Stoev, Stilian A.; Taqqu, Murad S. How rich is the class of multifractional Brownian motions? (English) Zbl 1094.60024 Stochastic Processes Appl. 116, No. 2, 200-221 (2006). The authors study two types of multifractional Brownian motion processes introduced by Peltier and Lévy-Vehel (1995) and A. Benassi, S. Jaffard and D. Roux [Rev. Mat. Iberoam. 13, No. 1, 19–90 (1997; Zbl 0880.60053)]. It is shown that these two types of processes have different correlation structures when their self-similarity parameters depend on time and are non-constants. This result contradicts with results in the literature on similarity of these two types of processes. The authors investigate the general class of multifractional Brownian motion processes, which includes the mentioned two types. Reviewer: Andrew Olenko (Kyïv) Cited in 34 Documents MSC: 60G15 Gaussian processes 60G18 Self-similar stochastic processes 60K99 Special processes Keywords:fractional Brownian motion; correlation structure; self-similarity; local self-similarity Citations:Zbl 0880.60053 PDFBibTeX XMLCite \textit{S. A. Stoev} and \textit{M. S. Taqqu}, Stochastic Processes Appl. 116, No. 2, 200--221 (2006; Zbl 1094.60024) Full Text: DOI References: [1] A. Ayache, M.S. Taqqu, Riemann-Stieltjes sums and generalized second order processes with applications to integration with respect to Fractional Brownian Motion, preprint, 2004.; A. Ayache, M.S. Taqqu, Riemann-Stieltjes sums and generalized second order processes with applications to integration with respect to Fractional Brownian Motion, preprint, 2004. [2] Ayache, A.; Cohen, S.; Véhel, J. L., The covariance structure of multifractional Brownian motion, (IEEE, International Conference on Acoustics Speech and Signal Processing (ICASSP) (2000)) [3] Benassi, A.; Jaffard, S.; Roux, D., Elliptic Gaussian random processes, Rev. Mat. Iber., 13, 1, 19-90 (1997) · Zbl 0880.60053 [4] Cohen, S., From self-similarity to local self-similarity: the estimation problem, (Dekking, M.; Véhel, J. L.; Lutton, E.; Tricot, C., Fractals: Theory and Applications in Engineering (1999), Springer: Springer Berlin) · Zbl 0965.60073 [5] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF series, vol. 61, SIAM, Philadelphia, PA, 1992.; I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF series, vol. 61, SIAM, Philadelphia, PA, 1992. · Zbl 0776.42018 [6] I.M. Gelfand, G.E. Shilov, Generalized functions: properties and operations, vol. 1, Academic Press, New York, 1964.; I.M. Gelfand, G.E. Shilov, Generalized functions: properties and operations, vol. 1, Academic Press, New York, 1964. [7] Gradshteyn, I. S.; Ryzhik, I. M., Table of Integrals, Series, and Products (1980), Academic Press: Academic Press New York · Zbl 0521.33001 [8] Meyer, Y.; Sellan, F.; Taqqu, M. S., Wavelets, generalized white noise and fractional integration: the synthesis of fractional Brownian motion, J. Fourier Anal. Appl., 5, 5, 465-494 (1999) · Zbl 0948.60026 [9] R.F. Peltier, J.L. Vehel, Multifractional Brownian motion: definition and preliminary results, Technical Report 2645, Institut National de Recherche en Informatique et an Automatique, INRIA, Le Chesnay, France, 1995.; R.F. Peltier, J.L. Vehel, Multifractional Brownian motion: definition and preliminary results, Technical Report 2645, Institut National de Recherche en Informatique et an Automatique, INRIA, Le Chesnay, France, 1995. [10] Pipiras, V.; Taqqu, M. S., Integration questions related to fractional Brownian motion, Probability Theory and Related Fields, 118, 2, 251-291 (2000) · Zbl 0970.60058 [11] S. Stoev, M.S. Taqqu, Path properties of the linear multifractional stable motion, Fractals 13 (2) (2005) 157-178.; S. Stoev, M.S. Taqqu, Path properties of the linear multifractional stable motion, Fractals 13 (2) (2005) 157-178. · Zbl 1304.28010 [12] S. Stoev, M.S. Taqqu, Stochastic properties of the linear multifractional stable motion, Adv. Appl. Probab. 36 (4) (2004) 1085-1115.; S. Stoev, M.S. Taqqu, Stochastic properties of the linear multifractional stable motion, Adv. Appl. Probab. 36 (4) (2004) 1085-1115. · Zbl 1068.60057 [13] Taqqu, M. S., Fractional Brownian motion and long-range dependence, (Doukhan, P.; Oppenheim, G.; Taqqu, M. S., Theory and Applications of Long-range Dependence (2003), Birkhäuser: Birkhäuser Basel), 5-38 · Zbl 1039.60041 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.