The Product of Independent Random Variables with Regularly Varying Tails

A distribution is said to have regularly varying tail with index – (0) if lim _x(kx,)/(x,)=k ^– for each k>0. Let X and Y be independent positive random variables with distributions and , respecitvely. The distribution of product XY is called Mellin–Stieltjes convolution (MS convolution) of and . It is known that D() (the class of distributions on (0,) that have regularly varying tails with index –) is closed under MS convolution. This paper deals with decomposition problem of distributions in D() related to MS convolution. A representation of a regularly varying function F of the following form is investigated: F(x)= _k=0 ^n–1 b _k f(a ^k x), where f is a measurable function and a and b _k (k=1,...,n–1) are real constants. A criterion is given for these constants in order that f be regularly varying. This criterion is applicable to show that there exist two distributions and such that neither nor belongs to D() (>0) and their MS convolution belongs to D().     

A Moment Matrix Approach to Multivariable Cubature

We develop an approach to multivariable cubature based on positivity, extension, and completion properties of moment matrices. We obtain a matrix-based lower bound on the size of a cubature rule of degree 2n + 1; for a planar measure , the bound is based on estimating where C:=C^# [ ] is a positive matrix naturally associated with the moments of . We use this estimate to construct various minimal or near-minimal cubature rules for planar measures. In the case when C = diag(c₁,...,c_n) (including the case when is planar measure on the unit disk), (C) is at least as large as the number of gaps c_k >c_k+1.     

The Existence of Kirkman Squares—Doubly Resolvable (v,3,1)-BIBDs

A Kirkman square with index , latinicity , block size k, and v points, KS _k (v;,), is a t×t array (t=(v–1)/(k–1)) defined on a v-set V such that (1) every point of V is contained in precisely cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V, and (3) the collection of blocks obtained from the non-empty cells of the array is a (v, k,)-BIBD. For =1, the existence of a KS _k (v; , ) is equivalent to the existence of a doubly resolvable (v, k, )-BIBD. The spectrum of KS ₂ (v; 1, 1) or Room squares was completed by Mullin and Wallis in 1975. In this paper, we determine the spectrum for a second class of doubly resolvable designs with =1. We show that there exist KS ₃ (v; 1, 1) for , v=3 and v27 with at most 23 possible exceptions for v.     

Stein-type improvements of confidence intervals for the generalized variance

Based on independent random matices X: p×m and S: p×p distributed, respectively, as N _pm (, I _m ) and W _p (n, ) with unknown and np, the problem of obtaining confidence interval for || is considered. Stein's idea of improving the best affine equivariant point estimator of || has been adapted to the interval estimation problem. It is shown that an interval estimator of the form |S|(b ^–1, a ^–1) can be improved by min{|S|, c|S +XX'|}(b ^–1, a ^–1) for a certain constant c depending on (a, b).     

A sharp partitioning-inequality for non-atomic probability measures based on the mass of the infimum of the measures

Summary If ₁, ... , are non-atomic probability measures on the same measurable space (S, ), then there is an -measurable partition {A _i } _{i = 1} ⁿ of S so that _i (A _i )(n – 1 + m)^–1 for all i=1, ..., n, where is the total mass of the largest measure dominated by each of the _i S; moreover, this bound is attained for all n1 and all m in [0, 1]. This result is an analog of the bound (n+1-M) ^-1of Elton et al. [5] based on the mass M of the supremum of the measures; each gives a quantative generalization of a well-known cake-cutting inequality of Urbanik [10] and of Dubins and Spanier [2].Research partly supported by NSF Grants DMS-84-01604 and DMS-86-01608     

Perturbation of Dirichlet forms by measures

Perturbations of a Dirichlet form by measures are studied. The perturbed form –_–+₊ is defined for _– in a suitable Kato class and ₊ absolutely continuous with respect to capacity. L _p-properties of the corresponding semigroups are derived by approximating _– by functions. For treating ₊, a criterion for domination of positive semigroups is proved. If the unperturbed semigroup has L _p -L _q -smoothing properties the same is shown to hold for the perturbed semigroup. If the unperturbed semigroup is holomorphic on L ₁ the same is shown to be true for the perturbed semigroup, for a large class of measures.     

Properties of a special class of doubly stochastic measures

Summary A measure on the unit squareI } I is doubly stochastic if(A } I) = (I } A) = the Lebesgue measure ofA for every Lebesgue measurable subsetA ofI = [0, 1]. By the hairpinL L ^–1, we mean the union of the graphs of an increasing homeomorphismL onI and its inverseL ^–1. By the latticework hairpin generated by a sequence {x _n :n Z} such thatx _n-1 < x_n (n Z), x _n = 0 and x _n = 1, we mean the hairpinL L ^–1 , whereL is linear on [x _n-1 ,x _n ] andL(n) =x _n-1 forn Z. In this note, a characterization of latticework hairpins which support doubly stochastic measures is given. This allows one to construct a variety of concrete examples of such measures. In particular, examples are given, disproving J. H. B. Kemperman's conjecture concerning a certain condition for the existence of doubly stochastic measures supported in hairpins.     

