Higher-Order Multifractional Stable Motion: Definition and Fundamental Properties

Abstract

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 ($\hat{I}\pm$-stable with $\hat{I}\pm \in (1,2]$), time-varying local regularity via a functional Hurst parameter $H(t)\in (n-1,n)$, and extended scaling behaviour of order $n\ge 1$. No existing framework combines all three. We establish rigorous existence via $L^{\hat{I}}\pm$-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 $\hat{I}{\pm }_X(t)=H(t)-1/\hat{I}\pm$, and characterize long-range dependence through codifference asymptotics. In particular, we obtain the precise decay exponent $(\hat{I}\pm -1)H_{+}+H(s)-n$ and the LRD criterion $(\hat{I}\pm -1)H_{+}+H(s)<n$, which generalizes the classical condition $H(s)+H_{+}<1$ for first-order Gaussian multifractional processes and reduces to $\hat{I}\pm H-1$ for LFSM with constant $H$.