Like Sparrows on a Clothes Line: The Self-Organization of Random Number Sequences

We study sequences of random numbers {Z[1],Z[2],Z[3],...,Z[n]} -- which can be considered random walks with reflecting barriers -- and define their "types" according to whether Z[i] > Z[i+1], (a down-movement), or Z[i] < Z[i+1] (up-movement). This paper examines the means, xi, to which the Zi converge, when a large number of sequences of the same type is considered. It is shown that these means organize themselves in such a way that, between two turning points of the sequence, they are equidistant from one another. We also show that m steps in one direction tend to offset one step in the other direction, as m -> infinity. Key words:random number sequence, self-organization, random walk, reflecting barriers.