The range of a self-similar additive gamma process is a scale invariant Poisson point process

A Poisson point process on R is called scale invariant if it has intensity measure $\theta x^{-1} dx$. It is known as the scale invariant Poisson spacing lemma that the spacing between consecutive points of a scale invariant Poisson point process are the points of another scale invariant Poisson point process with the same intensity. This paper investigated self-similar additive processes and give its hold-jump description. The range of a self-similar additive gamma processes is proved to be a scale invariant Poisson point process, and then the scale invariant Poisson spacing lemma follows. The uniqueness of the range of self-similar additive non-decreasing processes is given under certain conditions. The connection with extremal process and records are provided to give a broader context of the scale invariant Poisson spacing lemma.