Symmetric Distributions of Random Measures in Higher Dimensions

An infinite sequence of random variables X=(X ₁, X ₂,...) is said to be spreadable if all subsequences of X have the same distribution. Ryll-Nardzewski showed that X is spreadable iff it is exchangeable. This result has been generalized to various discrete parameter and higher dimensional settings. In this paper we show that a random measure on the tetrahedral space is spreadable, iff it can be extended to an exchangeable random measure on . The result is a continuous parameter version of a theorem by Kallenberg.     

Recursive method for arma model estimation (I)

In this paper a recursive method is given for estimating model under the natural conditions that the best linear predictor is the best predictor (in the mean square sense). Under these conditions we can prove the estimators ofp ₀ andq ₀ are strongly consistent. The asymptotic normality and the law of iterated logarithm for the estimators of _k 's and _j 's can also be proved.     

Hölder Regularity for Solutions of Ultraparabolic Equations in Divergence Form

We consider the second order differential equation , where (x,t) ^N+1, 0<m ₀N, the coefficients a _i,j belong to a suitable space of vanishing mean oscillation functions VMO _L and B=(b _i,j ) is a constant real matrix. The aim of this paper is to study interior regularity for weak solutions to the above equation assuming that F _j belong to a function space of Morrey type.     

A remark on almost sure convergence of weighted sums

Summary As a generalization of a theorem of Chow [1] it is shown by an elementary method that for i.i.d. r.v.'s X ₁,...,X _n, with expectation zero and finite p-th absolute moment (p2) the weighted sums converge to zero a.s.     

A Nonlocal Diffusion Equation whose Solutions Develop a Free Boundary

Let J:\mathbbR ? \mathbbRJ:\mathbb{R} \to \mathbb{R} be a nonnegative, smooth compactly supported function such that ò_\mathbbR J(r)dr = 1. \int_\mathbb{R} {J(r)dr = 1.} We consider the nonlocal diffusion problem

On the Structure of a Certain Class of Mixed Tsirelson Spaces

We study Banach spaces of the form We call such a space a p-space, p[1,), if for every k the space is isomorphic to _pk and the sequence (p_k) strictly decreases to p. We examine the finite block representability of the spaces _r in a p-space proving that it depends not only on p but also on the sequences (p_k) and (n_k). Assuming that _i n_i ^1/q decreases to 0, where q is the conjugate exponent of p, we prove the existence of an asymptotic biorthogonal system in X and also that c ₀ is finitely representable in X. Moreover we investigate the modified versions of p-spaces proving that, if n_km1/pkm-1/pk_m-1 increases to infinity for a subsequence (n_km) , then ₁ embeds into X. We also investigate complemented minimality for the class of spaces where is either a subsequence of the sequence of Schreier classes ( _n)_{n N} or a subsequence of ( _n)_{n N}.     

Estimation of a K-Functional of Higher Order in Terms of a K-Functional of Lower Order

Let U _j be a finite system of functionals of the form , and let be the subspace of the Sobolev space , 1 p +, that consists only of functions g such that U _j(g) = 0 for k _j < r. It is assumed that there exists at least one jump _j for every function _j , and if _j = _s for j s, then k _j k _s. For the K-functional

An algorithm for a linear diophantine equation and a problem of Frobenius

Summary We solve the diophantine equation for nonnegative variablesx _j , wherea _j andL are positive integers. We characterize both the values ofL that lead to solutions and those that do not lead to solutions. We solve the Frobenius problem of finding the largest value ofL for which no solution exists.     

On positive solutions of quasilinear elliptic systems

In this paper, we consider the existence and nonexistence of positive solutions of degenerate elliptic systems where –_p is the p-Laplace operator, p > 1 and is a C ^1,-domain in . We prove an analogue of [7, 16] for the eigenvalue problem with and obtain a non-existence result of positive solutions for the general systems.     

On computing the number of subgroups of a finite Abelian group

We consider the numberN _A (r) of subgroups of orderp ^r ofA, whereA is a finite Abelianp-group of type =₁,₂,..., _l ()), i.e. the direct sum of cyclic groups of order ⁱⁱ. Formulas for computingN _A (r) are well known. Here we derive a recurrence relation forN _A (r), which enables us to prove a conjecture of P. E. Dyubyuk about congruences betweenN _A (r) and the Gaussian binomial coefficient .     

A constructive method of solving the Liapounov equation for complex matrices

Summary This paper describes a method of solving the Liapounov equation (1)HM+M ^* H=2D, M in upper Hessenberg form,D diagonal. Initialising the first row of the matrixA arbitrarily, one can find (by solving equations with one unknown) the unknown elements ofA such that (2)AM+M ^* A ^* =2F, whereA differs from a Hermitian matrix only in that its diagonal elements need not be real.F is a diagonal matrix which is uniquely determined by the first row ofA. By solving Eq. (2) for several initial values one may generate several matricesA andF (in the most unfavourable case 2n–1A's andF's are needed) and superpose them to getn linearly independent Hermitian matricesH _j andD _j respectively for whichH _j M+M ^* H _j =2D _j is valid. Then one can solve the real system to obtain the solution of Eq. (1).This work was performed under the terms of the agreement on association between the Max-Planck-Institut für Plasmaphysik and Euratom.     

