an:00859945
Zbl 0846.60055
Cioczek-Georges, Renata; Mandelbrot, BenoĆ®t B.
A class of micropulses and antipersistent fractional Brownian motion
EN
Stochastic Processes Appl. 60, No. 1, 1-18 (1995).
0304-4149
1995
j
60G99 60J65
fractal sums of pulses; fractal sums of micropulses; fractional Brownian motion; Poisson random measure; self-similarity; self-affinity; stationarity of increments
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.