In probability theory, there are several different types of ergodicity,the ordinary ergodicity, the algebraic ergodicity, the exponential ergodicity, and the uniform (strong) ergodicity. In the non-ergodic case, there are in parallel different types of decay. A lot of criteria/sufficient conditions are known for these stabilities /instabilities. If one goes further asking for the speed of these stabilities/instabilities, then the known results are very limited. A subject knowing the stabilities only without knowing their speed is certainly incomplete. Motivated from the study on phase transitions and random algorithm, we have studied the speed estimation for more than two decades.