Chevet's theorem for stable processes II

LetB be a separable Banach space and let {:||1} denote the unit ball ofB ^*. LetX be a symmetricp-stableB-valued random variable and let {X _j } _j=1 ⁿ be i.i.d. copies ofX. LetB ₁ be a finite-dimensional Banach space with a symmetric unconditional basis {y _j } _j=1 ⁿ . An upper bound is obtained for that improves the one given by Giné, Marcus and Zinn [J. Functional Anal. 63, 47–73 (1985)].     

Existence of Minimizers for Nonconvex Variational Problems with Slow Growth

Consider the minimization problem

Investments in stochastic maximum flow networks

Each are (i, j) of the network has capacity _ij where _ij is a non-negative random variable. The capacity of any arc may be reduced increased by an amountu _ij 0 at a cost ofc _ij u _ij . The objective is to maximizev–Kc _ij u _ij wherev is the expected maximum flow. This problem is formulated as a two-stage linear program under uncertainty. Each feasible generates a constraint where is the probability arc (i, j) is in the minimum cut set and the expected value of the maximum flow under . The formulation is later generalized to include certain conditions under which the increase in capacity of an arc may be a non-deterministic function of the investmentc _ij u _ij .     

Central limit theorem for traces of large random symmetric matrices with independent matrix elements

We study Wigner ensembles of symmetric random matricesA=(a _ij ),i, j=1,...,n with matrix elementsa _ij ,ij being independent symmetrically distributed random variables

Linear programming bounds for codes in infinite projective spaces

We develop a technique for improving the universal linear programming bounds on the cardinality and the minimum distance of codes in projective spaces . We firstly investigate test functions P_j(m,n,s) having the property that P_j(m,n,s)<0 for somej if and only if the corresponding universal linear programming bound can be further improved by linear programming. Then we describe a method for improving the universal bounds. We also investigate the possibilities for attaining the first universal bounds.     

Positive Periodic Solutions of N-Species Neutral Delay Systems

In this paper, we employ some new techniques to study the existence of positive periodic solution of n-species neutral delay system N _i = N _i (t)[ _i (t) — _ij (t)N _j (t) — b _ij (t)N _j (t— _ij (t))— c _ij (t)N _j (t— _ij (t))].As a corollary, we answer an open problem proposed by Y. Kuang.     

On construction ofk-wise independent random variables

A 0–1probability space is a probability space (, 2,P), where the sample space -{0, 1} ⁿ for somen. A probability space isk-wise independent if, whenY _i is defined to be theith coordinate or the randomn-vector, then any subset ofk of theY _i 's is (mutually) independent, and it is said to be a probability spacefor p ₁,p ₂, ...,p _n ifP[Y _i =1]=p _i .We study constructions ofk-wise independent 0–1 probability spaces in which thep _i 's are arbitrary. It was known that for anyp ₁,p ₂, ...,p _n , ak-wise independent probability space of size always exists. We prove that for somep ₁,p ₂, ...,p _n [0,1],m(n,k) is a lower bound on the size of anyk-wise independent 0–1 probability space. For each fixedk, we prove that everyk-wise independent 0–1 probability space when eachp _i =k/n has size (n _k ). For a very large degree of independence —k=[n], for >1/2- and allp _i =1/2, we prove a lower bound on the size of . We also give explicit constructions ofk-wise independent 0–1 probability spaces.This author was supported in part by NSF grant CCR 9107349.This research was supported in part by the Israel Science Foundation administered by the lsrael Academy of Science and Humanities and by a grant of the Israeli Ministry of Science and Technology.     

Associated random variables and martingale inequalities

Summary Many of the classical submartingale inequalities, including Doob's maximal inequality and upcrossing inequality, are valid for sequences S _j such that the (S _j+1 -S _j's are associated (positive mean) random variables, and for more general demisubmartingales. The demisubmartingale maximal inequality is used to prove weak convergence to the two-parameter Wiener process of the partial sum processes constructed from a stationary two-parameter sequence of associated random variables X _ijwith .Alfred P. Sloan Research Fellow. Research supported in part by NSF Grants MCS 77-20683 and MCS 80-19384     

Remainder estimates for the approximation numbers of weighted Hardy operators acting onL 2

We consider the weighted Hardy integral operatorT:L ²(a, b) →L ²(a, b), −∞≤a<b≤∞, defined by . In [EEH1] and [EEH2], under certain conditions onu andv, upper and lower estimates and asymptotic results were obtained for the approximation numbersa _n(T) ofT. In this paper, we show that under suitable conditions onu andv, where ∥w∥_p=(∫ _a ^b |w(t)|^p dt)^1/p. Research supported by NSERC, grant A4021. Research supported by grant No. 201/98/P017 of the Grant Agency of the Czech Republic.