Divergent sums over excursions
Let \(X\) be a canonical standard Markov process, and let 0 be a recurrent point of the state space which is regular for itself (but is neither trap nor a holding point). The author proves criteria for the almost sure convergence or divergence of sums of functions of excursions away from 0. These criteria are in terms of the expectation of the integrated excursion measure, either in the natural time scale or in the scale of local time of \(X\) at 0. For certain diffusions these results lead, together with the explicit calculation of the excursion measure, to interesting 0-1 type laws for the small time behaviour of the sample paths of \(X\). If \(X\) is reflecting Brownian motion, the author obtains 0-1 laws for the maxima of \(X\) over excursion intervals and for the areas under the excursions.
scientific article; zbMATH DE number 713923