Location change in marginal distributions of linear functions of random vectors

Suppose X and Y are n × 1 random vectors such that l′X + f(l) and l′Y have the same marginal distribution for all n × 1 real vectors l and some real valued function f(l), and the existence of expectations of X and Y is not necessary. Under these conditions it is proven that there exists a vector M such that f(l) = l′M and X + M and Y have the same joint distribution. This result is extended to Banach-space valued random vectors.     

Sufficient conditions for existence of extrema in linear programming

It is shown that, on a closed convex subset X of a real Hausdorff locally convex space E, a continuous linear functional x′ on E has an extremum at an extreme point of X, provided X contains no line and X ∩ (x′)^?1 (λ₀) is non-empty and weakly compact for some real λ₀. It is also shown that any weakly locally compact closed convex subset of E that contains no line is the sum of its asymptotic cone and the closed convex hull of its extreme points.     

Uniqueness and value distribution of differences of entire functions

We consider the existence of transcendental entire solutions of certain type of non-linear difference equations. As an application, we investigate the value distribution of difference polynomials of entire functions. In particular, we are interested in the existence of zeros of fn(z)(λfm(z+c)+μfm(z))−a, where f is an entire function, n, m are two integers such that n?m>0, and λ, μ are non-zero complex numbers. We also obtain a uniqueness result in the case where shifts of two entire functions share a small function.     

Scaling limits for point random fields

If X is a point random field on $\text{R}$^d then convergence in distribution of the renormalization C_λ|X_λ ? α_λ| as λ → ∞ to generalized random fields is examined, where C_λ > 0, α_λ are real numbers for λ > 0, and X_λ(f) = λ^?dX(f_λ) for $\text{f}{}_{\mathrm{\lambda }}\text{(x) = f(}\text{x}\text{λ}\text{)}$. If such a scaling limit exists then C_λ = λ^θg(λ), where g is a slowly varying function, and the scaling limit is self-similar with exponent θ. The classical case occurs when $\text{θ =}\text{d}\text{2}$ and the limit process is a Gaussian white noise. Scaling limits of subordinated Poisson (doubly stochastic) point random fields are calculated in terms of the scaling limit of the environment (driving random field). If the exponent of the scaling limit is $\text{θ =}\text{d}\text{2}$ then the limit is an independent sum of the scaling limit of the environment and a Gaussian white noise. If $\text{θ <}\text{d}\text{2}$ the scaling limit coincides with that of the environment while if $\text{θ >}\text{d}\text{2}$ the limit is Gaussian white noise. Analogous results are derived for cluster processes as well.     

Martingale representation for Poisson processes with applications to minimal variance hedging

We consider a Poisson process η on a measurable space equipped with a strict partial ordering, assumed to be total almost everywhere with respect to the intensity measure λ of η. We give a Clark-Ocone type formula providing an explicit representation of square integrable martingales (defined with respect to the natural filtration associated with η), which was previously known only in the special case, when λ is the product of Lebesgue measure on R₊ and a σ-finite measure on another space X. Our proof is new and based on only a few basic properties of Poisson processes and stochastic integrals. We also consider the more general case of an independent random measure in the sense of Itô of pure jump type and show that the Clark-Ocone type representation leads to an explicit version of the Kunita-Watanabe decomposition of square integrable martingales. We also find the explicit minimal variance hedge in a quite general financial market driven by an independent random measure.     

Canonical row-column-exchangeable arrays

Consider a standard row-column-exchangeable array X = (X_ij : i,j ≥ 1), i.e., X_ij = f(a, ξ_i, η_j, λ_ij) is a function of i.i.d. random variables. It is shown that there is a canonical version of X, X′, such that X′, and α′, ξ′₁, ξ′₂,…, η′₁, η′₂,…, are conditionally independent given ∩_{n ≥ 1}σ(X′_ij : max(i,j) ≥ n). This result is quite a bit simpler to prove than the analogous result for the original array X, which is due to Aldous.     

Equivariant geometric K-homology for compact Lie group actions

Let G be a compact Lie-group, X a compact G-CW-complex. We define equivariant geometric K-homology groups $K^{G}_{*}(X)$ , using an obvious equivariant version of the (M,E,f)-picture of Baum-Douglas for K-homology. We define explicit natural transformations to and from equivariant K-homology defined via KK-theory (the “official” equivariant K-homology groups) and show that these are isomorphisms.     

