Convergence in distribution and Skorokhod Convergence for the general theory of processes

Summary This paper proves some Skorokhod Convergence Theorems for processes with filtration. Roughly, these are theorems which say that if a family of processes with filtration (X ⁿ , ⁿ ),n, converges in distribution in a suitable sense, then there exists a family of equivalent processes (Y ⁿ , ⁿ ),n, which converges almost surely. The notion of equivalence used is that of adapted distribution, which guarantees that each (X ⁿ , ⁿ ) has the same stochastic properties as (X ⁿ , ⁿ ), with respect to its filtration, such as the martingale property or the Markov property. The appropriate notion of convergence in distribution is convergence in adapted distribution, which is developed in the paper. Fortunately, any tight sequence of processes has a subsequence which converges in adapted distribution. For discrete time processes, (Y ⁿ , ⁿ ),n, and their limit (Y, ) may be taken as all having the same fixed filtration ⁿ =. In the continuous time case, theY ⁿ , ⁿ may require different filtrations ⁿ , which converge to. To handle this, convergence of filtrations is defined and its theory developed.During part of the time this work was in progress, it was supported by an NSERC operating grant, and the author was an NSERC University Research Fellow. The author wishes to thank the Steklov Mathematical Institute of the Soviet Academy of Sciences for its hospitality while the principle research in this paper was being begun, A.N. Shiryaev and P.C. Greenwood, who made the author's visit there possible, and Ján Miná for his hospitality while that research was being finished. We thank the referee who suggested the results in Sect. 12     

Some geometric applications of the beta distribution

Let be the angle between a line and a random k-space in Euclidean n-space R ⁿ. Then the random variable cos² has the beta distribution. This result is applied to show (1) in R ⁿthere are exponentially many (in n) lines going through the origin so that any two of them are nearly perpendicular, (2) any N-point set of diameter d in R ⁿlies between two parallel hyperplanes distance 2d{(log N)/(n-1)}^1/2 apart and (3) an improved version of a lemma of Johnson and Lindenstrauss (1984, Contemp. Math., 26, 189–206). A simple estimate of the area of a spherical cap, and an area-formula for a neighborhood of a great circle on a sphere are also given.     

Prolongations of Vector Fields and the Invariants-by-Derivation Property

For any given vector field X defined on some open set M ², we characterize the prolongations X _n ^* of X to the nth jet space M ⁽ⁿ⁾, n1, such that a complete system of invariants for X _n ^* can be obtained by derivation of lower-order invariants. This leads to characterizations of C -symmetries and to new procedures for reducing the order of an ordinary differential equation.     

Twisted derivations and distinct sets of lines that cover the same pairs of points

A partial projective plane of ordern consists of lines andn ² +n + 1 points such that every line hasn+1 points and distinct lines meet in a unique point. Suppose that two essentially different partial projective planes and of ordern, n a perfect square, that are defined on the same set of points cover the same pairs of points. For sufficiently largen we show that this implies that and have at leastn(n+1) lines. This bound is sharp and there exist essentially two different types of examples meeting the bound.As an application, we can show that derived planes provide an example for a pair of projective planes of square order with as much structure as possible in common, that is, as many lines as possible in common. Furthermore, we present a new method (twisted derivations) to obtain planes from one another by replacing the same number of lines as in a derivation.     

Completeness and non-speciality of linear series on space curves and deficiency of the corresponding linear series on normalization

Summary Let be a curve ofP ^r (r3) of degree d, C its normalization and , I() a saturated, homogeneous ideal of k[X₀, ...,X _r]. In this paper we show that, if N 0 is an integer such that, for nN, the linear series cut out on by the hypersurfaces of degree n is complete and non-special, then the deficiency of the linear series cut out on C by the hypersurfaces ofA _n,forn>N, is independent ofn and can be explicitly calculated;this is the case, for instance, whenN=d–r+1, and when N=n_i –r–1 (under suitable conditions) if is a component of the complete intersection of r–1 hypersurfaces of degrees n_i.Under financial support from the N.S.E.R.C. of Canada, the italian M.P.I. and the N.A.T.O. Fellowships Scheme Programme.The author wishes to thank R.Lazarsfeld for advice and the Curves Seminar group at Queen's, in particular A. V.Geramita and E.Davis, for fruitful and stimulating discussions on this subject.     

An extended continuous Newton method

This paper describes a numerical realization of an extended continuous Newton method defined by Diener. It traces a connected set of locally one-dimensional trajectories which contains all critical points of a smooth functionf: ⁿ . The results show that the method is effectively applicable.The authors would like to thank L. C. W. Dixon for pointing out some errors in the original version of this paper and for several suggestions of improvements.     

