We study a family of correlated one-dimensional random walks with a finite
memory range ni. These walks are extensions of the Taylor's walk as investi
gated by Goldstein, which has a memory range equal to one. At each step, wi
th a probability p(1) the random walker moves either to the right or to the
left with equal probabilities, or with a probability q = 1 - p, performs a
move, which is a stochastic Boolean function of the nl previous steps. We
first derive the most general form of this stochastic Boolean function, and
study some typical cases which ensure that the average value < Rn > of the
walker's location after n steps is zero for all values of n. In each case,
using a matrix technique, we provide a general method for constructing the
generating function of the probability distribution of R-n; we also establ
ish directly an exact analytic expression for the step-step correlations an
d the variance <R-n(2)) of the walk. From the expression of <R-n(2)>, which
is not straightforward to derive from the probability distribution, we sho
w that, for n approaching infinity, the variance of any of these walks beha
ves as n, provided p > 0. Moreover, in many cases, for a very small fixed v
alue of p, the variance exhibits a crossover phenomenon as n increases from
a not too large value. The crossover takes place for values of n around 1/
p. This feature may mimic the existence of a nontrivial Hurst exponent, and
induce a misleading analysis of numerical data issued from mathematical or
natural sciences experiments.