Two, more readily computable equivariant Nielsen numbers I. Nielsen theory for M-ads

In this, the first of two papers outlining a Nielsen theory for “two, more readily computable equivariant numbers”, we define and study two Nielsen type numbers N(f,k;X−{Xν}_ν∈M) and N(f,k;X,{Xν}_ν∈M), where f and k are M-ad maps. While a Nielsen theory of M-ads is of interest in its own right, our main motivation lies in the fact that maps of M-ads accurately mirror one of two fundamental structures of equivariant maps. Being simpler however, M-ad Nielsen numbers are easier to study and to compute than equivariant Nielsen numbers. In the sequel, we show our M-ad numbers can be used to form both upper and lower bounds on their equivariant counterparts.The numbers N(f,k;X−{Xν}_ν∈M) and N(f,k;X,{Xν}_ν∈M), generalize the generalizations to coincidences, of Zhao's Nielsen number on the complement N(f;X−A), respectively Schirmer's relative Nielsen number N(f;X,A). Our generalizations are from the category of pairs, to the category of M-ads. The new numbers are lower bounds for the number of coincidence points of all maps f^′ and k^′ which are homotopic as maps ofM-ads to f, respectively k firstly on the complement of the union of the subspaces Xν in the domain M-ad X, and secondly on all of X. The second number is shown to be greater than or equal to a sum of the first of our numbers. Conditions are given which allow for both equality, and Möbius inversion. Finally we show that the fixed point case of our second number generalizes Schirmer's triad Nielsen number N(f;X₁∪X₂).Our work is very different from what at first sight appears to be similar partial results due to P. Wong. The differences, while in some sense subtle in terms of definition, are profound in terms of commutability. In order to work in a variety of both fixed point and coincidence points contexts, we introduce in this first paper and extend in the second, the concept of an essentiality on a topological category. This allows us to give computational theorems within this diversity. Finally we include an introduction to both papers here.     

The Poisson equation on complete manifolds with positive spectrum and applications

In this paper we investigate the existence of a solution to the Poisson equation on complete manifolds with positive spectrum and Ricci curvature bounded from below. We show that if a function f has decay f=O(r−1−ε) for some ε>0, where r is the distance function to a fixed point, then the Poisson equation Δu=f has a solution u with at most exponential growth.We apply this result on the Poisson equation to study the existence of harmonic maps between complete manifolds and also existence of Hermitian-Einstein metrics on holomorphic vector bundles over complete manifolds, thus extending some results of Li-Tam and Ni.Assuming moreover that the manifold is simply connected and of Ricci curvature between two negative constants, we can prove that in fact the Poisson equation has a bounded solution and we apply this result to the Ricci flow on complete surfaces.     

Image of the Schwartz space under spectral projection

Let X?= G/K be a Riemannian symmetric space of non compact type and rank-one. The spectral projection P _λ f of a function f on X can be written P _λ f = f * φ _λ where φ _λ is the elementary spherical function corresponding to the complex parameter λ. We characterize the image of the Schwartz space ${\mathcal S^p(X)}$ under the spectral projection for 0 < p ≤ 2.     

Weak convergence of semimartingales and discretisation methods

Given a semimartingale one can construct a system (λ, A, B, C) where λ is the distribution of the initial value and (A, B, C) is the triple of global characteristics. Thus, given a process X and a system (λ, A, B, C) one can look for all probability measures P such that X is a P-semimartingale with initial distribution λ and global characteristics (A, B, C). We say that such a measure P is a solution to the semimartingale problem (λ, A, B, C).The paper is devoted to the study of a special type of semimartingale problem. We look for sufficient conditions to insure the existence of solutions and we develop a method to construct them by means of time-discretised schemes, using weak topology for probability measures.     

K-flows are forward deterministic,backward completely non-deterministic stationary point processes

Stationary processes with discrete time parameter and finitely many states are forward deterministic if and only if they are backward deterministic. In contrast to this we prove in the case of continuous time parameter: Every K-flow in a Lebesgue space is isomorphic to the flow of shifts of a stationary forward deterministic, backward completely non-deterministic process (X_t, t?R¹) with two states and with right-continuous paths having only finitely many jumps in any finite time-interval. The process may be considered as a point process. The result is obtained from a representation theorem for flows, describing increasing sub-σ-algebras of a flow up to “equivalence.”     

