Memory effects in fractional Brownian motion with Hurst exponent H<1/3

We study the regression to the origin of a walker driven by dynamically generated fractional Brownian motion (FBM) and we prove that when the FBM scaling, i.e., the Hurst exponent H<1/3, the emerging inverse power law is characterized by a power index that is a compelling signature of the infinitely extended memory of the system. Strong memory effects leads to the relation H=θ/2 between the Hurst exponent and the persistent exponent θ, which is different from the widely used relation H=1-θ. The latter is valid for 1/3<H<1 and is known to be compatible with the renewal assumption.