  We prove weak ergodicity of the inhomogeneous Markov process generated by the
generalized transition probability of Tsallis and Stariolo under power-law
decay of the temperature. We thus have a mathematical foundation to conjecture
convergence of simulated annealing processes with the generalized transition
probability to the minimum of the cost function. An explicitly solvable example
in one dimension is analyzed in which the generalized transition probability
leads to a fast convergence of the cost function to the optimal value. We also
investigate how far our arguments depend upon the specific form of the
generalized transition probability proposed by Tsallis and Stariolo. It is
shown that a few requirements on analyticity of the transition probability are
sufficient to assure fast convergence in the case of the solvable model in one
dimension.
