Asymptotic approximation of crossing probabilities of random sequences

Summary Let {X _i , i1} be a random sequence and {u _ni ,1in, n1} be an array of boundary values. We consider the asymptotic approximation of the probability P _n =P{X _i u _ni ,1in} by . We give sufficient conditions on X _i such that P _n–P _n ^* 0 as n. This generalizes the situation considered in extreme-value theory where the boundary is constant in i. The general theory is applied in particular to Gaussian cases.     

Small potential corrections for the discrete eigenvalues of the Sturm-Liouville problem

Summary The inverse Sturm-Liouville problem is the problem of finding a good approximation of a potential functionq such that the eigenvalue problem (^*)–y +qy=y holds on (0, ) fory(0)=y()=0 and a set of given eigenvalues . Since this problem has to be solved numerically by discretization and since the higher discrete eigenvalues strongly deviate from the corresponding Sturm-Liouville eigenvalues , asymptotic corrections for the 's serve to get better estimates forq. Let _k (1kn) be the first eigenvalues of (^*), let _k be the corresponding discrete eigenvalues obtained by the finite element method for (^*) and let _{k k} for the special caseq=0. Then, starting from an asymptotic correction technique proposed by Paine, de Hoog and Anderssen, new estimates for the errors of the corrected discrete eigenvalues are obtained and confirm and improve the knownO(kh ²)(h:=/(n+1)) behaviour. The estimates are based on new Sobolev inequalities and on Fourier analysis and it is shown that for 4+c ₂ k(n+1)/2, wherec ₁ andc ₂ are constants depending onq which tend to 0 for vanishingq.     

A relation between Chung's and Strassen's laws of the iterated logarithm

Summary Let W(t) be a standard Wiener process and let f(x) be a function from the compact class in Strassen's law of the iterated logarithm. We investigate the lim inf behavior of the variable sup ¦W(xT)(2T loglog T)^–1/2–f(x)¦, 0x1 suitably normalized as T.This extends Chung's result valid for f(x)0, stating that lim inf.[ sup ¦(2T loglogT)^–1/2 W(xT)¦(loglog T)^–1]=/4 a.s. T 0x1     

Dominated polynomials on $\mathcal{L}_P $ -spaces

It is well known that continuous bilinear forms on C(K) × C(K) are 2-dominated. This paper shows that generalizations of this result are not to be expected. The main result asserts that for every -space E(1 p ), every n 2, every r > 0 and every Banach space F , there exists an n-homogeneous polynomial P : E F such that P is not of type [_r], hence P is neither r-dominated nor r-semi-integral (if n = 2 and p = , F is supposed to contain an isomorphic copy of some , 1_q < ).Received: 24 November 2003     

On Rounding Error Sums Related to the Circle Problem

For a large real parameter t and 0 a b we consider sums where is the rounding error function, i.e. (z) = z - [z] - 1/2. We generalize Huxley's well known estimate by showing that holds uniformly in 0 a b . Fruther, we investigate an analogous question related to the divisor problem and show that the inequality , which (due to Huxley) holds uniformly in 0 a b , and which is in general not true for 1 a b t, is true uniformly in 0 a b .     

Holomorphe Funktionen auf Gebieten über Banach-Räumen zu vorgegebenen Konvergenzradien

Given a function: ₊ on a domain spread over an infinite dimensional complex Banach space E with a Schauder basis such that -log is plurisubharmonic and d (d denotes the boundary distance on ) one can find a holomorphic function f: with _f, where _f is the radius of convergence of f. If, in addition, is locally Lipschitz continuous with constant 1, f can be chosen so that (3M)^–1 _f, where M is the basis constant of E. In the particular case of E= ₁ there are holomorphic functions f on with= _f.     

