  The ensemble properties and time-averaged observables of a memory-induced
diffusive-superdiffusive transition are studied. The model consists in a random
walker whose transitions in a given direction depend on a weighted linear
combination of the number of both right and left previous transitions. The
diffusion process is nonstationary and its probability develops the phenomenon
of aging. Depending on the characteristic memory parameters, the ensemble
behavior may be normal, superdiffusive, or ballistic. In contrast, the
time-averaged mean squared displacement is equal to that of a normal undriven
random walk, which renders the process non-ergodic. In addition, and similarly
to Levy walks [Godec and Metzler, Phys. Rev. Lett. 110, 020603 (2013)], for
trajectories of finite duration the time-averaged displacement apparently
become random with properties that depend on the measurement time and also on
the memory properties. These features are related to the non-stationary
power-law decay of the transition probabilities to their stationary values.
Time-averaged response to a bias is also calculated. In contrast with Levy
walks [Froemberg and Barkai, Phys. Rev. E 87, 030104(R) (2013)], the response
always vanishes asymptotically.
