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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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