Extremal behavior of the heat random field

We study the exponential type inequalities for the distribution of the supremum of a random field that arises as the solution of the heat equation with a random initial condition that is a strictly sub-Gaussian random field. AMS 2000 Subject Classifications Primary—60G60; Secondary—60G17, 60F05     

Tail Behavior of a Threshold Autoregressive Stochastic Volatility Model

We consider a threshold autoregressive stochastic volatility model where the driving noises are sequences of iid regularly random variables. We prove that both the right and the left tails of the marginal distribution of the log-volatility process (α_t)_t are regularly varying with tail exponent −α with α > 0. We also determine the exact values of the coefficients in the tail behaviour of the process (α_t)_t. AMS 2000 Subject Classification. Primary—62G32, 62PO5     

Existence of Multivariate Max-Universal Laws

If suitably normalized maxima of an i.i.d. sample converge in distribution, the limiting distribution is known to be max-infinitely divisible and the common distribution of the sample is said to belong to its domain of attraction. We prove the existence of max-universal distributions belonging to the domain of attraction of every max-infinitely divisible law. The proof follows in the spirit of corresponding results for normalized sums of i.i.d. random variables originated by Doeblin and shows that necessarily the sampling size has to be rapidly increasing. Restricting the growth rate of the sampling size, we prove that one necessarily deals with max-semistable distributions and their domains of attraction. 2000 Mathematics subject classification Primary—60G70 Secondary—60E99, 60F05     

A large-deviations analysis of the GI/GI/1 SRPT queue

We consider a GI/GI/1 queue with the shortest remaining processing time discipline (SRPT) and light-tailed service times. Our interest is focused on the tail behavior of the sojourn-time distribution. We obtain a general expression for its large-deviations decay rate. The value of this decay rate critically depends on whether there is mass in the endpoint of the service-time distribution or not. An auxiliary priority queue, for which we obtain some new results, plays an important role in our analysis. We apply our SRPT results to compare SRPT with FIFO from a large-deviations point of view. 2000 Mathematics Subject Classification: Primary—60K25; Secondary—60F10; 90B22     

Euler’s constant, q-logarithms, and formulas of Ramanujan and Gosper

The aim of the paper is to relate computational and arithmetic questions about Euler’s constant γ with properties of the values of the q-logarithm function, with natural choice of~q. By these means, we generalize a classical formula for γ due to Ramanujan, together with Vacca’s and Gosper’s series for γ, as well as deduce irrationality criteria and tests and new asymptotic formulas for computing Euler’s constant. The main tools are Euler-type integrals and hypergeometric series. 2000 Mathematics Subject Classification Primary—11Y60; Secondary—11J72, 33C20, 33D15 The work of the second author is supported by an Alexander von Humboldt research fellowship Dedication: To Leonhard Euler on his 300th birthday.     

On the distribution of the number of customers in the symmetric M/G/1 queue

We consider an M/G/1 queue with symmetric service discipline. The class of symmetric service disciplines contains, in particular, the preemptive last-come-first-served discipline and the processor-sharing discipline. It has been conjectured in Kella et al. [1] that the marginal distribution of the queue length at any time is identical for all symmetric disciplines if the queue starts empty. In this paper we show that this conjecture is true if service requirements have an Erlang distribution. We also show by a counterexample, involving the hyperexponential distribution, that the conjecture is generally not true. AMS Subject Classifications Primary—60K25; Secondary—90B22     

Tail fit and the Zipf–Pareto law

A limit theorem with bounds on the rate of convergence is proven. The joint distribution of a fixed number of relative decrements of the top order statistics from a random sample converges to the limit as the sample size increases if and only if the underlying distribution is in essence a Pareto. In conjunction with a chi-square test of fit, it provides an asymptotically distribution-free test of fit to the family of distributions with regularly varying tails at infinity. When the limit distribution holds, rank-size plots obey Zipf’s law. The test can be used to detect departures from this Zipf–Pareto law.      

Benchmarking optimization software with performance profiles

We propose performance profiles — distribution functions for a performance metric — as a tool for benchmarking and comparing optimization software. We show that performance profiles combine the best features of other tools for performance evaluation. Received: February 2001 / Accepted: May 2001?Published online October 2, 2001     

Hereditary quasirandom properties of hypergraphs

Thomason and Chung, Graham and Wilson were the first to systematically investigate properties of quasirandom graphs. They have stated several quite disparate graph properties — such as having uniform edge distribution or containing a prescribed number of certain subgraphs — and proved that these properties are equivalent in a deterministic sense.     

Partition bijections, a survey

We present an extensive survey of bijective proofs of classical partitions identities. While most bijections are known, they are often presented in a different, sometimes unrecognizable way. Various extensions and generalizations are added in the form of exercises. 2000 Mathematics Subject Classification Primary—05A17; Secondary—05A30; 11P83 The author was partially supported by NSA and NSF grants.     

Some Asymptotic Results for the Number of Generalized Records

In this paper, we study asymptotic properties (large deviations and functional central limit theorem) of generalized record processes built on a triangular array of continuous and exchangeable random variables. As an application of these results, the links with the Kendall's rank correlation statistic are studied and testing exchangeability is discussed. AMS 2000 Subject Classification Primary—60F10 Secondary—60F17, 62G10     

OR/MS decision support models for the specialty crops industry: a literature review

Specialty crops (fruits, vegetables, grapes and wine, ornamentals—nursery and floriculture—tree nuts, berries, and dried fruits) comprise a substantial—and growing—portion of agribusiness. Still, the industry is facing a number of severe problems that must be resolved to sustain it and promote its continued growth. This paper is intended to provide growers and distribution managers insights into the variety of decision support possible and benchmarks for improvements they can help to achieve and to provide academic researchers with insights into industry operations and a vision of research needs.     