Radii of convexity and close-to-convexity of certain integral representations

Strict upper bounds are determined for ¦s(z)¦, ¦Re s(z)¦, and ¦Im s(z) ¦ in the class of functions s(z)=a _nzⁿ+a_n+1zⁿ⁺¹+... (n¹) regular in ¦z¦<1 and satisfying the condition ¦u (₁) –u (₂) ¦K¦ ₁- ₂¦, where U()=Re s (eⁱ ), K>0, and ₁ and ₂ are arbitrary real numbers. These bounds are used in the determination of radii of convexity and close-to-convexity of certain integral representations.Translated from Matematicheskie Zametki, Vol. 7, No. 5, pp. 581–592, May, 1970.The author wishes to thank L. A. Aksent'ev for his guidance in this work.     

Flag transitive planes of orderq n with a long cyclel ∞ as a collineation

Baker and Ebert [1] presented a method for constructing all flag transitive affine planes of orderq ² havingGF(q) in their kernels for any odd prime powerq. Kantor [6; 7; 8] constructed many classes of nondesarguesian flag transitive affine planes of even order, each admitting a collineation, transitively permuting the points at infinity. In this paper, two classes of non-desarguesian flag transitive affine planes of odd order are constructed. One is a class of planes of orderq ⁿ , whereq is an odd prime power andn 3 such thatq ⁿ 1 (mod 4), havingGF(q) in their kernels. The other is a class of planes of orderq ⁿ , whereq is an odd prime power andn 2 such thatq ⁿ 1 (mod 4), havingGF(q) in their kernels. Since each plane of the former class is of odd dimension over its kernel, it is not isomorphic to any plane constructed by Baker and Ebert [1]. The former class contains a flag transitive affine plane of order 27 constructed by Kuppuswamy Rao and Narayana Rao [9]. Any plane of the latter class of orderq ⁿ such thatn 1 (mod 2), is not isomorphic to any plane constructed by Baker ad Ebert [1].The author is grateful to the referee for many helpful comments.     

The Second Lacuna in the Orders of Movement Groups of the Spaces U n

We prove the nonexistence of symmetrically linearly connected spaces of hyperplane elements admitting a movement group G _r with n ² - n + 2 r n² - 1, n 5.     

Partitioning arrangements of lines II: Applications

In this paper we present efficient deterministic algorithms for various problems involving lines or segments in the plane, using the partitioning algorithm described in a companion paper [A3]. These applications include: (i) anO(m ^2/3 n ^2/3 · log^2/3 n · log^/3 (m/n)+(m+n) logn) algorithm to compute all incidences betweenm points andn lines, where is a constant <3.33; (ii) anO(m ^2/3 n ^2/3 · log^5/3 n · log^/3 (m/n)+(m+n) logn) algorithm to computem faces in an arrangement ofn lines; (iii) anO(n ^4/3 log^(+2)/3 n) algorithm to count the number of intersections in a set ofn segments; (iv) anO(n ^4/3 log^{( + 2)/3} n) algorithm to count red-blue intersections between two sets of segments, and (v) anO(n ^3/2 log^/3 n) algorithm to compute spanning trees with low stabbing number for a set ofn points. We also present an algorithm that, given set ofn points in the plane, preprocesses it, in timeO(nm log^+1/2 n), into a data structure of sizeO(m) forn lognmn ², so that the number of points ofS lying inside a query triangle can be computed inO((n/m) log^3/2 n) time.Work on this paper has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant DCR-83-20085, and by grants from the Digital Equipment Corporation and the IBM Corporation. A preliminary version of this paper appears in theProceedings of the 5th ACM Symposium on Computational Geometry, 1989, pp. 11–22.     

Mapping Properties of the Laplacian in Sobolev Spaces of Forms on Complete Hyperbolic Manifolds

For a complete manifold M with constant negative curvature, weprove that the rough Laplacian _R defines topological isomorphisms in the scale of Sobolev spaces H _p ^s (M) ofp-forms for all p, 0 < p< n. For the de Rham Laplacian and M= ⁿ , the Poincaréhyperbolic space, this is shown too for 0 pn,pn/2, p (n± 1)/2.     

Dehn twists and pseudo-Anosov diffeomorphisms

Summary We show that it is possible to obtain many pseudo-Anosov diffeomorphisms from Dehn twists. In particular, we generalize a theorem of Long and Morton to obtain that iff is a pseudo-Anosov diffeomorphism of an oriented surface andT is the Dehn twist around the simple closed curve , then the isotopy class ofT ⁿ f contains a pseudo-Anosov diffeomorphism except for at most 7 consecutive values ofn.     

