# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Ayache, A. and Lévy-Véhel, J. (1999). Generalized multifractional Brownian motion: definition and preliminary results. In Fractals: Theory and Applications in Engineering , eds M. Dekking, J. Lévy-Véhel, E. Lutton and C. Tricot, Springer, London, pp. 17–32. · Zbl 0964.60046 [2] Ayache, A. and Lévy-Véhel, J. (2000). The generalized multifractional Brownian motion. Statist. Infer. Stoch. Process. 3, 7–18. · Zbl 0979.60023 [3] Ayache, A. and Taqqu, M. S. (2003). Multifractional processes with random exponent. Preprint. Available as http://www.cmla.ens-cachan.fr/Cmla/Publications/2003/CMLA2003-19.ps.gz · Zbl 1050.60043 [4] Bardet, J.-M. and Bertrand, P. (2003). Definition, properties and wavelet analysis of multiscale fractional Brownian motion. · Zbl 1142.60329 [5] Benassi, A., Cohen, S. and Istas, J. (1998). Identifying the multifractional function of a Gaussian process. Statist. Prob. Lett. 39, 337–345. · Zbl 0931.60022 [6] Benassi, A., Cohen, S. and Istas, J. (2002). Identification and properties of real harmonizable fractional Lévy motions. Bernoulli 8, 97–115. · Zbl 1005.60052 [7] Benassi, A., Cohen, S. and Istas, J. (2004). On roughness indices for fractional fields. Bernoulli 10, 357–373. · Zbl 1062.60052 [8] Benassi, A., Jaffard, S. and Roux, D. (1997). Elliptic Gaussian random processes. Rev. Mat. Iberoamericana 13, 19–90. · Zbl 0880.60053 [9] Benassi, A., Bertrand, P., Cohen, S. and Istas, J. (2000). Identification of the Hurst index of a step fractional Brownian motion. Statist. Infer. Stoch. Process. 3, 101–111. · Zbl 0982.60081 [10] Cohen, S. (1999). From self-similarity to local self-similarity: the estimation problem. In Fractals: Theory and Applications in Engineering , eds M. Dekking, J. Lévy-Véhel, E. Lutton and C. Tricot, Springer, London, pp. 3–16. · Zbl 0965.60073 [11] Cohen, S. and Istas, J. (2004). A universal estimator of local self-similarity. Preprint. Available at http://brassens.<br/> upmf-grenoble.fr/$$\sim$$jistas/publications.html Falconer, K. J. (2002). Tangent fields and the local structure of random fields. J. Theoret. Prob. 15, 731–750. · Zbl 1013.60028 [12] Falconer, K. J. (2003). The local structure of random processes. J. London Math. Soc. 67, 657–672. · Zbl 1054.28003 [13] Leland, W. E., Taqqu, M. S., Willinger, W. and Wilson, D. V. (1994). On the self-similar nature of Ethernet traffic (extended version). IEEE/ACM Trans. Networking 2, 1–15. [14] Park, K. and Willinger, W. (eds) (2000). Self-Similar Network Traffic and Performance Evaluation . John Wiley, New York. [15] Paxson, V. and Floyd, S. (1995). Wide area traffic: the failure of Poisson modeling. IEEE/ACM Trans. Networking 3, 226–244. [16] Peltier, R. F. and Lévy-Véhel, J. (1995). Multifractional Brownian motion: definition and preliminary results. Tech. Rep. 2645, INRIA. [17] Pipiras, V. and Taqqu, M. S. (2004). Stable stationary processes related to cyclic flows. Ann. Prob. 32, 2222–2260. · Zbl 1054.60056 [18] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance . Chapman and Hall, New York. · Zbl 0925.60027 [19] Stoev, S. and Taqqu, M. S. (2004). Path properties of the linear multifractional stable motion. To appear in Fractals . · Zbl 1068.60057 [20] ski, Mandrekar and Cambanis1998surgailis:rosinski:mandrekar:cambanis:1998 Surgailis, D., Rosiński, J., Mandrekar, V. and Cambanis, S. (1998). On the mixing structure of stationary increment and self-similar $$S\alpha S$$ processes.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.