Sample Path Properties of Self-Similar Processes with Stationary Increments

Abstract

A real-valued process X=(X(t))t∈ℝX = (X(t))_{t\in\mathbb{R}} is self-similar with exponent H(HH (H-ss), if X(a⋅)=daHXX(a\cdot) =_d a^HX for all a>0a > 0. Sample path properties of HH-ss processes with stationary increments are investigated. The main result is that the sample paths have nowhere bounded variation if 0 1H > 1, in which case they are singular. However, nowhere bounded variation may occur also for H>1H > 1. Examples exhibiting this combination of properties are constructed, as well as many others. Most are obtained by subordination of random measures to point processes in ℝ2\mathbb{R}^2 that are Poincare, i.e., invariant in distribution for the transformations (t,x)↦(at+b,ax)(t, x) \mapsto (at + b, ax) of ℝ2\mathbb{R}^2. In a final section it is shown that the self-similarity and stationary increment properties are preserved under composition of independent processes: X1∘X2=(X1(X2(t)))t∈ℝX_1 \circ X_2 = (X_1(X_2(t)))_{t\in \mathbb{R}}. Some interesting examples are obtained this way.