On the significands of uniform random variables

Abstract

For all α > 0 and real random variables X, we establish sharp bounds for the smallest and the largest deviation of αX from the logarithmic distribution also known as Benford's law. In the case of uniform X, the value of the smallest possible deviation is determined explicitly. Our elementary calculation puts into perspective the recurring claims that a random variable conforms to Benford's law, at least approximately, whenever it has large spread.

The Mathematics of Benford's Law -- A Primer

“…No uniform random variable is close to Benford's law. In particular, by [5,Theorem 5.1], if X is a uniform random variable, i.e., X is uniformly distributed on [a, b] for some a < b, then for some 1 < t < 10, |P (S(X) ≤ t) − log t| > 0.0758 ; if X > 0 or X < 0 with probability one then the (lower) bound on the right is even larger, namely 0.134. Similar bounds away from Benford's law exist for normal and exponential distributions, for example, but for these distributions the corresponding sharp bounds are unknown [3, p. 40…”

“…Let U be uniformly distributed on [0, 1]. (i) U does not have scale-invariant digits since, for example, P (S(U) ≤ 2) = 1 9 but P (S(2U) ≤ 2) = 5 9 . (ii) As is easy to check directly, or follows immediately from the next theorem and Example 10 above, the random variable X = 10 U has scale-invariant significant digits.…”

“…Clearly, some random probability measures will not generate Benford behavior. For example, if P is P 1 half the time and P 2 half the time, where P 1 is uniformly distributed on [2,3] and P 2 is uniformly distributed on [4,5], then random samples from P will not have any entries with first significant digit 1, and hence cannot be close to Benford.…”

On Characterizations and Tests of Benford’s Law

“…Also in this case departure from Benford's law can be indexed by the shape parameter, since a Weibull random variable is reasonably close to be Benford when the shape parameter is not too large (Miller 2015, §3.5.3). Explicit error estimates for convergence of Weibull distributions to Benford's law can be found in Dümbgen andLeuenberger (2008, 2015), while Engel and Leuenberger (2003), Miller andNigrini (2008), andTwelves (2018) addressed the special case of the Exponential distribution. We take the scale parameter equal to 1, but similar results have been observed for different choices.…”

Best finite approximations of Benford's Law

“…Time and again, Benford's Law has emerged as a perplexingly prevalent phenomenon. One popular approach to understand this prevalence seeks to establish (mild) conditions on a probability measure that make (1.1) or (1.2) hold with good accuracy, perhaps even exactly [7,13,14,15,29]. It is the goal of the present article to provide precise quantitative information for this approach.…”