Higher-Order Multifractional Stable Motion: Definition and Fundamental Properties

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 ($\alpha$-stable with $\alpha \in (1,2]$), 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 $L^{\alpha }$-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 ${\alpha }_X(t)=H(t)-1\mathrm{/}\alpha$, and characterize long-range dependence through codifference asymptotics. In particular, we obtain the precise decay exponent $(\alpha -1)H_{+}+H(s)-n$ and the LRD criterion $(\alpha -1)H_{+}+H(s)<n$, which generalizes the classical condition $H(s)+H_{+}<1$ for first-order Gaussian multifractional processes and reduces to $\alpha H-1$ for LFSM with constant $H$.