Abstract:

The authors researched the shift operators of map family in vague ge neralizati on. Given a parameter set T=［0，+∞) or T=｛0，1，2，…｝,and two arbit rary non empty sets Ω and E,which are not disposed any measurable σ algebra and  any measure.The set of all maps φ： T∩［0，β)→E,where β∈T∪｛＋∞｝, is deno ted by  Ω (E). The map X:   Ω→Ω(E) is a map family X=｛X(t),t∈T∩［0， σ)｝ in fact. It is followed that the necessary and sufficient condition of existence for shift  operators of a map family X=｛X(t),t∈T∩［0，σ)｝.  Some properties  of shift operators and some applications to stochastic processes for shift opera tors are discussed.

Key words: Map family, Shift operators, Existence and property, Stationary process, Markov , process