Fundamental Flaws in Feller's Classical Derivation of Benford's Law

Feller's classic text 'An Introduction to Probability Theory and its Applications' contains a derivation of the well known significant-digit law called Benford's law. More specifically, Feller gives a sufficient condition ("large spread") for a random variable $X$ to be approximately Benford distributed, that is, for $\log_{10}X$ to be approximately uniformly distributed modulo one. This note shows that the large-spread derivation, which continues to be widely cited and used, contains serious basic errors. Concrete examples and a new inequality clearly demonstrate that large spread (or large spread on a logarithmic scale) does not imply that a random variable is approximately Benford distributed, for any reasonable definition of "spread" or measure of dispersionComment: 7 page

Bayesian Posteriors Without Bayes' Theorem

The classical Bayesian posterior arises naturally as the unique solution of several different optimization problems, without the necessity of interpreting data as conditional probabilities and then using Bayes' Theorem. For example, the classical Bayesian posterior is the unique posterior that minimizes the loss of Shannon information in combining the prior and the likelihood distributions. These results, direct corollaries of recent results about conflations of probability distributions, reinforce the use of Bayesian posteriors, and may help partially reconcile some of the differences between classical and Bayesian statistics.Comment: 6 pages, no figure

Regularity of Digits and Significant Digits of Random Variables

A random variable X is digit-regular (respectively, significant-digit-regular) if the probability that every block of k given consecutive digits (significant digits) appears in the b-adic expansion of X approaches b &supk; as the block moves to the right, for all integers b > 1 and k ? 1. Necessary and sufficient conditions are established, in terms of convergence of Fourier coefficients, and in terms of convergence in distribution modulo 1, for a random variable to be digit-regular (significant-digit regular), and basic relationships between digit-regularity and various classical classes of probability measures and normal numbers are given. These results provide a theoretical basis for analyses of roundoff errors in numerical algorithms which use floating-point arithmetic, and for detection of fraud in numerical data via using goodness-of-fit of the least significant digits to uniform, complementing recent tests for leading significant digits based on Benford's law.normal numbers, significant digits, Benford's law, digit-regular random variable, significant-digit-regular random variable, law of least significant digits, floating-point numbers, nonleading digits, trailing digits

Monotonically Improving Limit-Optimal Strategies in Finite-State Decision Processes

Suppose you are in a casino with a number of dollars you wish to gamble. You may quit whenever you please, and your objective is to find a strategy which will maximize the probability that you reach some goal, say $1000. In formal gambling-theoretic terminology, since there are only a finite number of dollars in the world, and since you may quit and leave whenever you wish, this is a finite-state leavable gambling problem [4], and the classical result of Dubins and Savage [4, Theorem 3.9.2.] says that for each e \u3e 0 there is always a stationary strategy which is uniformly e-optimal. That is, there is always a strategy for betting in which the bet you place at each play depends only on your current fortune, and using this strategy your expected fortune at the time you quit gambling is within e of the most you could expect under any strategy. In general, optimal stationary strategies do not always exist, even in finite-state leavable gambling problems [4, Example 3.9.2.] although they do if the number of bets available for each fortune is also finite [4, Theorem 3.9.1.], an assumption which certainly does not hold in a casino with an oddsmaker (someone who will let you bet any amount on practically any future event - he simply sets odds he considers favourable to the house). An e-optimal stationary strategy is by definition quite good, but it does have the disadvantage that it is not getting any better, and in general always remains e away from optimal at some states. The purpose of this paper is to introduce the notion of a strategy which is monotonically improving and optimal in the limit, and to prove that such strategies exist in all finite-state leavable gambling problems and in all finite-state Markov decision processes with positive, negative, and discounted pay-offs; in fact even Markov strategies [6] with these properties are shown to exist. The questions of whether monotonically improving limit-optimal (MILO) strategies exist in nonleavable finite-state gambling problems, in finite-state average reward Markov decision processes, or in countable state problems (with various pay-offs) are left open

A Proportionality Principle for Partitioning Problems

In a general class of measure-partitioning or fair-division problems, the extremal case occurs when the measures are proportional. Applications are given to classical and recent fair-division problems, and to statistical decision theory, mathematical physics, Banach space theory, and inequalities for continuous random variables

On the Oval Shapes of Beach Stones

This article introduces a new stochastic non-isotropic frictional abrasion model, in the form of a single short partial integro-differential equation, to show how frictional abrasion alone of a stone on a planar beach might lead to the oval shapes observed empirically. The underlying idea in this theory is the intuitive observation that the rate of ablation at a point on the surface of the stone is proportional to the product of the curvature of the stone at that point and the likelihood the stone is in contact with the beach at that point. Specifically, key roles in this new model are played by both the random wave process and the global (non-local) shape of the stone, i.e., its shape away from the point of contact with the beach. The underlying physical mechanism for this process is the conversion of energy from the wave process into the potential energy of the stone. No closed-form or even asymptotic solution is known for the basic equation, which is both non-linear and non-local. On the other hand, preliminary numerical experiments are presented in both the deterministic continuous-time setting using standard curve-shortening algorithms and a stochastic discrete-time polyhedral-slicing setting using Monte Carlo simulation