An identity on the maximum of a set of random variables

It is shown that if X₁, X₂, …, X_n are symmetric random variables and max(X₁, …, X_n)⁺ = max(0, X₁, …, X_n), then E[max(X₁,…,X_n)⁺]=[max(X₁,X₁,+X₂,+X₁,+X₃,…X₁,+X_n)⁺], and in the case of independent identically distributed symmetric random variables, E[max(X₁, X₂)⁺] = E[(X₁)⁺] + (1/2)E[(X₁ + X₂)⁺], so that for independent standard normal random variables, E[max(X₁, X₂)⁺] = (1/√2π)[1 + (1/√2)].     

Stronger Lagrangian bounds by use of slack variables: Applications to machine scheduling problems

Lagrangian relaxation is a powerful bounding technique that has been applied successfully to manyNP-hard combinatorial optimization problems. The basic idea is to see anNP-hard problem as an easy-to-solve problem complicated by a number of nasty side constraints. We show that reformulating nasty inequality constraints as equalities by using slack variables leads to stronger lower bounds. The trick is widely applicable, but we focus on a broad class of machine scheduling problems for which it is particularly useful. We provide promising computational results for three problems belonging to this class for which Lagrangian bounds have appeared in the literature: the single-machine problem of minimizing total weighted completion time subject to precedence constraints, the two-machine flow-shop problem of minimizing total completion time, and the single-machine problem of minimizing total weighted tardiness.     

Better saddlepoint confidence intervals via bootstrap calibration

Confidence interval construction for parameters of lattice distributions is considered. By using saddlepoint formulas and bootstrap calibration, we obtain relatively short intervals and bounds with coverage errors, in contrast with and coverage errors for normal theory intervals and bounds when the population distribution is absolutely continuous. Closed form solutions are also provided for the cases of binomial and Poisson distributions. The method is illustrated by some simulation results.

     

Co-countable sets of uniqueness for series of independent random variables

Given a sequence of independent random variables (fk) on a standard Borel space Ω with probability measure μ and a measurable set F, the existence of a countable set S⊂F is shown, with the property that series k_∑ckfk which are constant on S are constant almost everywhere on F. As a consequence, if the functions fk are not constant almost everywhere, then there is a countable set S⊂Ω such that the only series k_∑ckfk which is null on S is the null series; moreover, if there exists b<1 such that for every k and every α, then the set S can be taken inside any measurable set F with μ(F)>b.     

A note on summability of multiple Laguerre expansions

A simple structure of the multiple Laguerre polynomial expansions is used to study the Cesàro summability above the critical index for the convolution type Laguerre expansions. The multiple Laguerre polynomial expansion of an -radial function is shown to be an -radial function that coincides with the Laguerre polynomial expansion of , which allows us to settle the problem of summability below the critical index for the -radial functions.

     

A maximal inequality for partial sums of finite exchangeable sequences of random variables

Let be a finite exchangeable sequence of Banach space valued random variables, i.e., a sequence such that all joint distributions are invariant under permutations of the variables. We prove that there is an absolute constant such that if , then

for all . This generalizes an inequality of Montgomery-Smith and Lata{\l}a for independent and identically distributed random variables. Our maximal inequality is apparently new even if is an infinite exchangeable sequence of random variables. As a corollary of our result, we obtain a comparison inequality for tail probabilities of sums of arbitrary random variables over random subsets of the indices.

     

A characterization of probability distributions by mean values

Let (X) be a measurable complex function onR;X, Y, Z be i.i.d. random variables; and (t, u, v)=E(tX+uY+nZ), wheret, u, vR. In this paper we describe a class of function (x) such that the distribution ofX, Y, Z is determined by the funetion (t, u, v). The main result is a generalization of the author's characterization of normal and stable distributions.     

Which I.I.D. Sums are recurrently dominated by their maximal terms?

