On the structure of stationary flat processes

The paper contains some results related to the fundamental question of Davidson whether all rotationally stationary line processes in the plane which have a.s. no parallel lines are Cox (i.e. doubly stochastic Poisson processes). This problem is shown to be equivalent to the corresponding one for stationarity under translations only. The partial solutions by Papangelou are improved in various directions, and they are further extended to the case of marked k-dimensional flats (hyperplanes) in R ^d for arbitrary k and d with 0<k<d. It turns out that the main result of Papangelou carries over to the case k d/2, while the opposite case seems to require stronger regularity assumptions. In the former case, stationarity is typically needed in 2(d–k) directions only. The present treatment (like the one of Papangelou) proceeds in two steps, in proving first that sufficiently smooth stationary random measures are invariant, and second that point processes without parallel atoms and with invariant conditional intensities are Cox. In the final section, some related problems are discussed which provide some further insight into the structure of the basic Davidson problem (which remains open).     

Three theorems on ρ* random fields

This paper generalizes results by Bradley.⁽³⁾ Suppose that for 1=1,2,...X _k ¹ :k ^d is a centered, weakly stationary ^*-mixing random field, and suppose lim_l Cov(X ₀ ¹ ,x _k ¹ ) exists, anyk ^d . Then the successive spectral densities converge uniformly to a continuous function. For a sequence of strictly stationary random fields that are uniformly ^*-mixing and satisfy a indeberg condition, a CLT is proved for sequences of sums from the fields. This result is then applied: given a centered strictly stationary ^*-mixing random field whose probability density and joint densities are continuous, then a kernel estimator for the probability density obeys the CLT.     

Random walks on the torus with several generators

Given n vectors {_i} ∈ [0, 1)^d, consider a random walk on the d‐dimensional torus ??^d = ?^d/?^d generated by these vectors by successive addition and subtraction. For certain sets of vectors, this walk converges to Haar (uniform) measure on the torus. We show that the discrepancy distance D(Q*^k) between the kth step distribution of the walk and Haar measure is bounded below by D(Q*^k) ≥ C₁k^?n/2, where C₁ = C(n, d) is a constant. If the vectors are badly approximated by rationals (in a sense we will define), then D(Q*^k) ≤ C₂k^?n/2d for C₂ = C(n, d, _j) a constant. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Isomorphism and Measure Rigidity for Algebraic Actions on Zero-Dimensional Groups

We consider mixing ^d-actions on compact zero-dimensional abelian groups by automorphisms. Rigidity of invariant measures does not hold for such actions in general; we present conditions which force an invariant measure to be Haar measure on an affine subset. This is applied to isomorphism rigidity for such actions. We develop a theory of halfspace entropies which plays a similar role in the proof to that played by invariant foliations in the proof of rigidity for smooth actions.     

On infinitely divisible self-similar random fields

Summary A class of self-similar stationary random fields in ^d , d1 with finite variance is constructed by means of multiple stochastic integrals with respect to the Poisson random measure in ^d+1. Various topics associated with these fields such as subordination, ergodicity, existence of higher order moments, uniqueness of stochastic integral representation, renormalized powers of linear generalized fields and some limit theorems are studied. A Lévy-Hinin type formula for the characteristic functional of general infinitely divisible self-similar random fields with finite variance is obtained.     

On the structure of stationary flat processes. II

Summary The present paper continues the work by Davidson, Krickeberg, Papangelou, and the author on proving, under weakest possible assumptions, that a stationary random measure or a simple point process on the space of k-flats in R ^d is a.s. invariant or a Cox process respectively. The problems for and are related by the fact that is Cox whenever the Papangelou conditional intensity measure of (a thinning of) is a.s. invariant. In particular, is shown to be a.s. invariant, whenever it is absolutely continuous with respect to some fixed measure and has no (so called) outer degeneracies. When k=d–22, no absolute continuity is needed, provided that the first moments exist and that has no inner degeneracies either. Under a certain regularity condition on , it is further shown that and are simultaneously non-degenerate in either sense.     

Membership of distributions of polynomials in the Nikolskii–Besov class

The main result of this paper asserts that the distribution density of any non-constant polynomial f(ξ₁,ξ₂,...) of degree d in independent standard Gaussian random variables ξ₁ (possibly, in infinitely many variables) always belongs to the Nikol’skii–Besov space B^1/d (R¹) of fractional order 1/d (see the definition below) depending only on the degree of the polynomial. A natural analog of this assertion is obtained for the density of the joint distribution of k polynomials of degree d, also with a fractional order that is independent of the number of variables, but depends only on the degree d and the number of polynomials. We also give a new simple sufficient condition for a measure on R^k to possess a density in the Nikol’skii–Besov class B^α(R)^k. This result is applied for obtaining an upper bound on the total variation distance between two probability measures on R^k via the Kantorovich distance between them and a certain Nikol’skii–Besov norm of their difference. Applications are given to estimates of distributions of polynomials in Gaussian random variables.     

