  The aim of this paper is to present a result of discrete approximation of
some class of stable self-similar stationary increments processes. The
properties of such processes were intensively investigated, but little is known
on the context in which such processes can arise. To our knowledge,
discretisation and convergence theorems are available only in the case of
stable L\'evy motions and fractional Brownian motions. This paper yields new
results in this direction. Our main result is the convergence of the random
rewards schema, which was firstly introduced by Cohen and Samorodnitsky, and
that we consider in a more general setting. Strong relationships with Kesten
and Spitzer's random walk in random sceneries are evidenced. Finally, we study
some path properties of the limit process.
