Sample path properties of self-similar processes with stationary increments.

*(English)*Zbl 0555.60025A 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 PoincarĂ©, 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.

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 PoincarĂ©, 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 |