Fractal Characteristics of Measures; An Approach Via Function Spaces

Let be a positive Radon measure in R ⁿ with compact support . Let Q _jm be cubes with side-length 2^-j+1 originating from the canonical tiling of R ⁿ where j\in N ₀ and m\in Z ⁿ. If \in R, 0 < p \le , 0 < q \le , then _pq is the mixed _q – _p -quasi-norm of the sequence 2 ^j (Q _jm ). Quantities of this type are considered in fractal geometry (multifractal formalism) and in the theory of the function spaces B ^s _pq (R ⁿ) and F ^s _pq (R ⁿ). In Theorem 1 we deal with the question when _pq is an equivalent quasi-norm in some of these spaces (-property). If || = 0, then _S consists of those points (t,s) in the t–s-diagram in Figure 1 for which belongs to B ^s _p (R ⁿ) with pt = 1. Theorem 2 deals with the interrelation of S and _pq . Some applications to truncated Riesz potentials, Bessel potentials and Fourier transforms of are given.     

On the existence of wave operators for the Klein-Gordon equation

The problem of existence of wave operators for the Klein-Gordon equation ( _t ² –+²+iV_1t+V₂)u(x,t)=0 (x R ⁿ,t R, n3, >0) is studied where V₁ and V₂ are symmetric operators in L₂(R ⁿ) and it is shown that conditions similar to those of Veseli-Weidmann (Journal Functional Analysis 17, 61–77 (1974)) for a different class of operators are also sufficient for the Klein-Gordon equation.     

Necessary and sufficient conditions for the convergence of convolution products of non-identical distributions on finite abelian semigroups

Given a sequence of probability measures ( _n ) on a finite abelian semigroup, we present necessary and sufficient conditions which guarantee the weak convergence of the convolution products _{k,n k+1}*···* _n (k<n), asn for allk0. These conditions are verifiable in the sense that they are based entirely on the individual measures in the sequence ( _n ).     

Number of no nisomorphic subgraphs in an n-point graph

LetG(n) be the set of all nonoriented graphs with n enumerated points without loops or multiple lines, and let v_k(G) be the number of mutually nonisomorphic k-point subgraphs of G G(n). It is proved that at least |G(n)| (1–1/n) graphs G G(n) possess the following properties: a) for any k [6log₂n], where c=–c log₂c–(1–c)×log₂(1–c) and c>1/2, we havev _k(G) > C _n ^k (1–1/n²); b) for any k [cn + 5 log₂n] we havev _k(G) = C _n ^k . Hence almost all graphs G G(n) containv(G) 2ⁿ pairwise nonisomorphic subgraphs.Translated from Matematicheskie Zametki, Vol. 9, No. 3, pp. 263–273, March, 1971.     

Operator stable laws: Series representations and domains of normal attraction

LetX,X ₁,X ₂,... be i.i.d. random vectors in ^d. The limit laws that can arise by suitable affine normalizations of the partial sums,S _n=X ₁+...+X _n, are calledoperator-stable laws. These laws are a natural extension to ^d of the stable laws on. Thegeneralized domain of attraction of [GDOA()] is comprised of all random vectorsX whose partial sums can be affinely normalized to converge to . If the linear part of the affine transformation is restricted to take the formn ^–B for some exponent operatorB naturally associated to thenX is in thegeneralized domain of normal attraction of [GDONA()]. This paper extends the theory of operator-stable laws and their domains of attraction and normal attraction.     

Distribution of Values of Additive Functions with Respect to the Logarithmic Frequency

We study the weak convergence of distribution functions _x(n x: f _x (n) < u). Here _x denotes the logarithmic frequency and f _x , x 6, is a set of integer-valued strongly additive functions. The method of factorial moments is basic in the proofs.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 44, No. 4, pp. 546–557, October–December, 2004.     

An Extension of the Roots Separation Theorem

Let T _n be an n×n unreduced symmetric tridiagonal matrix with eigenvalues ₁<₂<< _n and W _k is an (n–1)×(n–1) submatrix by deleting the kth row and the kth column from T _n , k=1,2,...,n. Let _{12 n–1} be the eigenvalues of W _k . It is proved that if W _k has no multiple eigenvalue, then ₁<₁<₂<₂<< _n–1< _n–1< _n ; otherwise if _i = _i+1 is a multiple eigenvalue of W _k , then the above relationship still holds except that the inequality _i < _i+1< _i+1 is replaced by _i = _i+1= _i+1.     

Problèmes de Sturm-Liouville: bornes pour les valeurs propres et les zéros des fonctions propres

Résumé Pour évaluer ₂, on partage l'intervalleI=[a,b] en deux:I=[a,y] etI=[y,b]. A l'aide de l'inégalité de Barta et du processus d'itération de Schwarz, on détermine, en partant d'une fonction particulière, des bornes inférieuresv _n ^– (y) et _n ^– (y) pour la première valeur propre des nouveaux problèmes définis dansI etI. Auk-ième pas de l'itération, la meilleure borne possible pour ₂ est donnée parv _k ^– (y _k ), oùy _k est la racine de l'équationv _k ^– – _k ^– . De plus,v _k ^– (y _k ) ₂ ety _k tend vers le zéro de la deuxième fonction propre.

