  We study a family of correlated one-dimensional random walks with a finite
memory range M.These walks are extensions of the Taylor's walk as investigated
by Goldstein, which has a memory range equal to one. At each step, with a
probability p, 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 M 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 <R_n> 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 establish directly an exact analytic
expression for the step-step correlations and 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 show that, for n going to infinity, the
variance of any of these walks behaves as n, provided p>0. Moreover, in many
cases, for a very small fixed value 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
non-trivial Hurst exponent, and induce a misleading analysis of numerical data
issued from mathematical or natural sciences experiments.
