Alternative micropulses and fractional Brownian motion

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 H < 12 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 (H>12) 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.