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- Theodore P. Hill
- 1996

The history, empirical evidence, and classical explanations of the Significant-Digit (or Benford's) Law are reviewed, followed by a summary of recent invariant-measure characterizations and then a new statistical derivation of the law in the form of a CLT-like theorem for significant digits. If distributions are selected at random (in any " unbiased " way),… (More)

- Theodore P Hill
- 2008

A derivation of Benford's Law or the First-Digit Phenomenon is given assuming only base-invariance of the underlying law. The only base-invariant distributions are shown to be convex combinations of two extremal probabilities, one corresponding to point mass and the other a log-Lebesgue measure. The main tools in the proof are identification of an… (More)

The pointwise limit S of a sequence of Stieltjes transforms (S n) of real Borel probability measures (P n) is itself the Stieltjes transform of a Borel p.m. P if and only if iy S(iy) → −1 as y → ∞, in which case P n converges to P in distribution. Applications are given to several problems in mathematical physics. probability measures. Lévy's classical… (More)

- David Gllat, T P Hill
- 2006

Dedicated to Lester Dubins on his seventieth birthday. A one-sided refinement of the strong law of large numbers is found for which the partial weighted sums not only converge almost surely to the expected value, but also the convergence is such that eventually the partial sums all exceed the expected value. The new weights are distribution-free, depending… (More)

Near a stable fixed point at 0 or ∞, many real-valued dynamical systems follow Benford's law: under iteration of a map T the proportion of values in {x, T (x), T 2 (x),. .. , T n (x)} with mantissa (base b) less than t tends to log b t for all t in [1, b) as n → ∞, for all integer bases b > 1. In particular , the orbits under most power, exponential, and… (More)

- T. P. Hill
- 2008

If asked whether certain digits in numbers collected randomly from, for example, the front pages of newspapers or from stock-market prices should occur more often than others, most people would think not. Nonetheless, in 1881, the astronomer and mathematician Simon Newcomb published a two-page article in the American Journal of Mathematics reporting an… (More)

Starting with a Borel probability measure P on X (where X is a separable Banach space or a compact metrizable convex subset of a locally convex topological vector space), the class Y(P), called the fusions of P, consists of all Borel probability measures on X which can be obtained from P by fusing parts of the mass of P, that is, by collapsing parts of the… (More)

(1/4)(b-a), and this bound is best possible. Probabilis-tically, this says that if a prophet (player with complete foresight) makes a side payment of (b-a)/8 to a gambler (player using nonanticipating stop rules), the game becomes at least fair for the gambler.

- Theodore P. Hill
- 2010

If Ill' ... , ~tn are non-atomic probability measures on the same measurable space (S, ff"), then there is an ff"-measurable partition {AJ7= 1 of S so that IlJAi)~(n-l+m)-l for all i=I, ... ,n, where m=l l i 01 lli l l is the total mass of the largest measure dominated by each of the Il/s; moreover, this bound is attained for all n ~ 1 and all m in [0,1].… (More)

Drawing from a large, diverse body of work, this survey presents a comprehensive and unified introduction to the mathematics underlying the prevalent logarithmic distribution of significant digits and significands, often referred to as Benford's Law (BL) or, in a special case, as the First Digit Law. The invariance properties that characterize BL are… (More)