Abstract: Traditionally, stationarity refers to shift invariance of the distribution of a stochastic process. In this paper, we rediscover stationarity as a path property instead of a distributional property. More precisely, we characterize a set of paths, denoted by , which corresponds to the notion of stationarity. On one hand, the set is shown to be large enough, so that for any stationary process, almost all of its paths are in . On the other hand, we prove that any path in will behave in the optimal way under any stationarity test satisfying some mild conditions.

2000 AMS Mathematics Subject Classification: Primary: 60G10; Secondary: 60G17, 62M10.

Keywords and phrases: Stationarity, path property.