Temps d'occupation de (0, ϵ) pour les marches aléatoires

Let be a real-valued random walk and the time spent by S in (0, ϵ) before time n. The asymptotic behaviour of An has been determined by Spitzer [16] who pointed out the role of the condition The purpose of this paper is to analyse the asymptotic sample path behaviour of An whenever (S) holds. Typically, we show that given a nondecreasing function such that , then almost surely lim inf according as the integral diverges or converges, where . In the special case when the step distribution X has zero mean and finite variance, (S) holds with , and our main result extends a theorem of Chung and Erdős [5] for the simple symmetric random walk. The key ingredient of the proof is an estimate for the tail distribution of An in terms of Φ which follows from classical results of fluctuation theory. Similar arguments apply in continuous time to Levy processes.