## The Annals of Probability

- Ann. Probab.
- Volume 13, Number 1 (1985), 1-27.

### Sample Path Properties of Self-Similar Processes with Stationary Increments

#### Abstract

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.

#### Article information

**Source**

Ann. Probab., Volume 13, Number 1 (1985), 1-27.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993063

**Digital Object Identifier**

doi:10.1214/aop/1176993063

**Mathematical Reviews number (MathSciNet)**

MR770625

**Zentralblatt MATH identifier**

0555.60025

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G10: Stationary processes

Secondary: 60G17: Sample path properties 60K99: None of the above, but in this section 60G55: Point processes 60G57: Random measures 60E07: Infinitely divisible distributions; stable distributions

**Keywords**

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

#### Citation

Vervaat, Wim. Sample Path Properties of Self-Similar Processes with Stationary Increments. Ann. Probab. 13 (1985), no. 1, 1--27. doi:10.1214/aop/1176993063. https://projecteuclid.org/euclid.aop/1176993063