The Annals of Probability

Sample Path Properties of Self-Similar Processes with Stationary Increments

Wim Vervaat

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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

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

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

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


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.

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