Majorization of Gaussian processes and geometric applications

Summary Let (x _i ^* ) _i=1 ⁿ denote the decreasing rearrangement of a sequence of real numbers (x _i ) _i=1 ⁿ . Then for everyij, and every 1kn, the 2-nd order partial distributional derivatives satisfy the inequality, . As a consequence we generalize the theorem due to Fernique and Sudakov. A generalization of Slepian's lemma is also a consequence of another differential inequality. We also derive a new proof and generalizations to volume estimates of intersecting spheres and balls in ⁿ proved by Gromov.Supported by NSF grant # DMS 8909745, and the USA-Israel Binational Science Foundation grant # 86-00074, and grant for the Promotion of Research at the Technion     

Concerning the solution set ofAx=b whereP≦A≦Q andp≦b≦q

Summary LetPQ ben×n real matrices so that ifPAQ for some matrixA, thenA is nonsingular. Letp andq ben-dimensional real column vectors. This paper determines the set of all solutionsx to the equationAx=b for allA andb so thatPAQ andpbq.     

A partial ordering of some of the elements of Algol

Them _Algol productions of Algol are of the formX _p0 ::=X _p1 X _p2 ...X _{pn p} , where 1pm _Algol, 1n _p,X _p0 is a defined type andX _pj, 1jn _p is either a defined type or a basic symbol or possibly, representing the empty string ifn _p=1 [1]. A partial ordering of that subset of Algol's basic symbols and defined types which for somep are eitherX _p0 orX _p1 is exhibited. This ordering is of interest in implementing the syntax-oriented translator described by Ingerman.     

Stabilitätsungleichungen für Diskretisierungen von Randwertaufgaben gewöhnlicher Differentialgleichungen

Summary We consider two-point boundary value problems for linear differential equations (of ordern). The main part consists in proving stronger apriori estimates for (mn–1) using discreteL ₁-norms of the involved right-hand sides; thus extending known results for discreteL _p -norms with 2p. These estimates are the main tool in proving sharp estimates for the error \y–y _h \ _h, as well asfor \P _y –P _h y _h \ _h, , whereP _h denotes a consistent approximation of the differential operatorP of ordermn–1, in terms of the discretL ₁-norm of the truncation error. By means of an interpolation technique these estimates yield a correct estimate of the order of convergence, also in case that the solution satisfies locally some Lipschitz-conditions.     

Representations of integers as sums of powers with increasing exponents

Let(n) be the least integer such thatn may be represented in the formn=x ₁ ² +x ₂ ³ +...+x _(n) ⁽ⁿ⁾⁺¹ wherex ₁,x ₂, ...,x _(n) are natural numbers. We computed(n) forn 250 000 and found that(n) 5 for all thesen exceptn=56, 160 for which(n)=6. Also(n) 4 for 41542<n<=250 000.     

Cubature formulae which are exact on spacesP,intermediate betweenP k andQ k′

Summary In this paper we search, from the orthogonal polynomial theory, for conditions which allow to obtain cubature formulae on compacts of ⁿ , with weight function, and which are exact on the spaceR( _{k 1}, k₂, ..., k_n) of all polynomials of degree k _i respectively to each variablex _i , 1in.     

On a Problem of P. Gallagher

Let p be an odd prime. For each integer a with x < a x + u and (a,p) = 1, we define by a 1 (mod p) and 1 p - 1. Let r(p,u,x) be the number of integers with x < a x + u and (a,p) = 1 for which a and are of opposite parity, and let E(n,u,x) = r(n,u,x) - 1/2 1, where denotes summation over all a such that (a,p) = 1. The main purpose of this paper is to prove that for any positive integer 1 u we have the asymptotic formula

How non-uniform can a uniform sample be: a histogram approach

Summary Let _n be the empirical probability measure associated with n i.i.d. random vectors each having a uniform distribution in the unit square S of the plane. After _n is known, take the worst partition of the square into kn rectangles R _i, each with its short side at least times as long as the long side, and let Z= n|_n(R _j)–(R _j)|. We prove distribution inequalities for Z implying the right half of c _p,(n,k)^p/2 EZ ^p C _p,(n,k ^p/2, p > 0. (The left half follows easily by considering non-random partitions.) Similar results are obtained in other dimensions, and for population distributions other than uniform, and our results are related to data based histogram density estimation.Supported by NSF Grant MCS 8201128Supported by NSF Grant DMS-8401996     

