Order Statistics and Benford's Law

Fix a base B > 1 and let ζ have the standard exponential distribution; the distribution of digits of ζ base B is known to be very close to Benford’s law. If there exists a C such that the distribution of digits of C times the elements of some set is the same as that of ζ, we say that set exhibits shifted exponential behavior base B. Let X1, . . . , XN be i.i.d.r.v. If the Xi’s are Unif, then as N → ∞ the distribution of the digits of the differences between adjacent order statistics converges to shifted exponential behavior. If instead Xi’s come from a compactly supported distribution with uniformly bounded first and second derivatives and a second-order Taylor series expansion at each point, then the distribution of digits of any N consecutive differences and all N − 1 normalized differences of the order statistics exhibit shifted exponential behavior. We derive conditions on the probability density which determine whether or not the distribution of the digits of all the unnormalized differences converges to Benford’s law, shifted exponential behavior, or oscillates between the two, and show that the Pareto distribution leads to oscillating behavior.