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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)
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)
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)
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)
This paper describes a new mathematical method called conflation for consolidating data from independent experiments that measure the same physical quantity. Conflation is easy to calculate and visualize and minimizes the maximum loss in Shannon information in consolidating several independent distributions into a single distribution. A formal mathematical(More)
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)