A class of micropulses and antipersistent fractional Brownian motion. (English) Zbl 0846.60055

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.


60G99 Stochastic processes
60J65 Brownian motion
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