  Cauchy's formula was originally established for random straight paths
crossing a body $B \subset \mathbb{R}^{n}$ and basically relates the average
chord length through $B$ to the ratio between the volume and the surface of the
body itself. The original statement was later extended in the context of
transport theory so as to cover the stochastic paths of Pearson random walks
with exponentially distributed flight lengths traversing a bounded domain. Some
heuristic arguments suggest that Cauchy's formula may also hold true for
Pearson random walks with arbitrarily distributed flight lengths. For such a
broad class of stochastic processes, we rigorously derive a generalized
Cauchy's formula for the average length travelled by the walkers in the body,
and show that this quantity depends indeed only on the ratio between the volume
and the surface, provided that some constraints are imposed on the entrance
step of the walker in $B$. Similar results are obtained also for the average
number of collisions performed by the walker in $B$, and an extension to
absorbing media is discussed.
