A simple construction of the Fractional Brownian motion

In this work we introduce correlated random walks on $\mathbb{Z}$. When picking suitably at random the coefficient of correlation, and taking the average over a large number of walks, we obtain a discrete Gaussian process, whose scaling limit is the fractional Brownian motion. We have to use two radically different models for both cases $\frac{1}{2}\le H<1$ and $0<H<\frac{1}{2}$. This result provides an algorithm for the simulation of the fractional Brownian motion, which appears to be quite efficient.