Correlated Random Walks With A Finite Memory Range

We study a family of correlated one-dimensional random walks with a finite memory rangeM. 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 probabilityp, the random walker moves either to the right or to the left with equal probabilities, or with a probabilityq = 1 -pperforms a move, which is a stochastic Boolean function of theMprevious steps. We first derive the most general form of this stochastic Boolean function, and study some typical cases which ensure that the average value of the walker's location afternsteps is zero for all values ofn. In each case, using a matrix technique, we provide a general method for constructing the generating function of the probability distribution ofRn; we also establish directly an exact analytic expression for the step–step correlations and the variance$\langle R^2_n\rangle$of the walk. From the expression of$\langle R^2_n\rangle$, which is not straightforward to derive from the probability distribution, we show that, fornapproaching infinity, the variance of any of these walks behaves asn, providedp > 0. Moreover, in many cases, for a very small fixed value ofp, the variance exhibits a crossover phenomenon asnincreases from a not too large value. The crossover takes place for values ofnaround1/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.