. For normed isotropic convex bodies in R n we investigate the behaviour of the (n \Gamma 1)-dimensional volume of intersections with hyperplanes orthogonal to a fixed direction, considered as a function of the distance of the hyperplane to the origin. It is a conjecture that for arbitrary normed isotropic convex bodies and random directions this function -- with high probability -- is close to a Gaussian density, for large dimension n. This would be a kind of central limit theorem. We determine this function explicitly for several families of convex bodies and several directions and obtain results concerning the asymptotic behaviour supporting the conjecture. MSC 2000: 52A21 (primary), 60F25 (secondary) Keywords: convex body, isotropic, cross section, central limit theorem, marginal distribution Introduction The main topic of the present paper is a version of the central limit theorem in the geometric context of convex bodies. A normed convex body K ` R n is a convex compact se...

We analyze the natural process of flipping a coin which is caught in the hand. We prove that vigorously-flipped coins are biased to come up the same way they started. The amount of bias depends on a single parameter, the angle between the normal to the coin and the angular momentum vector. Measurements of this parameter based on high-speed photography are reported. For natural flips, the chance of coming up as started is about.51.

. A perturbation technique can be used to simplify and sharpen A. C. Yao's theorems about the behavior of shellsort with increments (h; g; 1). In particular, when h = \Theta(n 7=15 ) and g = \Theta(h 1=5 ), the average running time is O(n 23=15 ). The proof involves interesting properties of the inversions in random permutations that have been h-sorted and g-sorted. Shellsort, also known as the "diminishing increment sort" [7, Algorithm 5.2.1D], puts the elements of an array (X 0 ; : : : ; X n\Gamma1 ) into order by successively performing a straight insertion sort on larger and larger subarrays of equally spaced elements. The algorithm consists of t passes defined by increments (h t\Gamma1 ; : : : ; h 1 ; h 0 ), where h 0 = 1; the jth pass makes X k X l whenever l \Gamma k = h t\Gammaj . A. C. Yao [11] has analyzed the average behavior of shellsort in the general three-pass case when the increments are (h; g; 1). The most interesting part of his analysis dealt with the third ...

Improved mixing time bounds for the Thorp shuffle and L-reversal chain Ben Morris ∗ We prove a theorem that reduces bounding the mixing time of a card shuffle to verifying a condition that involves only pairs of cards, then we use it to obtain improved bounds for two previously studied models. E. Thorp introduced the following card shuffling model in 1973. Suppose the number of cards n is even. Cut the deck into two equal piles. Drop the first card from the left pile or from the right pile according to the outcome of a fair coin flip. Then drop from the other pile. Continue this way until both piles are empty. We obtain a mixing time bound of O(log 4 n). Previously, the best known bound was O(log 29 n) and previous proofs were only valid for n a power of 2. We also analyze the following model, called the L-reversal chain, introduced by Durrett. There are n cards arrayed in a circle. Each step, an interval of cards of length at most L is chosen uniformly at random and its order is reversed. Durrett has conjectured that the mixing time is O(max(n, n3 L3)log n). We obtain a bound that is within a factor O(log 2 n) of this, the first bound within a poly log factor of the conjecture. 1

Abstract. We analyze the natural process of flipping a coin which is caught in the hand. We show that vigorously flipped coins tend to come up the same way they started. The limiting chance of coming up this way depends on a single parameter, the angle between the normal to the coin and the angular momentum vector. Measurements of this parameter based on high-speed photography are reported. For natural flips, the chance of coming up as started is about.51. Key words. Berry phase, randomness, precession, image analysis

Kepler attempted to prove Divine design in the system of the world but actually had to attribute the eccentricities of the planetary orbits to randomness. Kant and even Laplace supported Kepler’s conclusion although Newton had proved that the eccentricities depended on the velocities of planetary motion. However, the velocities themselves are random; the system of the world does not exclude randomness. Key words: eccentricities of planetary orbits; randomness in nature; system of the world. 1. Randomness: General Information Aristotle 1 and other early scientists and philosophers attempted to define, or at least to throw light upon randomness. His examples of random events are a sudden meeting of two acquaintances (Phys. 196b30) and a sudden unearthing of a buried treasure (Metaphys. 1025a). In both cases the event occurred without being aimed at. Many ancient authors

This, now slightly revised text, is intended for a broader circle of readers. It appeared in Italian, although with suppressed references, as Lo sviluppo della teoria della probabilità e della statistica in Storia della

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L’Angleterre et les statistiques pendant l’entre-deux-guerres est un sujet qui a été abondamment traité mais l’Angleterre et les probabilités dans la même période semble avoir été peu considéré. Cela peut sembler paradoxal. Cet article considère la scène probabiliste anglaise et examine les travaux de Turing, Paley et Linfoot, en phase avec la manière de l’Europe continentale. Nous examinons également l’attitude des statisticiens aux travaux continentaux exprimés dans les réactions à la thèse de Turing sur le théorème central de la limite et au traité de Harald Cramér Random Variables and Probability Distributions. Quelques documents originaux sont reproduits et fournissent des éléments pour la discussion. England and statistical theory in the inter-war period is a subject that abounds with material but England and probability theory may seem empty. This may seem a paradoxical situation. This paper considers the English probability scene and examines work in the continental manner by Turing, Paley and Linfoot. It also considers the response of statisticians to continental work as manifested in the reactions to Turing’s fellowship dissertation on the central limit theorem and Harald Cramér’s tract Random Variables and Probability Distributions. Some documents from the time are reproduced and they provide the focus for the discussion. Acknowledgement: I am grateful to two anonymous referees for their comments.

This paper presents a considerably improved version of the notion of event-differentiability from Machina (1992). An alternative definition has been independently developed by Epstein (1999) in his analysis of the concept of uncertainty aversion.