Refined upper bounds for the linear Diophantine problem of Frobenius

We study the Frobenius problem: given relatively prime positive integers a_1,...,a_d, find the largest value of t (the Frobenius number g(a_1,...,a_d)) such that m_1 a_1 + ... m_d a_d = t has no solution in nonnegative integers m_1,...,m_d. We introduce a method to compute upper bounds for g(a_1,a_2,a_3), which seem to grow considerably slower than previously known bounds. Our computations are based on a formula for the restricted partition function, which involves Dedekind-Rademacher sums, and the reciprocity law for these sums.Comment: 12 pages, 5 figure

Compound Poisson process with a Poisson subordinator

A compound Poisson process whose randomized time is an independent Poisson process is called compound Poisson process with Poisson subordinator. We provide its probability distribution, which is expressed in terms of the Bell polynomials, and investigate in detail both the special cases in which the compound Poisson process has exponential jumps and normal jumps. Then for the iterated Poisson process we discuss some properties and provide convergence results to a Poisson process. The first-crossing-time problem for the iterated Poisson process is finally tackled in the cases of (i) a decreasing and constant boundary, where we provide some closed-form results, and (ii) a linearly increasing boundary, where we propose an iterative procedure to compute the first-crossing-time density and survival functions.Comment: 16 pages, 7 figure

Higher-dimensional Dedekind sums and their bounds arising from the discrete diagonal of the n-cube

AbstractHigher-dimensional Dedekind sums are defined as a generalization of a recent one-dimensional probability model of Dilcher and Girstmair to a d-dimensional cube. The analysis of the frequency distribution of diagonal lattice points leads to new formulae in certain special cases, and also to new bounds for the classical Dedekind sums. We define a new correspondence between n-dimensional Dedekind sums and certain convex n-dimensional cones, and we conjecture that these cones have a largest spacial angle of π/6. Bounds on n-dimensional Dedekind sums are important in the enumeration of lattice points in polytopes, since they are the building blocks for the lattice point enumerator of a polytope. Here, upper bounds for n-dimensional Dedekind sums are expressed in terms of 1-dimensional moments, and various relations among the moments are derived using statistical methods

Testing the Independence of Poisson Variates under the Holgate Bivariate Distribution: The Power of a New Evidence Test.

A new Evidence Test is applied to the problem of testing whether two Poisson random variables are dependent. The dependence structure is that of Holgate’s bivariate distribution. These bivariate distribution depends on three parameters, 0 \u3c theta_1, theta_2 \u3c infty, and 0 \u3c theta_3 \u3c min(theta_1, theta_2). The Evidence Test was originally developed as a Bayesian test, but in the present paper it is compared to the best known test of the hypothesis of independence in a frequentist framework. It is shown that the Evidence Test is considerably more powerful when the correlation is not too close to zero, even for small samples

Sample path analysis and distributions of boundary crossing times

This monograph is focused on the derivations of exact distributions of first boundary crossing times of Poisson processes, compound Poisson processes, and more general renewal processes.  The content is limited to the distributions of first boundary crossing times and their applications to various stochastic models. This book provides the theory and techniques for exact computations of distributions and moments of level crossing times. In addition, these techniques could replace simulations in many cases, thus providing more insight about the phenomenona studied. This book takes a general approach for studying telegraph processes and is based on nearly thirty published papers by the author and collaborators over the past twenty five years.  No prior knowledge of advanced probability is required, making the book widely available to students and researchers in applied probability, operations research, applied physics, and applied mathematics. 

Examples and problems in mathematical statistics

This book presents examples that illustrate the theory of mathematical statistics and details how to apply the methods for solving problems.  While other books on the topic contain problems and exercises, they do not focus on problem solving. This book fills an important niche in the statistical theory literature by providing a theory/example/problem approach.  Each chapter is divided into four parts: Part I provides the needed theory so readers can become familiar with the concepts, notations, and proven results; Part II presents examples from a variety of fields including engineering, mathe