Higher-Order Multifractional Stable Motion: Definition and Fundamental Properties
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910281292054528 |
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| author | Lechiheb, Atef |
| author_facet | Lechiheb, Atef |
| contents | This paper introduces the $n$-th order multifractional stable motion ($n$-MFSM), a novel stochastic process that simultaneously unifies three key modelling features: heavy-tailed distributions ($α$-stable with $α\in(1,2]$), time-varying local regularity via a functional Hurst parameter $H(t)\in(n-1,n)$, and extended scaling behaviour of order $n\geq1$. No existing framework combines all three. We establish rigorous existence via $L^α$-integrability analysis, derive both moving-average and harmonizable representations with explicit constants, prove local asymptotic self-similarity with complete identification of the limit process, determine the exact pointwise Hölder regularity $α_X(t)=H(t)-1/α$, and characterize long-range dependence through codifference asymptotics. In particular, we obtain the precise decay exponent $(α-1)H_+ + H(s)-n$ and the LRD criterion $(α-1)H_++H(s)<n$, which generalizes the classical condition $H(s)+H_+<1$ for first-order Gaussian multifractional processes and reduces to $αH-1$ for LFSM with constant $H$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_02003 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Higher-Order Multifractional Stable Motion: Definition and Fundamental Properties Lechiheb, Atef Probability 60G18, 60G52, 60G17, 60E07, 60G15 This paper introduces the $n$-th order multifractional stable motion ($n$-MFSM), a novel stochastic process that simultaneously unifies three key modelling features: heavy-tailed distributions ($α$-stable with $α\in(1,2]$), time-varying local regularity via a functional Hurst parameter $H(t)\in(n-1,n)$, and extended scaling behaviour of order $n\geq1$. No existing framework combines all three. We establish rigorous existence via $L^α$-integrability analysis, derive both moving-average and harmonizable representations with explicit constants, prove local asymptotic self-similarity with complete identification of the limit process, determine the exact pointwise Hölder regularity $α_X(t)=H(t)-1/α$, and characterize long-range dependence through codifference asymptotics. In particular, we obtain the precise decay exponent $(α-1)H_+ + H(s)-n$ and the LRD criterion $(α-1)H_++H(s)<n$, which generalizes the classical condition $H(s)+H_+<1$ for first-order Gaussian multifractional processes and reduces to $αH-1$ for LFSM with constant $H$. |
| title | Higher-Order Multifractional Stable Motion: Definition and Fundamental Properties |
| topic | Probability 60G18, 60G52, 60G17, 60E07, 60G15 |
| url | https://arxiv.org/abs/2606.02003 |