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Mixed Moving Averages and Self-Similarity

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  • First Online:
Stable Non-Gaussian Self-Similar Processes with Stationary Increments

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Abstract

The focus of the chapter is on a large class of symmetric stable self-similar processes with stationary increments, known as self-similar mixed moving averages. Minimal representations of self-similar mixed moving averages are related to nonsingular flows. Based on the structure of the underlying flows, self-similar mixed moving averages are decomposed uniquely into four major classes that are associated with dissipative, fixed, cyclic and conservative nonperiodic flows. These classes are also derived based on nonminimal representations, without referring to the underlying flows. Finally, canonical representations of self-similar mixed moving averages are derived for the dissipative, fixed and cyclic classes, and an example of a conservative nonperiodic self-similar mixed moving average is also provided.

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Notes

  1. 1.

    ​​The process is called “mixed” because of the presence of the X dimension which gives rise to a “mixture.”

  2. 2.

    ​​Indeed, under mild assumptions (e.g., Doob [11], Section 11, Chapter XI), such a process has a spectral domain representation \(\int _{\mathbb{R}}\widehat{f}_{t}(x)\widehat{B}(dx)\), where \(\widehat{f}_{t}(x) = (e^{itx} - 1)f(x)\) and \(\widehat{B}(dx)\) is a Gaussian Hermitian measure with the control measure dx. Then, under additional mild assumptions, the postulated representation \(\int _{\mathbb{R}}(G(t + u) - G(u))B(du)\) follows by setting G to be the inverse Fourier transform of the function f.

  3. 3.

    ​​Unique modulo μ means, for example, that if {ψ c } c > 0 and \(\{\widetilde{\psi }_{c}\}_{c>0}\) are two such flows, then for all c > 0, \(\psi _{c} =\widetilde{\psi } _{c}\) μ-a.e.

  4. 4.

    ​​To see this easily, express (3.33) as

    $$\displaystyle{G(x,u) = h_{c_{1}}(x)\ c_{1}^{\kappa }G(\varPhi _{c_{ 1}}^{1}(x),c_{ 1}^{-1}u +\varPhi _{ c_{ 1}}^{2}(x)) + c_{ 1}^{\kappa }\varPhi _{c_{ 1}}^{3}(x)}$$

    and iterate.

  5. 5.

    ​​Do not confuse this function F with the one defined in ( 3.50 ).

  6. 6.

    ​​In Pipiras and Taqqu [37], dissipative fractional stable motions (DFSMs) are called dilated fractional stable motions.

  7. 7.

    ​​Do not confuse the process X α with the set X.

  8. 8.

    ​In Pipiras and Taqqu [36] (see Definition 5.1), the fixed fractional stable motion was called mixed linear fractional stable motion because of Theorem 3.11 below.

  9. 9.

    ​​Cyclic fractional stable motion is abbreviated as cLFSM, since the abbreviation LFSM is commonly used for linear fractional stable motion.

  10. 10.

    ​​We use (2.101) and (2.102), but the h is replaced by lnc since we are considering here a multiplicative flow instead of an additive flow. Note also that β 2 = κ for the 2-semi-additive functional in Definition 3.2.

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Authors and Affiliations

  1. Statistics and Operations Research, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA

    Vladas Pipiras

  2. Department of Mathematics and Statistics, Boston University, Boston, MA, USA

    Murad S. Taqqu

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  1. Vladas Pipiras
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  2. Murad S. Taqqu
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Pipiras, V., Taqqu, M.S. (2017). Mixed Moving Averages and Self-Similarity. In: Stable Non-Gaussian Self-Similar Processes with Stationary Increments. SpringerBriefs in Probability and Mathematical Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-62331-3_3

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