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Summary In order to evaluate ₂, we cut the intervalI=[a,b] into two parts:I=[a,y] andI=[y,b]. Using Barta's inequality and Schwarz's iteration procedure and starting from a particular function, we determine lower boundsv _k ^– (y) and _k ^– (y) for the first eigenvalue of the new problems defined inI andI. Afterk iterations, the best possible bound for ₂ isv _k ^– (y _k ), wherey _k is the root of the equationv _k ^– – _k ^– . Moreoverv _k ^– (y _k ) ₂ andy _k tends to the zero of the second eigenfunction.

     

Covering a convex body by smaller homothetic copies

Let a convex bodyAE ⁿ be covered bys smaller homothetic copies with coefficients ₁, ..., _s , respectively. It is conjectured that ₁ + ...+ _s n. This conjecture is confirmed in two cases:n is arbitrary ands=n+1;s is arbitrary andn=2.     

A survey of partial difference sets

LetG be a finite group of order . Ak-element subsetD ofG is called a (,k, , )-partial difference set if the expressionsgh ^–1, forg andh inD withgh, represent each nonidentity element inD exactly times and each nonidentity element not inD exactly times. IfeD andgD iffg ^–1D, thenD is essentially the same as a strongly regular Cayley graph. In this survey, we try to list all important existence and nonexistence results concerning partial difference sets. In particular, various construction methods are studied, e.g., constructions using partial congruence partitions, quadratic forms, cyclotomic classes and finite local rings. Also, the relations with Schur rings, two-weight codes, projective sets, difference sets, divisible difference sets and partial geometries are discussed in detail.     

On a class of partially ordered sets and their linear invariants

Let (, <) be a finite partially ordered set with rank function. Then is the disjoint union of the classes _k of elements of rank k and the order relation between elements in _k and _k+1 can be represented by a matrix S _k. We study partially ordered sets which satisfy linear recurrence relations of the type S _k (S _k ^T ) – c _k (S _{k – 1})^T S _{k – 1} = d _k ⁺ – c _k d _k ^– ) Id for all k and certain coefficients d _k ⁺, d _k ^- and c _k.     

The birth and death processes with zero as their absorbing barrier

LetE=(0, 1,...), Q ^b=(q_ij), i, j=0, 1, ..., whereq _{i, i–1}=a_i, q_{i, i+1}=b_i, q_ii=–(a_i+b_i), q_ij=0, when|i–j|>1. a ₀=0, b₀=b>0, a_i, b_i>0 (i>0). Lettingb=0 inQ ^b, we get the matrixQ ⁰.The time homogeneous Markov processX ^b ={x ^b (t,w), 0t< ^b (w)} (X ⁰={x⁰(t,w), 0t<⁰(w)}), withQ ^b (Q ⁰, respectively) as its density matrix and withE as its state space, is calledQ ^b (Q ⁰, respectively) process in this paper.Q ^b andQ ⁰ processes are all called the birth and death processes, with zero being the reflecting barrier ofQ ^b processes, the absorbing barrier ofQ ⁰ processes.AllQ ^b processes have been constructed by both probability and analytical methods (Wang [2], Yang [1]). In this paper, theQ ⁰ processes are imbedded intoQ ^b processes and all theQ ⁰ processes are directly constructed from theQ ^b processes. The main results are:Letb>0 be arbitrarily fixed, then there is a one to one correspondence between theQ ⁰ processes and theQ ^b processes (in the sense of transition probability).TheQ ⁰ process is unique iffR ^*=. SupposingR<, then:IfX ⁰={x⁰(t,w), 0t<⁰(w)} is a non-minimalQ ⁰ process, then its eigensequence (p, q, r _n, n–1) satisfies § 4(7).Conversely, let a non-negative number sequence (p, q, r _n, n–1) satisfying § 4(7) be arbitrarily given, then there exists a unique non-minimalQ ⁰ processX ⁰ with eigensequence (p, q, r _n, n–1). The Laplace transform of the transition probability (p _ij ⁰ (t)) ofX ⁰ is determined by § 4(15). X ⁰ is honest iffr _–1=0.X ⁰ satisfies the forward equation iffp=0.     

Minimal Fine Limits for a Class of Potentials

We consider potentials G _k associated with the Weinstein equation with parameter k in , _j=1 ⁿ (² u/ x ² _j ) + (k/x _n ) ( u/ x _n ) = 0, on the upper half space in ⁿ . We show that if the representing measure satisfies the growth condition y _n /(1+|y|) ^n-k < , where max(k, 2 – n) < 1, then G _k has a minimal fine limit of 0 at every boundary point except for a subset of vanishing (n – 2 + ) dimensional Hausdorff measure. We also prove this exceptional set is best possible.