STATIONARITY AS A PATH PROPERTY

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.