Approximation of multivariate distribution functions

In the paper the unknown distribution function is approximated with a known distribution function by means of Taylor expansion. For this approximation a new matrix operation — matrix integral — is introduced and studied in [PIHLAK, M.: Matrix integral, Linear Algebra Appl. 388 (2004), 315–325]. The approximation is applied in the bivariate case when the unknown distribution function is approximated with normal distribution function. An example on simulated data is also given.      

Elliptic grids, rational functions, and the Padé interpolation

We consider rational functions with n prescribed poles for which there exists a divided difference operator transforming them to rational functions with n−1 poles. The poles of such functions are shown to lie on the elliptic grids. There is a one-to-one correspondence between this problem of admissible grids and the Poncelet problem on two quadrics. Additionally, we outline an explicit scheme of the Padé interpolation with prescribed poles and zeros on the elliptic grids. Dedicated to Richard Askey on the occasion of his seventieth birthday. 2000 Mathematics Subject Classification Primary—42C05; Secondary—39A13, 41A05, 41A21.     

On the asymptotic distribution of the likelihood ratio under the regularity conditions due to doob

Summary The concepts of the asymptotic maximum likelihood estimates—AMLEs in short—and their asymptotic identity are introduced in section 1. They seem to be more adequate than the usual one for uses in the large sample theory. The AMLE is a slightly weakened version of the usual maximum likelihood estimate and therefore it should have a bit wider applicability than the original one. The asymptotic normality of a consistent AMLE and Wilks’ theorem concerning the asymptotic distribution of the statistic —2 log λ, where λ is the likelihood ratio, can be obtained under the regularity conditions due to Doob in section 2. A set of conditions which assure the existence of a unique and consistent AMLE is presented in section 3 and in the final section 4 the proof of the existence of the unique and consistent AMLE under those conditions is given. This work has been motivated by the work of Ogawa, Moustafa and Roy [3].     

On the normality a posteriori for exponential distributions,using the Bayesian estimation

Summary The Bayesian estimation problem for the parameter θ of an exponential probability distribution is considered, when it is assumed that θ has a natural conjugate prior density and a loss-function depending on the squared error is used. It is shown that, with probability one, the posterior density of the Bayesian—centered and scaled parameter converges pointwise to the normal probability density. The weak convergence of the posterior distributions to the normal distribution follows directly. Both correct and incorrect models are studied and the asymptotic normality is stated respectively.     

Regular Score Tests of Independence in Multivariate Extreme Values

The score tests of independence in multivariate extreme values derived by Tawn (Tawn, J.A., “Bivariate extreme value theory: models and estimation,” Biometrika 75, 397–415, 1988) and Ledford and Tawn (Ledford, A.W. and Tawn, J.A., “Statistics for near independence in multivariate extreme values,” Biometrika 83, 169–187, 1996) have non-regular properties that arise due to violations of the usual regularity conditions of maximum likelihood. Two distinct types of regularity violation are encountered in each of their likelihood frameworks: independence within the underlying model corresponding to a boundary point of the parameter space and the score function having an infinite second moment. For applications, the second form of regularity violation has the more important consequences, as it results in score statistics with non-standard normalisation and poor rates of convergence. The corresponding tests are difficult to use in practical situations because their asymptotic properties are unrepresentative of their behaviour for the sample sizes typical of applications, and extensive simulations may be needed in order to evaluate adequately their null distribution. Overcoming this difficulty is the primary focus of this paper. We propose a modification to the likelihood based approaches used by Tawn (Tawn, J.A., “Bivariate extreme value theory: models and estimation,” Biometrika 75, 397–415, 1988) and Ledford and Tawn (Ledford, A.W. and Tawn, J.A., “Statistics for near independence in multivariate extreme values,” Biometrika 83, 169–187, 1996) that provides asymptotically normal score tests of independence with regular normalisation and rapid convergence. The resulting tests are straightforward to implement and are beneficial in practical situations with realistic amounts of data. AMS 2000 Subject Classification Primary—60G70 Secondary—62H15     

Partial Sum Process for Records

Suppose the upper records from a sequence of i.i.d. random variables is in the domain of attraction of a normal distribution. Consider the D(0,1]-valued process {Z_n(·)} constructed by usual interpolation of the partial sums of the records. We prove that under some mild conditions, {Z_n} converges to a limiting Gaussian process in D(0,1]. As a consequence, the partial sums of records is asymptotically normal. AMS 2000 Subject Classification Primary—60F17 Secondary—60G70     

Markov model of connection between the distribution of static strength and fatigue life of a fibrous composite

Within the framework of a unified mathematical model based on the Markov chain theory, an attempt is made to describe the distribution of static strength, the fatigue curve, and the accumulation of fatigue damages. It is assumed that the fatigue failure of a test specimen occurs after the destruction of some its critical microvolume consisting of two — elastic (brittle fibers) and plastic (matrix) — parts. In the second part, plastic strains accumulate as soon as the cyclic load exceeds some level. Numerical examples are presented. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 5, pp. 615–630, September–October, 2006.     

On 2-stage robust LP with RHS uncertainty: complexity results and applications

We investigate here the class—denoted R-LP-RHSU—of two-stage robust linear programming problems with right-hand-side uncertainty. Such problems arise in many applications e.g: robust PERT scheduling (with uncertain task durations); robust maximum flow (with uncertain arc capacities); robust network capacity expansion problems; robust inventory management; some robust production planning problems in the context of power production/distribution systems. It is shown that such problems can be formulated as large scale linear programs with associated nonconvex separation subproblem. A formal proof of strong NP-hardness for the general case is then provided, and polynomially solvable subclasses are exhibited. Differences with other previously described robust LP problems (featuring row-wise uncertainty instead of column wise uncertainty) are highlighted.