  We present a study of both the ``Iterative Procedure'' and the ``$\beta \to
\beta'$ transformation'', proposed by Tsallis et al (Physica A261, 534) to find
the probabilities $p_i$ of a system to be in a state with energy $\epsilon_i$,
within the framework of a generalized statistical mechanics. Using stability
and convexity arguments, we argue that the iterative procedure does not always
provide the right temperature dependence of thermodynamic observables. In
addition, we show how to get the correct answers from the ``$\beta \to \beta'$
transformation''. Our results provide an evidence that the Tsallis statistics
with normalized q-expectation values is stable for all ranges of temperatures.
We also show that the cut-off in the computation of probabilities is required
to achieve the stable solutions.
