Open Access
February, 1985 Sample Path Properties of Self-Similar Processes with Stationary Increments
Wim Vervaat
Ann. Probab. 13(1): 1-27 (February, 1985). DOI: 10.1214/aop/1176993063


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.


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Wim Vervaat. "Sample Path Properties of Self-Similar Processes with Stationary Increments." Ann. Probab. 13 (1) 1 - 27, February, 1985.


Published: February, 1985
First available in Project Euclid: 19 April 2007

zbMATH: 0555.60025
MathSciNet: MR770625
Digital Object Identifier: 10.1214/aop/1176993063

Primary: 60G10
Secondary: 60E07 , 60G17 , 60G55 , 60G57 , 60K99

Keywords: bounded variation of sample paths , composition of random functions , Poincare point processes , Polynomial processes , Random measures , Self-similar processes , stable processes fractional processes , Stationary increments , subordination to point processes

Rights: Copyright © 1985 Institute of Mathematical Statistics

Vol.13 • No. 1 • February, 1985
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