Flat points of two-dimensional Brownian motion

Summary SupposeZ(·) is a two-dimensional Brownian motion. It is shown that a.s. there existt ₀ and >0 such thatZ(t ₀) is an extremal point of the convex hull of {Z(t)|t ₀–tt₀} and also an extremal point of the convex hull of {Z(t)|t ₀tt₀+} and, moreover, the tangent lines to the convex hulls atZ(t ₀) form a non-zero angle.The result is related to the following unsolved problem of S.J. Taylor. Do there exist a.s.t ₀ and >0 such that the intersection of the convex hulls of {Z(t)|t ₀–tt₀} and {Z(t)|t ₀tt₀+} contains onlyZ(t ₀)?This research was partially supported by Grant-in-Aid for Scientific Research (No. 400101540202), Ministry of Education, Science and Culture     

On the asymptotic equivalence ofL pmetrics for convergence to normality

Summary This paper is concerned with the rate of convergence to zero of theL _pmetrics _np1p, constructed out of differences between distribution functions, for departure from normality for normed sums of independent and identically distributed random variables with zero mean and unit variance. It is shown that the _np are, under broad conditions, asymptotically equivalent in the strong sense that, for 1p, p, _np/_np is universally bounded away from zero and infinity asn.     

On Tabor's problem concerning a certain quasi-ordering of iterative roots of functions

Summary Using the Isaacs-Zimmermann's theory of iterative roots of functions, we prove a theorem concerning the problemP 250 posed by J. Tabor:Letf: E E be a given mapping. Denote byF the set of all iterative roots off. InF we define the following relation: if and only if is an iterative root of. The relation is obviously reflexive and transitive. The question is: Is it also antisymmetric? If we consider iterative roots of a monotonic function the answer is yes. But in general the question is open.Here we prove that there exists a three-element decomposition { _i ;i = 1, 2, 3} of the setE ^E with blocks _i of the same cardinality 2^cardE such that the functions from ₁ do not possess any proper iterative root, the quasi-ordering is not antisymmetric onF(f) for anyf ₂, and is an ordering onF(f) for anyf ₃. Iff is a strictly increasing continuous self-bijection ofE, then the relation is an ordering onF(f) ifff is different from the identity mapping of the setE.     

The law of the iterated logarithm for normalized empirical distribution function

This paper deals with the law of the iterated logarithm and its analogues for sup , where the sup is taken on an interval of the form (a _n ,b _n ),(0a _n <b _n 1). Under certain conditions on a _n and b _n the corresponding lim sup results will be proved.     

Remarks on ergodicity of stationary irreducible transient Markov chains

Summary Consider a stationary process {X _n(), – < n < . If the measure of the process is finite (the measure of the whole sample space finite), it is well known that ergodicity of the process {X _n(), - < n < and of each of the subprocesses {X _n(), 0 n < , {X _n(), – < n 0 are equivalent (see [3]). We shall show that this is generally not true for stationary processes with a sigma-finite measure, specifically for stationary irreducible transient Markov chains. An example of a stationary irreducible transient Markov chain {X _n(), - < n <} with {itX_n(), 0 n < < ergodic but {X _n(), < n 0 nonergodic is given. That this can be the case has already been implicitly indicated in the literature [4]. Another example of a stationary irreducible transient Markov chain with both {X _n(), 0 n < and {itX _n(),-< < n 0} ergodic but {X _n(), - < n < nonergodic is presented. In fact, it is shown that all stationary irreducible transient Markov chains {X _n(), - < n < < are nonergodic.This research was supported in part by the Office of Naval Research.John Simon Guggenheim Memorial Fellow.     

Semi-groupes sous-Markoviens engendrés par des opérateurs matriciels

We characterize generators of sub-Markovian semigroups onL ^p () by a version of Kato's inequality. This will be used to show (under precise assumptions) that the semigroup generated by a matrix operatorA=(A _ij )_1i,jn on (L ^p ()) ⁿ is sub-Markovian if and only if the semigroup generated by the sum of each rowA _i 1+...+A _in (1in), is sub-Markovian. The corresponding result on (C ₀(X)) ⁿ characterizes dissipative operator matrices.