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Alternative micropulses and fractional Brownian motion

R. Cioczek-Georges and Benoît Mandelbrot

Stochastic Processes and their Applications, 1996, vol. 64, issue 2, 143-152

Abstract: We showed in an earlier paper (1995a) that negatively correlated fractional Brownian motion (FBM) can be generated as a fractal sum of one kind of micropulses (FSM). That is, FBM of exponent is the limit (in the sense of finite-dimensional distributions) of a certain sequence of processes obtained as sums of rectangular pulses. We now show that more general pulses yield a wide range of FBMs: either negatively (as before) or positively () correlated. We begin with triangular (conical and semi-conical) pulses. To transform them into micropulses, the base angle is made to decrease to zero, while the number of pulses, determined by a Poisson random measure, is made to increase to infinity. Then we extend our results to more general pulse shapes.

Keywords: Fractal; sums; of; pulses; Fractal; sums; of; micropulses; Fractional; Brownian; motion; Poisson; random; measure; Self-similarity; Self-affinity; Stationarity; of; increments (search for similar items in EconPapers)
Date: 1996
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