Fractional Brownian Density Process and Its Self-Intersection Local Time of Order k

The fractional Brownian density process is a continuous centered Gaussian ( ^d )-valued process which arises as a high-density fluctuation limit of a Poisson system of independent d-dimensional fractional Brownian motions with Hurst parameter H. ( ( ^d ) is the space of tempered distributions). The main result proved in the paper is that if the intensity measure of the (initial) Poisson random measure on ^d is either the Lebesgue measure or a finite measure, then the density process has self-intersection local time of order k 2 if and only if Hd < k/(k – 1). The latter is also the necessary and sufficient condition for existence of multiple points of order k for d-dimensional fractional Brownian motion, as proved by Talagrand¹². This result extends to a non-Markovian case the relationship known for (Markovian) symmetric -stable Lévy processes and their corresponding density processes. New methods are used in order to overcome the lack of Markov property. Other properties of the fractional Brownian density process are also given, in particular the non-semimartingale property in the case H 1/2, which is obtained by a general criterion for the non-semimartingale property of real Gaussian processes that we also prove.     

On the Central Limit Theorem for Nonuniform φ-Mixing Random Fields

The partial-sum processes, indexed by sets, of a stationary nonuniform -mixing random field on the d-dimensional integer lattice are considered. A moment inequality is given from which the convergence of the finite-dimensional distributions to a Brownian motion on the Borel subsets of [0, 1] ^d is obtained. A Uniform CLT is proved for classes of sets with a metric entropy restriction and applied to certain Gibbs fields. This extends some results of Chen⁽⁵⁾ for rectangles. In this case and when the variables are bounded a simpler proof of the uniform CLT is given.     

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.     

A strong approximation for logarithmic averages of partial sums of random variables

LetS _n be the partial sums of -mixing stationary random variables and letf(x) be a real function. In this note we give sufficient conditions under which the logarithmic average off(S _n / _n ) converges almost surely to _– f(x)d(x). We also obtain strong approximation forH(n)= _k=1 ⁿ k ^–1 f(S _k /k)=logn _– f(x)d(x) which will imply the asymptotic normality ofH(n)/log^1/2 n. But for partial sums of i.i.d. random variables our results will be proved under weaker moment condition than assumed for -mixing random variables.     

Blocking Structures of Hermitian Varieties

Given a hermitian variety H(d,q²) and an integer k (d–1)/2, a blocking set with respect to k-subspaces is a set of points of H(d,q²) that meets all k-subspaces of H(d,q²). If H(d,q²) is naturally embedded in PG(d,q²), then linear examples for such a blocking set are the ones that lie in a subspace of codimension k of PG(d,q²). Up to isomorphism there are k+1 non-isomorphic minimal linear blocking sets, and these have different cardinalities. In this paper it is shown for 1 k< (d–1)/2 that all sufficiently small minimal blocking sets of H(d,q²) with respect to k-subspaces are linear. For 1 k< d/2–3, it is even proved that the k+1 minimal linear blocking sets are smaller than all minimal non-linear ones.AMS Classification: 1991 MSC: 51E20, 51E21     

Large deviations and the maximum entropy principle for marked point random fields

Summary We establish large deviation principles for the stationary and the individual empirical fields of Poisson, and certain interacting, random fields of marked point particles in ^d . The underlying topologies are induced by a class of not necessarily bounded local functions, and thus finer than the usual weak topologies. Our methods yield further that the limiting behaviour of conditional Poisson distributions, as well as certain distributions of Gibbsian type, is governed by the maximum entropy principle. We also discuss various applications and examples.Supported by the Deutsche Forschungsgemeinschaft     

Minkowski functionals of Poisson processes

Summary This paper treats Poisson processes N with the convex ring S as state space, having a translation and rotation invariant intensity measure . Fixing a test set S ₀S, denoting by Q^k the random set of all points covered by N at least k times and collecting the Minkowski functionals in a generating function V, a simple formula is derived for the expectation of V(S ₀ Q ^k ) under an adequate moment condition on . This formula reflects the Poisson character of the process and extends to intersections of finite families of independent Poisson processes.     

