×

zbMATH — the first resource for mathematics

Alternative micropulses and fractional Brownian motion. (English) Zbl 0879.60076
The generation of fractional Brownian motion (FBM) as a fractal sum of micropulses is considered [cf. B. B. Mandelbrot and J. W. Van Ness, SIAM Rev. 10, 422-437 (1968; Zbl 0179.47801)]). In an earlier paper of the authors [Stochastic Process Appl. 60, 1-8 (1995; Zbl 0846.60055)], rectangular pulses leading to negatively correlated (the self-affinity exponent \(H<1/2\)) FMB’s have been examined. More general pulse shapes are treated here starting with conical and semiconical pulses and ending with Lévy staircase, Cantor pyramid and multifractal staircase. The location of the pulse, its width and height are random, the transformation of a pulse proceeds by letting the tangent of the base angle go to zero. It is shown that only pulses without jumps at their starting and ending points generate positively correlated and ordinary Brown motion (\(H\geq 1/2\)).
Reviewer: Ivan Saxl (Praha)

MSC:
60J65 Brownian motion
60G18 Self-similar stochastic processes
28A80 Fractals
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Cioczek-Georges, R.; Mandelbrot, B.B., A class of micropulses and antipersistent fractional Brownian motion, Stochastic process. appl., 60, 1-8, (1995) · Zbl 0846.60055
[2] Cioczek-Georges, R.; Mandelbrot, B.B., Stable fractal sums of pulses: the general case, (1995), preprint · Zbl 0844.60017
[3] Cioczek-Georges, R.; Mandelbrot, B.B.; Samorodnitsky, G.; Taqqu, M., Stable fractal sums of pulses: the cylindrical case, Bernoulli, 1, 3, 201-216, (1995) · Zbl 0844.60017
[4] Evertsz, C.J.G.; Mandelbrot, B.B., Multifractal measures, (), Appendix B
[5] Mandelbrot, B.B., LES objets fractals: forme, hasard et dimension, (1975), Flammarion Paris · Zbl 0900.00018
[6] Mandelbrot, B.B., The fractal geometry of nature, (1982), W.H. Freeman New York · Zbl 0504.28001
[7] Mandelbrot, B.B., Introduction to fractal sums of pulses, (), 110-123 · Zbl 0829.60032
[8] Mandelbrot, B.B., Fractal sums of pulses: self-affine global dependence and lateral limit theorems, (1995), Preprint
[9] Mandelbrot, B.B.; Van Ness, J.W., Fractional Brownian motions, fractional noises and applications, SIAM rev., 10, 422-437, (1968) · Zbl 0179.47801
[10] Resnick, S.I., Extreme values, regular variation and point processes, (1987), Springer New York · Zbl 0633.60001
[11] Voss, R., Fractals in nature: from characterization to simulation, (), 21-70
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.