Theodore P. Hill

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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)
A derivation of Benford's Law or the First-Digit Phenomenon is given assuming only base-invariance of the underlying law. The only baseinvariant 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 individual elements of the vertebrate skeleton are separated by three different types of joints, fibrous, cartilaginous and synovial joints. Synovial joint formation in the limbs is coupled to the formation of the prechondrogenic condensations, which precede the formation of the joint interzone. We are beginning to understand the signals involved in the(More)
Let XI, X2, . . . be independent random variables taking values in [a, b], and let T denote the stop rules for X1, X2, Then E(sup,, X,,) sup{EX,: t E T) < (1/4)(b a), and this bound is best possible. Probabilistically, 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(More)
The pointwise limit S of a sequence of Stieltjes transforms (Sn) of real Borel probability measures (Pn) is itself the Stieltjes transform of a Borel p.m. P if and only if iy S(iy) → −1 as y → ∞, in which case Pn converges to P in distribution. Applications are given to several problems in mathematical physics.
For fair-division or cake-cutting problems with value functions which are normalized positive measures (i.e., the values are probability measures) maximin-share and minimax-envy inequalities are derived for both continuous and discrete measures. The tools used include classical and recent basic convexity results, as well as ad hoc constructions. Examples(More)
Implicitly defined (and easily approximated) universal constants 1.1 < an < 1.6, n = 2, 3, ... , are found so that if XI, X 2 , ••• are i.i.d. non-negative random variables and if Tn is the set of stop rules for Xl, "', Xn, then E(max{Xl , ••• ,Xn}) ~ an sup {EX, : tE Tn}, and the bound an is best possible. Similar universal constants 0 < bn < Y. are found(More)
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 only on the relative ranks of the observations. A(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 (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)