semigroup. Under some technical conditions, if $g$ is slowly varying at infinity, then a.s. $$\align \limsup\sb{t\to\infty} {{L\sp 0\sb t} \over {g(t/ \log\log g(t))\log \log g(t)}} &=1\\ \intertext{and} \limsup\sb{t\to\infty} {{L\sp 0\sb t- L\sp x\sb t} \over {g\sp{1/2} (t/\log \log g(t))\log \log g(t)}} &= \sqrt{2} \sigma(x), \endalign$$ where $L\sp y\sb s$ denotes the local time at level $y$ and time $s$ and $\sigma\sp 2(x)= \int\sp \infty\sb 0 (p\sb t(0)- p\sb t(x))\sp 2 dt$. There is a similar result for symmetric random walks in dimensions 1 and 2.\par It was recently observed by {\it 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 {\it B. E. Fristedt} and {\it W. E. Pruitt} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 18, 167-182 (1971; Zbl 0197.442)].
[J.Bertoin (Paris)]

MSC 2000:: *60J55 Additive functionals

Keywords: laws of the iterated logarithm; local times of symmetric Lévy processes; domain of attraction of a stable law; Green function; slowly varying at infinity

Citations: Zbl 0207.487; Zbl 0197.442

Cited in: Zbl 0835.60068

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