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

TL;DR

This paper characterizes when the range of a self-similar additive process forms a scale-invariant Poisson point process, linking it to gamma-distributed increments and revealing properties of spacings in such processes.

Contribution

It provides a necessary and sufficient condition for the range of a self-similar additive process to be a scale-invariant Poisson point process, connecting gamma distributions to process structure.

Findings

Range of process forms a Poisson point process iff increments are gamma-distributed.

Spacings in scale-invariant Poisson processes are themselves scale-invariant Poisson points.

Characterizes self-similar processes with independent increments leading to Poisson ranges.

Abstract

It is shown that for a non-decreasing self-similar stochastic process $T$ with independent increments, the range of $T$ forms a Poisson point process with $\sigma$-finite intensity if and only if the one-dimensional distribution of $T(1)$ is of the gamma type. This follows from a general hold-jump description of such processes $T$, and implies the known result that the spacings between consecutive points of a scale invariant Poisson point process, with intensity $\theta x^{-1} dx$, are the points of another scale invariant Poisson point process with the same intensity.