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 ($\alpha$-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 $n\geq1$. 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 H\"older regularity $\alpha_X(t)=H(t)-1/\alpha$, 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 $H$.