  An apparently ideal way to generate continuous bounded stochastic processes
is to consider the stochastically perturbed motion of a point of small mass in
an infinite potential well, under overdamped approximation. Here, however, we
show that the aforementioned procedure can be fallacious and lead to incorrect
results. We indeed provide a counter-example concerning one of the most
employed bounded noises, hereafter called Tsallis-Stariolo-Borland (TSB) noise,
which admits the well known Tsallis q-statistics as stationary density. In
fact, we show that for negative values of the Tsallis parameter q
(corresponding to sufficiently large diffusion coefficient of the stochastic
force), the motion resulting from the overdamped approximation is unbounded. We
then investigate the cause of the failure of Kramers first type approximation,
and we formally show that the solutions of the full Newtonian non-approximated
model are bounded, following the physical intuition. Finally, we provide a new
family of bounded noises extending the TSB noise, the boundedness of whose
solutions we formally show.
