Sample Path Properties of Self-Similar Processes with Stationary Increments

Abstract

A real-valued process X=(X(t))telR is self-similar with exponent H (H-ss), if X(a.)d aHX for all a>0. Sample path properties of H-ss processes with stationary increments are investigated. The main result is that the sample paths have nowhere bounded variation if 0<H<1, unless X(t) tX(1) and H=1, and apart from this can have locally bounded variation only for H>1, in which case they are singular. Surprisingly, nowhere bounded variation may occur also for H>1. The first example in the literature exhibiting this combination properties is constructed, as well as many others. All examples are obtained by subordination of random measures to point processes in in R2 that are Poincare, i.e., invariant in distribution for the transformations (t,x)->(at+b,ax) of R2. In a final section two ways of combining two ss processes with stationary increments into new ones are studied, one of them being composition of random functions X1oX2=(X1(X1X2(t)))teR.