Research Scholars in Residence

A Prophet Inequality Related to the Secretary Problem

Theodore P. Hill et al. — Tue, 06 Mar 2012 14:50:29 PST

Let Z1, Z2 , .. . , Zn be independent 0-1-valued random variables. A gambler gels a. reward 1 if he stop8 a.t the time of the last success and otherwise gets no reward. A simple comparison with a Poisson process is used to show that a prophet can do at most e times as well as the gambler using an optimal stopping time. For fixed n, the best constant is (n/(n -l ))"-1.

Finite-state Markov Chains Obey Benford’s Law

Arno Berger et al. — Tue, 07 Feb 2012 08:42:21 PST

A sequence of real numbers(x_n)is Benford if the significands, i.e. the fraction parts in the floating-point representation of (x_n), are distributed logarithmically. Similarly, a discrete-time irreducible and aperiodic finite-state Markov chain with probability transition matrix P and limiting matrix P* is Benford if every component of both sequences of matrices (Pⁿ −P*)and (P ⁿ⁺¹-Pⁿ) is Benford or eventually zero. Using recent tools that established Benford behavior both for Newton’s method and for finite-dimensional linear maps, via the classical theories of uniform distribution modulo 1 and Perron-Frobenius, this paper derives a simple sufficient condition (“nonresonance”) guaranteeing that P, or the Markov chain associated with it, is Benford. This result in turn is used to show that almost all Markov chains are Benford, in the sense that if the transition probabilities are chosen independently and continuously, then the resulting Markov chain is Benford with probability one. Concrete examples illustrate the various cases that arise, and the theory is complemented with several simulations and potential applications.

A Basic Theory of Benford’s Law

Arno Berger et al. — Thu, 26 Jan 2012 13:50:53 PST

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 developed in detail. Special attention is given to the emergence of BL in a wide variety of deterministic and random processes. Though mainly expository in nature, the article also provides strengthened versions of, and simplified proofs for, many key results in the literature. Numerous intriguing problems for future research arise naturally.

Conflations of Probability Distributions

Theodore P. Hill — Thu, 26 Jan 2012 13:49:37 PST

The conflation of a finite number of probability distributions P₁,...,P_n is a consolidation of those distributions into a single probability distribution Q = Q(P₁,...,P_n), where intuitively Q is the conditional distribution of independent random variables X1,...,X_n with distributions P₁,...,P_n, respectively, given that X₁ = ···= X_n. Thus, in large classes of distributions the conflation is the distribution determined by the normalized product of the probability density or probability mass functions. Q is shown to be the unique probability distribution that minimizes the loss of Shannon information in consolidating the combined information from P₁,...,P_n into a single distribution Q, and also to be the optimal consolidation of the distributions with respect to two minimax likelihood-ratio criteria. In that sense, conflation may be viewed as an optimal method for combining the results from several different independent experiments. When P₁,...,P_n are Gaussian, Q is Gaussian with mean the classical weighted-mean-squares reciprocal of variances. A version of the classical convolution theorem holds for conflations of a large class of a.c. measures.

Criticisms of the proposed “new SI”

Theodore P. Hill — Mon, 17 Oct 2011 16:48:36 PDT

Obituary for Lester Eli Dubins, 1921-2010

David Gilat et al. — Wed, 13 Jul 2011 11:34:58 PDT

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

Arno Berger et al. — Thu, 02 Jun 2011 12:49:03 PDT

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, Fellergives a sufficient condition (“large spread”) for a random variable X to be approximately Benford distributed, that is, for log₁₀X to be approximately uniformly distributed moduloone. 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 demonstratethat larges pread (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 dispersion.

