A characterization of temporal homogeneity for additive processes

Abstract:

In this paper, temporal homogeneity of an $\mathbb {R}^{d}$-valued additive process is studied. When an additive process has the stationary independent increments property, the process is called a temporally homogeneous additive process or Lévy process. Moreover, an additive process is said to have the independent increments property in a strong sense if the process has the independent increments property at every finite stopping time (that is, its increments starting at any finite stopping time and events before the stopping time are independent). This paper shows that if an additive process has the independent increments property in a strong sense, then the process is temporally homogeneous, provided the process is immediately random. In particular, in the case of additive processes with Poisson distributed independent increments, it follows that, under some non-degeneracy conditions, the temporal homogeneity is equivalent to the independent increments property at the first jumping time and that, in the degenerate cases, whether each process has the independent increments property at the first jumping time is determined.