In a recent paper we proposed a non-Markovian random walk model with memory
of the maximum distance ever reached from the starting point (home). The
behavior of the walker is at variance with respect to the simple symmetric
random walk (SSRW) only when she is at this maximum distance, where, having the
choice to move either farther or closer, she decides with different
probabilities. If the probability of a forward step is higher then the
probability of a backward step, the walker is bold and her behavior turns out
to be super-diffusive, otherwise she is timorous and her behavior turns out to
be sub-diffusive. The scaling behavior vary continuously from sub-diffusive
(timorous) to super-diffusive (bold) according to a single parameter γ∈R. We investigate here the asymptotic properties of the bold case in the
non ballistic region γ∈[0,1/2], a problem which was left partially
unsolved in \cite{S}. The exact results proved in this paper require new
probabilistic tools which rely on the construction of appropriate martingales
of the random walk and its hitting times
