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Stochastic properties of the linear multifractional stable motion. (English) Zbl 1068.60057
The linear fractional stable motion is given by the integral representation \[ X(t)=\int_{\mathbb R}a\big((t+s)_+^{H-1/\alpha}- s_+^{H-1/\alpha}\big)+b\big((t+s)_-^{H-1/\alpha}- s_-^{H-1/\alpha}\big) \,M_{\alpha,\beta}(ds), \] where \((a,b)\in\mathbb R^2\setminus\{(0,0)\}\), \(0<H<1\), and \(M_{\alpha,\beta}\) is an \(\alpha\)-stable independently scattered random measure with \(0<\alpha<2\), skewness \(-1\leq\beta\leq1\) and Lebesgue control measure. Linear multifractional stable motion (LMSM) is obtained when replacing the Hurst index \(H\) by a deterministic function with \(0<H(t)<1\) and further introducing a skewness intensity function with \(-1\leq\beta(s)\leq1\). The authors present some interesting properties of LMSM. For every \(t>0\) they show that LMSM is stochastically continuous if and only if \(H\) is continuous. Moreover, necessary and sufficient conditions in terms of \(H\) for stochasic continuity at \(t=0\) are given. These results are extended to the case \(H(t)\in(0,1)\cup\{1/\alpha\}\) for \(0<\alpha\leq1\). Further, the local asymptotic behaviour is investigated in case the skewness intensity \(\beta\) is continuous and \(H\) fulfills a certain Hölder continuity. The main tool for the proofs is to consider LMSM as the \(\alpha\)-stable random field \(X(t,H(t))\) and to study properties of the field \(X(u,v)\) and of its derivative fields \(\partial_v^nX(u,v)\), \(n\in\mathbb N\).

MSC:
60G18 Self-similar stochastic processes
60E07 Infinitely divisible distributions; stable distributions
60K99 Special processes
60G17 Sample path properties
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