Differences between Independent Variables and Almost Benford Behavior

Fix a base B and let X1, . . . , XN be independent identically distributed random variables. If the Xi’s are drawn from a uniform distribution, then as N → ∞ the distribution of the digits of the differences between adjacent Xi’s tends to a universal distribution which is almost Benford’s Law; we call this Almost Benford behavior. For each base we develop a rapidly convergent Fourier series expansion. In base e one term yields five digits of accuracy; in base 10 two terms yield three digits. Fix a δ ∈ (0, 1) and choose N independent random variables from a nice probability density. The distribution of digits of any N consecutive differences and all N −1 normalized differences of the Xi’s exhibit Almost Benford behavior. We derive conditions on the probability density which determine whether or not the distribution of the digits of all the un-normalized differences converges to Benford’s Law, Almost Benford behavior, or oscillates between the two. As an example the Pareto distribution leads to oscillating behavior. We introduce a new technique to study equidistribution questions modulo 1; such questions have long been known to be related to Benford’s Law. By differentiating the cumulative distribution function of the logarithms modulo 1, applying Poisson Summation and then integrating the resulting expression, we derive rapidly converging explicit formulas measuring the deviations from Benford’s Law.