Random Walks with Decreasing Steps

We consider a one-dimensional random walk beginning at the origin and moving ± cn at time n. If ∑ cn < ∞ then with probability one the random walk has a final location. We describe the distribution of the final location and find the cumulants and the moments in terms of the power sums of the step sizes. Some interesting definite integrals involving infinite products of trigonometric functions can be evaluated probabilistically. Similar results hold for random walks with steps chosen uniformly in [−bn, bn], assuming that ∑ bn <∞. 1 Harmonic Random Walk Imagine a random walker who is not only staggering but also tiring. In contrast to a simple random walker who takes steps of equal length but random direction, our harmonic random walker first takes a step of length 1, then a step of length 1/2, then a step of length 1/3, and so on. As in a simple random walk the direction of each step is equally likely to be left as right and independent of the other steps. Assume the walk begins at the origin at time zero. What is the distribution of the location as time goes to infinity? For the simple random walk the location at time n has a binomial distribution with variance n (assuming steps of unit length), and so there is not a limiting distribution unless rescaling is done as in the Central Limit Theorem in which the location at time n is scaled to units of √ n so that the distribution tends to a standard normal distribution. However, the harmonic random walk does have a limiting distribution for the location as time goes to infinity, as do many other random walks with variable steps. The fundamental probability space Ω consists of countable binary sequences to represent the sequence of left-right choices in a random walk. We identify Ω with the interval [0, 1] by means of the binary representation of numbers in [0, 1] and we use Lebesgue measure on [0, 1] as the probability measure.