The quality of the Diophantine approximations found by the Jacobi-Perron algorithm and related algorithms

Given a vector of real numbers=(₁,... _d ) ^d , the Jacobi-Perron algorithm and related algorithms, such as Brun's algorithm and Selmer's algorithm, produce a sequence of (d+1)×(d+1) convergent matrices {C⁽ⁿ⁾():n1} whose rows provide Diophantine approximations to . Such algorithms are specified by two mapsT:[0, 1] ^d [0, 1] ^d and A:[0,1] ^d GL(d+1,), which compute convergent matrices C⁽ⁿ⁾())...A(T())A(). The quality of the Diophantine approximations these algorithms find can be measured in two ways. The best approximation exponent is the upper bound of those values of for which there is some row of the convergent matrices such that for infinitely many values ofn that row of C⁽ⁿ⁾() has . The uniform approximation exponent is the upper bound of those values of such that for all sufficiently large values ofn and all rows of C⁽ⁿ⁾⁽⁾ one has . The paper applies Oseledec's multiplicative ergodic theorem to show that for a large class of such algorithms and take constant values and on a set of Lebesgue measure one. It establishes the formula where are the two largest Lyapunov exponents attached by Oseledec's multiplicative ergodic theorem to the skew-product (T, A,d), whered is aT-invariant measure, absolutely continuous with respect to Lebesgue measure. We conjecture that holds for a large class of such algorithms. These results apply to thed-dimensional Jacobi-Perron algorithm and Selmer's algorithm. We show that; experimental evidence of Baldwin (1992) indicates (nonrigorously) that. We conjecture that holds for alld2.     

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.     

Entire functions of exponential type,increasing slowly along the real hyperplane

Entire functionsf(z), zC ⁿ , of exponential type at most a and bounded on subsets E of the real hyperplane, are investigated. It is known that if E is relatively dense with respect to the Lebesgue measure or it is an -net inR ⁿ , then such f(z) are bounded on all ofR ⁿ (for e-nets in the case of sufficiently small ). It is shown that if E is close in a certain sense either to a relatively dense subset ofR ⁿ , or to an -net, then f(z) cannot increase fast alongR ⁿ . Similar estimates are established for integral metrics.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 50, pp. 74–76, 1988.     

On the Influence of Weights on Extremes

Let X ₁, ..., X_n be an i.i.d. sequence of random variables, from an unknown distribution F, and X ₁ ^W , ... X _n ^W be a sample from , the weighted empirical distribution function of X ₁, ..., X_n. We define the order statistics X _1,n ^W ... X _n,n ^W of X ₁ ^W , ..., X _n ^W . Under suitable assumptions on weights, we study the influence of the maxima in the construction of limit theorems. We choose a resample size m(n) and we derive conditions on m(n) for the in probability and with probability 1 consistency of X _m(n),m(n) ^W . The presence of weights has an influence on the resample size and requires the use of new tools. When X _n,n is in the domain of attraction of an extreme value distribution, m(n) , and , as n , all our results hold.     

Semi-infinite optimization: Structure and stability of the feasible set

The problem of the minimization of a functionf: ⁿ under finitely many equality constraints and perhaps infinitely many inequality constraints gives rise to a structural analysis of the feasible setM[H, G]={xⁿ¦H(x)=0,G(x, y)0,yY} with compactY^r. An extension of the well-known Mangasarian-Fromovitz constraint qualification (EMFCQ) is introduced. The main result for compactM[H, G] is the equivalence of the topological stability of the feasible setM[H, G] and the validity of EMFCQ. As a byproduct, we obtain under EMFCQ that the feasible set admits local linearizations and also thatM[H, G] depends continuously on the pair (H, G). Moreover, EMFCQ is shown to be satisfied generically.The authors would like to thank Rainer Hettich and Doug Ward for fruitful discussions. Moreover, the authors are indebted to the anonymous referees for their valuable comments.     

Duality and multiplication operators

We describe a space of functions contained inxxLx (D)C(D G) but not necessarily inU. We give a representation of these functions as bounded multiplication operators on the Bergman spacexxLx _a ² and identify the subspace consisting of functions which induce compact multiplication operators. We also describe a newC ^*-subalgebra ofxxLx (D) which we conjecture to be a proper super-set ofU.Most of this research was done while the second author was visiting Cleveland State University. He would like to thank the Mathematics Department for its hospitality. He would also like to thank the NNSFC for partial support.     

The complexity and construction of many faces in arrangements of lines and of segments

We show that the total number of edges ofm faces of an arrangement ofn lines in the plane isO(m ^2/3– n ^2/3+2 +n) for any>0. The proof takes an algorithmic approach, that is, we describe an algorithm for the calculation of thesem faces and derive the upper bound from the analysis of the algorithm. The algorithm uses randomization and its expected time complexity isO(m ^2/3– n ^2/3+2 logn+n logn logm). If instead of lines we have an arrangement ofn line segments, then the maximum number of edges ofm faces isO(m ^2/3– n ^2/3+2 +n (n) logm) for any>0, where(n) is the functional inverse of Ackermann's function. We give a (randomized) algorithm that produces these faces and takes expected timeO(m ^2/3– n ^2/3+2 log+n(n) log² n logm).The first author is pleased to acknowledge partial support by the Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and the National Science Foundation under Grant CCR-8714565. Work on this paper by the third author has been supported by Office of Naval Research Grant N00014-82-K-0381, by National Science Foundation Grant DCR-83-20085, by grants from the Digital Equipment Corporation, and the IBM Corporation, and by a research grant from the NCRD-the Israeli National Council for Research and Development. A preliminary version of this paper has appeared in theProceedings of the 4th ACM Symposium on Computational Geometry, 1988, pp. 44–55.