We compute the average shape of trajectories of some one--dimensional
stochastic processes x(t) in the (t,x) plane during an excursion, i.e. between
two successive returns to a reference value, finding that it obeys a scaling
form. For uncorrelated random walks the average shape is semicircular,
independently from the single increments distribution, as long as it is
symmetric. Such universality extends to biased random walks and Levy flights,
with the exception of a particular class of biased Levy flights. Adding a
linear damping term destroys scaling and leads asymptotically to flat
excursions. The introduction of short and long ranged noise correlations
induces non trivial asymmetric shapes, which are studied numerically.Comment: 16 pages, 16 figures; accepted for publication in Phys. Rev.