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Sample path properties of self-similar processes with stationary increments. (English) Zbl 0555.60025
A real-valued process $$X=(X(t))_{t\in {\mathbb{R}}}$$ is self-similar with exponent H (H-ss), if $$X(a\cdot)=_ da^ HX$$ 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 $$0<H\leq 1$$, unless X(t)$$\equiv tX(1)$$ and $$H=1$$, and apart from this can have locally bounded variation only for $$H>1$$, in which case they are singular. However, nowhere bounded variation may occur also for $$H>1.$$
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 $${\mathbb{R}}^ 2$$ that are Poincaré, i.e., invariant in distribution for the transformations $$(t,x)\mapsto (at+b,ax)$$ of $${\mathbb{R}}^ 2$$. In a final section it is shown that the self-similarity and stationary increment properties are preserved under composition of independent processes: $$X_ 1\circ X_ 2=(X_ 1(X_ 2(t)))_{t\in {\mathbb{R}}}$$. Some interesting examples are obtained this way.

##### MSC:
 60G17 Sample path properties 60G10 Stationary stochastic processes 60K99 Special processes 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60G57 Random measures 60E07 Infinitely divisible distributions; stable distributions
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