Cioczek-Georges, Renata; Mandelbrot, Benoît B. A class of micropulses and antipersistent fractional Brownian motion. (English) Zbl 0846.60055 Stochastic Processes Appl. 60, No. 1, 1-18 (1995). Summary: We begin with stochastic processes obtained as sums of “up-and-down” pulses with random moments of birth \(\tau\) and random lifetime \(w\) determined by a Poisson random measure. When the pulse amplitude \(\varepsilon \to 0\), while the pulse density \(\delta\) increases to infinity, one obtains a process of “fractal sum of micropulses.” A CLT style argument shows convergence in the sense of finite-dimensional distributions to a Gaussian process with negatively correlated increments. In the most interesting case the limit is fractional Brownian motion (FBM), a self-affine process with the scaling constant \(0 < H < 1/2\). The construction is extended to the multidimensional FBM field as well as to micropulses of more complicated shape. Cited in 1 ReviewCited in 37 Documents MSC: 60G99 Stochastic processes 60J65 Brownian motion Keywords:fractal sums of pulses; fractal sums of micropulses; fractional Brownian motion; Poisson random measure; self-similarity; self-affinity; stationarity of increments PDFBibTeX XMLCite \textit{R. Cioczek-Georges} and \textit{B. B. Mandelbrot}, Stochastic Processes Appl. 60, No. 1, 1--18 (1995; Zbl 0846.60055) Full Text: DOI References: [1] Cioczek-Georges, R.; Mandelbrot, B. B., Alternative micropulses and fractional Brownian motion (1995), Preprint · 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 (1995), To appear in Bernoulli · Zbl 0844.60017 [4] Itô, K., Stochastic Processes, Aarhus University Lecture Notes Series, Vol. 16 (1969) · Zbl 0226.60053 [5] Mandelbrot, B. B., Fractal sum of pulses, and new random variables and functions (1984), Unpublished [6] Mandelbrot, B. B., Introduction to fractal sums of pulses, (Zaslawsky, G.; Schlesinger, M. F.; Frisch, U., Lévy Flights and Related Phenomena in Physics (Nice, 1994). Lévy Flights and Related Phenomena in Physics (Nice, 1994), Lecture Notes in Physics (1995), Springer: Springer New York) · Zbl 0829.60032 [7] Mandelbrot, B. B., Fractal sums of pulses: self-affine global dependence and lateral limit theorems (1995), Preprint [8] Mandelbrot, B. B.; Van Ness, J. W., Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10, 422-437 (1968) · Zbl 0179.47801 [9] Resnick, S. I., Extreme Values, Regular Variation and Point Processes (1987), Springer: Springer New York · Zbl 0633.60001 [10] Takács, L., On secondary processes generated by a Poisson process and their applications in physics, Acta Math. Acad. Sci. Hungar., 5, 203-235 (1954) · Zbl 0059.12102 [11] Westcott, M., On the existence of a generalized shot-noise process, (Studies in Probability and Statistics (1976), North Holland: North Holland Amsterdam), 73-88 · Zbl 0342.60043 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.