A class of micropulses and antipersistent fractional Brownian motion

We begin with stochastic processes obtained as sums of “up-and-down” pulses with random moments of birth τ and random lifetime w determined by a Poisson random measure. When the pulse amplitude ε → 0, while the pulse density δ 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 < 12. The construction is extended to the multidimensional FBM field as well as to micropulses of more complicated shape.