arXiv subjects

Murad S. Taqqu

Publications and source records attributed to Murad S. Taqqu.

57 records · Page 4Linked to original sources

arXiv2004

Integral representations of periodic and cyclic fractional stable motions

Stable non-Gaussian self-similar mixed moving averages can be decomposed into several components. Two of these are the periodic and cyclic fractional stable motions which are the subject of this study. We focus on the structure of their integral representations and show that the periodic fractional stable motions have, in fact, a canonical representation. We study several examples and discuss questions of uniqueness, namely how to determine whether two given integral representations of periodic or cyclic fractional stable motions give rise to the same process.

math.PR↗

arXiv2004

Identification of periodic and cyclic fractional stable motions

Self-similar stable mixed moving average processes can be related to nonsingular flows through their minimal representations. Self-similar stable mixed moving averages related to dissipative flows have been studied, as well as processes associated with identity flows which are the simplest type of conservative flows. The focus here is on self-similar stable mixed moving averages related to periodic and cyclic flows. Periodic flows are conservative flows such that each point in the space comes back to its initial position in finite time, either positive or null. The flow is cyclic if the return time is positive. Self-similar mixed moving averages are called periodic, respectively cyclic, fractional stable motions if their minimal representations are generated by periodic, respectively cyclic, flows. These processes, however, are often defined by a nonminimal representation. We provide a way to detect whether they are periodic or cyclic even if their representation is nonminimal. By using these identification results, we obtain a more refined decomposition of self-similar mixed moving averages.

math.PR↗

arXiv2004

Semi-additive functionals and cocycles in the context of self-similarity

Self-similar symmetric $α$-stable, $α\in(0,2)$, mixed moving averages can be related to nonsingular flows. By using this relation and the structure of the underlying flows, one can decompose self-similar mixed moving averages into distinct classes and then examine the processes in each of these classes separately. The relation between processes and flows involves semi-additive functionals. We establish a general result about semi-additive functionals related to cocycles, and identify the presence of a new semi-additive functional in the relation between processes and flows. This new functional is useful for finding the kernel function of self-similar mixed moving averages generated by a given flow. It also sheds new light on previous results on the subject.

math.PR↗