Self-Similar Processes with Stationary Increments Generated by Point Processes

Abstract

A real-valued process $X = (X(t))_{t\in \mathbb{R}}$ is self-similar with exponent $H(H$-ss), if $X(a\cdot) =_d a^HX$ for all $a > 0$. The present paper studies $H$-ss processes $X_H$ with stationary increments that can be represented for $t > 0$ by $X_H(t) := \int |x|^H \operatorname{sgn} x \Pi((0, t\rbrack, dx) =: \int x \Pi^H((0, t\rbrack, dx)$, where $\Pi$ is a point process in $\mathbb{R}^2$ that is Poincare, i.e., invariant in distribution under the transformations $(t, x) \mapsto (\text{at} + b, ax)$ of $\mathbb{R}^2$. In particular, $X_H$ allows such a representation if it is a jump process, $\Pi^H$ being the graph of its jumps. Several examples of Poincare processes $\Pi$ are presented. These lead in many cases to new examples of $H$-ss processes $X_H$ with stationary increments. Furthermore, it is investigated for which $H$ the integral expression for $X_H$ converges, conditionally or absolutely. If $\Pi$ has finite intensity $\mathbb{E}\Pi$, then $(1, \infty)$ is wp1 the set of $H$ for which $X_H$ converges absolutely. If $\mathbb{E}\Pi$ is not finite, then the situation is more complicated, as is the case for conditional convergence. Several examples illustrate this. In the final section the integrator $\pi$ in the expression for $X_H$ is replaced by $\Pi - \mathbb{E}\Pi$, which gives conditional convergence for more $H$ in $(0, 1)$.