Equivariant Kähler compactifications of homogeneous manifolds

A compact complex manifoldX is an equivariant compactification of a homogeneous manifoldG/H (G a connected complex Lie group,H a closed complex subgroup ofG), if there exists a holomorphic action ofG onX such that theG-orbit of some pointx inX is open and H is the isotropy group ofx. GivenG andH, for some groups (e.g.,G nilpotent) there are necessary and sufficient conditions for the existence of an equivariant Kähler compactification which are proven in this paper.     

Equivariant Chow Ring and Chern Classes of Wonderful Symmetric Varieties of Minimal Rank

We describe the equivariant Chow ring of the wonderful compactification X of a symmetric space of minimal rank, via restriction to the associated toric variety Y. Also, we show that the restrictions to Y of the tangent bundle T _X and its logarithmic analogue S _X decompose into a direct sum of line bundles. This yields closed formulas for the equivariant Chern classes of T _X and S _X , and, in turn, for the Chern classes of reductive groups considered by Kiritchenko.     

Geometric ergodicity of Harris recurrent Marcov chains with applications to renewal theory

Let (X_n) be a positive recurrent Harris chain on a general state space, with invariant probability measure π. We give necessary and sufficient conditions for the geometric convergence of λPⁿf towards its limit π(f), and show that when such convergence happens it is, in fact, uniform over f and in L¹(π)-norm. As a corollary we obtain that, when (Xn) is geometrically ergodic, ∝ π(dx)6Pⁿ(x,·)-π6 converges to zero geometrically fast. We also characterize the geometric ergodicity of (X_n) in terms of hitting time distributions. We show that here the so-called small sets act like individual points of a countable state space chain. We give a test function criterion for geometric ergodicity and apply it to random walks on the positive half line. We apply these results to non-singular renewal processes on [0,∞) providing a probabilistic approach to the exponencial convergence of renewal measures.     

Smooth actions of finite Oliver groups on spheres

In this article, we deal with the following two questions. For smooth actions of a given finite group G on spheres S, which smooth manifolds F occur as the fixed point sets in S, and which real G-vector bundles ν over F occur as the equivariant normal bundles of F in S? We focus on the case G is an Oliver group and answer both questions under some conditions imposed on G, F, and ν. We construct smooth actions of G on spheres by making use of equivariant surgery, equivariant thickening, and Oliver's equivariant bundle extension method modified by an equivariant wegde sum construction and an equivariant bundle subtraction procedure.     

Poisson cluster measures: Quasi-invariance, integration by parts and equilibrium stochastic dynamics

The distribution μcl of a Poisson cluster process in X=Rd (with i.i.d. clusters) is studied via an auxiliary Poisson measure on the space of configurations in X=n_?Xn, with intensity measure defined as a convolution of the background intensity of cluster centres and the probability distribution of a generic cluster. We show that the measure μcl is quasi-invariant with respect to the group of compactly supported diffeomorphisms of X and prove an integration-by-parts formula for μcl. The corresponding equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms.     

On an asymptotic series of Ramanujan

An asymptotic series in Ramanujan’s second notebook (Entry 10, Chap. 3) is concerned with the behavior of the expected value of φ(X) for large λ where X is a Poisson random variable with mean λ and φ is a function satisfying certain growth conditions. We generalize this by studying the asymptotics of the expected value of φ(X) when the distribution of X belongs to a suitable family indexed by a convolution parameter. Examples include the binomial, negative binomial, and gamma families. Some formulas associated with the negative binomial appear new.     

A van Kampen theorem for equivariant fundamental groupoids

The equivariant fundamental groupoid of a G-space X is a category which generalizes the fundamental groupoid of a space to the equivariant setting. In this paper, we prove a van Kampen theorem for these categories: the equivariant fundamental groupoid of X can be obtained as a pushout of the categories associated to two open G-subsets covering X. This is proved by interpreting the equivariant fundamental groupoid as a Grothendieck semidirect product construction, and combining general properties of this construction with the ordinary (non-equivariant) van Kampen theorem. We then illustrate applications of this theorem by showing that the equivariant fundamental groupoid of a G-CW complex only depends on the 2-skeleton and also by using the theorem to compute an example.