In the real world, many evolving stochastic processes appear heavy tails, excess kurtosis, and other non-normal evidences, though, they eventually converge to normals due to the central limit theorem, and the augment effect. So far many studies focusing on the normal cases, such as Brownian Motion, or Geometric Brownian Motion etc, have shown their restrictions in dealing with non-normal phenomena, although they have achieved a great deal of success. Moreover, in many studies, the statistical properties, such as the distributional parameters of an evolving process, have been studied at a special time spot, not having grasped the whole picture during the whole evolving time period. In this paper, we propose to approximate an evolving stochastic process based upon a process characterized by a time-varying mixture distribution family to grasp the whole evolving picture of its evolution behavior. Good statistical properties of such a time-varying process are well illustrated and discussed. The parameters in such a time-varying mixture distribution family are optimized by the Genetic Methods, namely, the Genetic Algorithm (GA) and Genetic Programming (GP). Numerical experiments are carried out and the results prove that our proposed approach works well in dealing with a non-normal evolving stochastic process.