Gordon Simons

On the theory of elliptically contoured distributions

The theory of elliptically contoured distributions is presented in an unrestricted setting, with no moment restrictions or assumptions of absolute continuity. These distributions are defined parametrically through their characteristic functions and then studied primarily through the use of stochastic representations which naturally follow from the work of Schoenberg [5] on spherically symmetric distributions. It is shown that the conditional distributions of elliptically contoured distributions are elliptically contoured, and the conditional distributions are precisely identified. In addition, a number of the properties of normal distributions (which constitute a type of elliptically contoured distributions) are shown, in fact, to characterize normality.

Inequalities for Ek(X,Y) When the Marginals are Fixed.

When k(x, y) is a quasi-monotone function and the random variables X and Y have fixed distributions, it is shown under some further mild conditions that ℰ k(X, Y) is a monotone functional of the joint distribution function of X and Y. Its infimum and supremum are both attained and correspond to explicitly described joint distribution functions.

On [alpha]-symmetric multivariate distributions

A random vector is said to have a 1-symmetric distribution if its characteristic function is of the form [phi](t1 + ... + tn). 1-Symmetric distributions are characterized through representations of the admissible functions [phi] and through stochastic representations of the radom vectors, and some of their properties are studied.

Unbiased Coin Tossing with a Biased Coin

PrQcedures are exhibited and analyzed for converting a sequence of iid Bernoulli variables with unknown mean p into a Bernoulli variable with mean 1/2. The efficiency of several procedures is studied.

A strong law of large numbers for trimmed sums, with applications to generalized St. Petersburg games

Extending a result of Einmahl, Haeusler and Mason (1988), a characterization of the almost sure asymptotic stability of lightly trimmed sums of upper order statistics is given when the right tail of the underlying distribution with positive support is surrounded by tails that are regularly varying with the same index. The result is motivated by applications to cumulative gains in a sequence of generalized St. Petersburg games in which a fixed number of the largest gains of the player may be withheld.

Probability and expectation inequalities

SummaryThis paper introduces a mathematical framework within which a wide variety of known and new inequalities can be viewed from a common perspective. Probability and expectation inequalities of the following types are considered: (a)P(ZεA)≧ P(Z′εA) for some class of setsA, (b)ℰk(Z)≧ℰk(Z′) for some class of functionsk, and (c)ℰl(Z)≧ℰk(Z′) for some class of pairs of functionsl andk. It is shown, sometimes using explicit constructions ofZ andZ′, that, in several cases, (a) ⇔ (b) ⇔ (c); included here are cases of normal and elliptically contoured distributions. A case where (a) ⇒ (b) ⇔ (c) is studied and is expressed in terms of“n monotone” functions for (some of) which integral representations are obtained. Also, necessary and sufficient conditions for (c) are given.

Approximating the inverse of a symmetric positive definite matrix

Abstract It is shown for an n × n symmetric positive definite matrix T = ( t i , j with negative off-diagonal elements, positive row sums and satisfying certain bounding conditions that its inverse is well approximated, uniformly to order l/ n 2 , by a matrix S = ( s i , j ), where s i , j = δ i , j / t i , j + 1/ t.. , δ i , j being the Kronecker delta function, and t .. being the sum of the elements of T . An explicit bound on the approximation error is provided.

A discrete analogue and elementary derivation of ‘Lévy's equivalence’ for Brownian motion

The present note presents a discrete analogue of Levys extended equivalence for symmetric simple random walks and provides an elementary derivation of Levys basic and extended based upon this analogue. Finally, it describes an almost sure version of the extended equivalence depending on an Ito-type stochastic integral.

SOME RESULTS ON THE BOMBER PROBLEM

The problem of optimally allocating partially effective, defensive weapons against randomly arriving enemy aircraft so that a bomber maximizes its probability of reaching its designated target is considered in the usual continuous-time context, and in a discrete-time context. The problem becomes that of determining the optimal number of missiles K(n, t) to use against an enemy aircraft encountered at time (distance) t away from the target when n is the number of remaining weapons (missiles) in the bombers arsenal. Various questions associated with the properties of the function K are explored including the long-standing, unproven conjecture that it is a non-decreasing function of its first variable. POISSON PROCESS; OPTIMAL ALLOCATION; TOTAL POSITIVITY

LAWS OF LARGE NUMBERS FOR COOPERATIVE ST. PETERSBURG GAMBLERS

SummaryGeneral linear combinations of independent winnings in generalized \St~Petersburg games are interpreted as individual gains that result from pooling strategies of different cooperative players. A weak law of large numbers is proved for all such combinations, along with some almost sure results for the smallest and largest accumulation points, and a considerable body of earlier literature is fitted into this cooperative framework. Corresponding weak laws are also established, both conditionally and unconditionally, for random pooling strategies.