Sample Path Properties of Self-Similar Processes with Stationary Increments

Vervaat, Wim

Ann. Probab., Tome 13 (1985) no. 4, p. 1-27 / Harvested from Project Euclid

Résumé

A real-valued process $X = (X(t))_{t\in\mathbb{R}}$ is self-similar with exponent $H (H$-ss), if $X(a\cdot) =_d a^HX$ 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 Poincare, 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.

Détail

Publié le : 1985-02-14
Classification:  Self-similar processes,  stationary increments,  bounded variation of sample paths,  subordination to point processes,  Poincare point processes,  random measures,  stable processes fractional processes,  polynomial processes,  composition of random functions,  60G10,  60G17,  60K99,  60G55,  60G57,  60E07