Counterexamples in the Theory of Fair Division

Theodore P. Hill et al. — Fri, 20 May 2011 08:41:12 PDT

How to Publish Counterexamples in 1 2 3 Easy Steps

Theodore P. Hill — Fri, 20 May 2011 08:41:09 PDT

Hoisting the Black Flag

Theodore P. Hill — Fri, 20 May 2011 08:41:05 PDT

Cutting Cakes Carefully

Theodore P. Hill et al. — Fri, 20 May 2011 08:41:00 PDT

Benford’s Law Strikes Back: No Simple Explanation in Sight for Mathematical Gem

Arno Berger et al. — Fri, 20 May 2011 08:40:57 PDT

Towards a Better Definition of the Kilogram

Theodore P. Hill et al. — Fri, 20 May 2011 08:40:53 PDT

It is widely accepted that improvement of the current International System of Units (SI) is necessary, and that central to this problem is redefinition of the kilogram. This paper compares the relative advantages of two main proposals for a modern scientific definition of the kilogram: an ‘electronic kilogram’ based on a fixed value of Planck’s constant, and an ‘atomic kilogram’ based on a fixed value for Avogadro’s number. A concrete and straightforward atomic definition of the kilogram is proposed. This definition is argued to be more experimentally neutral than the electronic kilogram, more realizable by school and university laboratories than the electronic kilogram, and more readily comprehensible than the electronic kilogram.

A Stronger Conclusion to the Classical Ham Sandwich Theorem

John H. Elton et al. — Fri, 20 May 2011 08:40:49 PDT

The conclusion of the classical ham sandwich theorem for bounded Borel sets may be strengthened, without additional hypotheses – there always exists a common bisecting hyperplane that touches each of the sets, that is, that intersects the closure of each set. In the discrete setting, where the sets are finite (and the measures are counting measures), there always exists a bisecting hyperplane that contains at least one point in each of the sets. Both these results follow from the main theorem of this note, which says that for n compactly supported positive finite Borel measures in Rⁿ, there is always an (n − 1)-dimensional hyperplane that bisects each of the measures and intersects the support of each measure. Thus, for example, at any given instant of time, there is one planet, one moon and one asteroid in our solar system and a single plane touching all three that exactly bisects the total planetary mass, the total lunar mass, and the total asteroidal mass of the solar system. In contrast to the bisection conclusion of the classical ham sandwich theorem, this bisection-and-intersection conclusion does not carry over to unbounded sets of finite measure.

Goal Problems in Gambling Theory

Theodore P. Hill — Fri, 17 Sep 2010 12:17:46 PDT

A short introduction to goal problems in abstract gambling theory is given, along with statements of some of the main theorems and a number of examples, open problems and references. Emphasis is on the finite-state, countably-additive setting with such classical objectives as reaching a goal, hitting a goal infinitely often, staying in the goal, and maximizing the average time spent at a goal.

A characterisation of Newton maps

Arno Berger et al. — Mon, 23 Aug 2010 10:31:52 PDT

Conditions are given for a C^k map T to be a Newton map, that is, the map associated with a differentiable real-valued function via Newton’s method. For finitely differentiable maps and functions, these conditions are only necessary, but in the smooth case, i.e. for k = ∞ , they are also sufficient. The characterisation rests upon the structure of the fixed point set of T and the value of the derivative T¹ there, and it is best possible as is demonstrated through examples.

Regularity of digits and significant digits of random variables

Theodore P. Hill et al. — Thu, 19 Aug 2010 13:09:11 PDT

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^-k 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.

On the Existence of Good Markov Strategies

Theodore P. Hill — Thu, 19 Aug 2010 12:47:31 PDT

In contrast to the known fact that there are gambling problems based on a finite state space for which no stationary family of strategies is at all good, in every such problem there always exist ε-optimal Markov families (in which the strategy depends only on the current state and time) and also ε-optimal tracking families (in which the strategy depends only on the current state and the number of times that state has been previously visited). More generally, this result holds for all finite state gambling problems with a payoff which is shift and permutation invariant.

Additive Comparisons of Stop Rule and Supremum Expectations of Uniformly Bounded Independent Random Variables

Theodore P. Hill et al. — Thu, 19 Aug 2010 12:47:30 PDT

Let XI, X2, . . . be independent random variables taking values in [a, b], and let T denote the stop rules for X1, X2, Then E(supn>1 Xn) - sup{ EXt t ≡ 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 rules), the game becomes at least fair for the gambler.

Ratio Comparisons of Supremum and Stop Rule Expectations

Theodore P. Hill et al. — Thu, 19 Aug 2010 12:47:29 PDT

Suppose X₁,X₂,...,X_n are independent non-negative random variables with finite positive expectations. Let T_n denote the stop rules for X₁,...,X_n. The main result of this paper is that E(max{X₁,...,X_n }) <2 >sup{EX_t t ε T_n }. The proof given is constructive, and sharpens the corresponding weak inequalities of Krengel and Sucheston and of Garling.