We present a generalization of the persistent random-walk model in which th
e step at time n depends on the stare of the step at time n-T, for arbitrar
y T. This gives rise to arbitrarily long memory effects, yet by an appropri
ate transformation the model is tractable by essentially the same technique
s applicable to the usual persistent random-walk problem. We apply our resu
lts to the specific case of delayed "step'' persistence, and analyze its as
ymptotic statistical properties.