For i.i.d. sequences (X _n) we characterize lim sup _n X _n/sup₁i<n|S _i |= in terms of the distribution function. Previous results of Kesten⁽³⁾ and Wittmann⁽⁶⁾ are immediate consequences. We also determine the typical magnitude of sup₁in|S _i |.Partially supported by NSF Grant DMS-9007416     

An asymptotic representation for products of random matrices

Let {X_k} be a stationary ergodic sequence of nonnegative matrices. It is shown in this paper that, under mild additional conditions, the logarithm of the i, jth element of X_t···X₁ is well approximated by a sum of t random variables from a stationary ergodic sequence. This representation is very useful for the study of limit behaviour of products of random matrices. An iterated logarithm result and an estimation result of use in the theory of demographic population projections are derived as corollaries.     

幂次为2,3,4,5的素变量非线性型的整数部分

考虑了一个混合幂次为2,3,4,5的素变量非线性型的整数部分表示无穷多素数的问题.运用Davenport-Heilbronn方法证明了:如果λ_1,λ_2,λ_3,λ_4是正实数,至少有一个λ_i/λ_j(1≤ij≤4)是无理数,那么存在无穷多素数p_1,p_2,p_3,p_4,p,使得[λ_1p_1~2+λ_2p_2~3+λ_3p_3~4+λ_4p_4~5]=p.     

次线性期望下的一般中心极限定理

受Peng-中心极限定理的启发,本文主要应用G-正态分布的概念,放宽Peng-中心极限定理的条件,在次线性期望下得到形式更为一般的中心极限定理.首先,将均值条件E[X_n]=ε[X_n]=0放宽为|E[X_n]|+|ε[X_n]|=O(1/n);其次,应用随机变量截断的方法,放宽随机变量的2阶矩与2+δ阶矩条件;最后,将该定理的Peng-独立性条件进行放宽,得到卷积独立随机变量的中心极限定理.     

A family of Sobolev orthogonal polynomials on the unit ball

A family of orthonormal polynomials on the unit ball B^d of with respect to the inner product

where Δ is the Laplace operator, is constructed explicitly.     

Bivariate version of the Hahn-Sonine theorem

We consider orthogonal polynomials in two variables whose derivatives with respect to are orthogonal. We show that they satisfy a system of partial differential equations of the form where , , is a vector of polynomials in and for , and is an eigenvalue matrix of order for . Also we obtain several characterizations for these polynomials. Finally, we point out that our results are able to cover more examples than Bertran's.

     

A Continuous Genetic Algorithm Designed for the Global Optimization of Multimodal Functions

Genetic algorithms are stochastic search approaches based on randomized operators, such as selection, crossover and mutation, inspired by the natural reproduction and evolution of the living creatures. However, few published works deal with their application to the global optimization of functions depending on continuous variables.A new algorithm called Continuous Genetic Algorithm (CGA) is proposed for the global optimization of multiminima functions. In order to cover a wide domain of possible solutions, our algorithm first takes care over the choice of the initial population. Then it locates the most promising area of the solution space, and continues the search through an intensification inside this area. The selection, the crossover and the mutation are performed by using the decimal code. The efficiency of CGA is tested in detail through a set of benchmark multimodal functions, of which global and local minima are known. CGA is compared to Tabu Search and Simulated Annealing, as alternative algorithms.     

Self-normalized central limit theorem for sums of weakly dependent random variables

Let {X _n,n1} be a strictly stationary sequence of weakly dependent random variables satisfyingEX _n=,EX _n ² <,Var S _n /n² and the central limit theorem. This paper presents two estimators of ². Their weak and strong consistence as well as their rate of convergence are obtained for -mixing, -mixing and associated sequences.Supported by a NSF grant and a Taft travel grant. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025.Supported by a Taft Post-doctoral Fellowship at the University of Cincinnati and by the Fok Yingtung Education Foundation of China. Hangzhou University, Hangzhou, Zhejiang, P.R. China and Department of Mathematics, National University of Singapore, Singapore 0511.     

On the order and type of basic and composite sets of polynomials in complete Reinhardt domains

In this paper we will define the order and type of basic and composite sets of polynomials of several complex variables in complete Reinhardt domains. Also, the property of basic and composite sets of polynomials of several complex variables in complete Reinhardt domains is discussed.