Stochastic domination in space‐time for the contact process

Liggett and Steif (2006) proved that, for the supercritical contact process on certain graphs, the upper invariant measure stochastically dominates an i.i.d. Bernoulli product measure. In particular, they proved this for and (for infection rate sufficiently large) d‐ary homogeneous trees T_d. In this paper, we prove some space‐time versions of their results. We do this by combining their methods with specific properties of the contact process and general correlation inequalities. One of our main results concerns the contact process on T_d with . We show that, for large infection rate, there exists a subset Δ of the vertices of T_d, containing a “positive fraction” of all the vertices of T_d, such that the following holds: The contact process on T_d observed on Δ stochastically dominates an independent spin‐flip process. (This is known to be false for the contact process on graphs having subexponential growth.) We further prove that the supercritical contact process on observed on certain d‐dimensional space‐time slabs stochastically dominates an i.i.d. Bernoulli product measure, from which we conclude strong mixing properties important in the study of certain random walks in random environment.     

A Functional Central Limit Theorem for Asymptotically Negatively Dependent Random Fields

Let X _k ; k N ^d be a random field which is asymptotically negative dependent in a certain sense. Define the partial sum process in the usual way so that , where . Under some suitable conditions, we show that W _n (·) converges in distribution to a Brownian sheet. Direct consequences of the result are functional central limit theorems for negative dependent random fields. The result is based on some general theorems concerning asymptotically negative dependent random fields, which are of independent interest.     

An interior point method, based on rank-1 updates, for linear programming

We propose a polynomial time primal—dual potential reduction algorithm for linear programming. The algorithm generates sequencesd ^k andv ^k rather than a primal—dual interior point (x ^k ,s ^k ), where and fori = 1, 2,,n. Only one element ofd ^k is changed in each iteration, so that the work per iteration is bounded by O(mn) using rank-1 updating techniques. The usual primal—dual iteratesx ^k ands ^k are not needed explicitly in the algorithm, whereasd ^k andv ^k are iterated so that the interior primal—dual solutions can always be recovered by aforementioned relations between (x ^k, s^k) and (d ^k, v^k) with improving primal—dual potential function values. Moreover, no approximation ofd ^k is needed in the computation of projection directions. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

Dualisierung der Hauptkurven in allgemeinen Cayley/Klein-Räumen

Dualizing the osculating hyperplanes of a curvek of an n-dimensional Cayley/Klein space (CK space) according to the duality principle of projective spaces yields a curve of the dual CK space, the dual curvek _d ofk. A point ofk with torsion 0 corresponds to a stationary point ofk _d . The dualization of the osculating hyperplanes ofk _d yieldsk. There are simple relations between the moving simplexes, the CK arc lengths and the CK curvatures of the two curves which are valid for all CK spaces, even for those which are not self-dual.

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Herrn Prof. Dr.Johannes Böhm zum 70. Geburtstag gewidmet     

Krasnosel'skii numbers for bounded finitely starlike sets inR d

For eachk andd, 1kd, definef(d, d)=d+1 andf(d, k)=2d if 1kd–1. The following results are established:Let be a uniformly bounded collection of compact, convex sets inR ^d . For a fixedk, 1kd, dim {MM in }k if and only if for some > 0, everyf(d, k) members of contain a commonk-dimensional set of measure (volume) at least.LetS be a bounded subset ofR ^d . Assume that for some fixedk, 1kd, there exists a countable family of (k–l)-flats {H _i :i1} inR ^d such that clS S {H_i i 1 } and for eachi1, (clS S) H _i has (k–1) dimensional measure zero. Every finite subset ofS sees viaS a set of positivek-dimensional measure if and only if for some>0, everyf(d,k) points ofS see viaS a set ofk-dimensional measure at least .The numbers off(d,d) andf(d, 1) above are best possible.Supported in part by NSF grant DMS-8705336.     

Applications of random sampling in computational geometry,II

We use random sampling for several new geometric algorithms. The algorithms are Las Vegas, and their expected bounds are with respect to the random behavior of the algorithms. These algorithms follow from new general results giving sharp bounds for the use of random subsets in geometric algorithms. These bounds show that random subsets can be used optimally for divide-and-conquer, and also give bounds for a simple, general technique for building geometric structures incrementally. One new algorithm reports all the intersecting pairs of a set of line segments in the plane, and requiresO(A+n logn) expected time, whereA is the number of intersecting pairs reported. The algorithm requiresO(n) space in the worst case. Another algorithm computes the convex hull ofn points inE ^d inO(n logn) expected time ford=3, andO(n ^[d/2]) expected time ford>3. The algorithm also gives fast expected times for random input points. Another algorithm computes the diameter of a set ofn points inE ³ inO(n logn) expected time, and on the way computes the intersection ofn unit balls inE ³. We show thatO(n logA) expected time suffices to compute the convex hull ofn points inE ³, whereA is the number of input points on the surface of the hull. Algorithms for halfspace range reporting are also given. In addition, we give asymptotically tight bounds for (k)-sets, which are certain halfspace partitions of point sets, and give a simple proof of Lee's bounds for high-order Voronoi diagrams.