Some inverse problems involving conditional expectations

Let (Ω, F, P) be a probability space, let H be a sub-σ-algebra of F, and let Y be positive and H-measurable with E[Y] = 1. We discuss the structure of the convex set CE(Y; H) = {X ∈ pF: Y = E[X|H]} of random variables whose conditional expectation given H is the prescribed Y. Several characterizations of extreme points of CE(Y; H) are obtained. A necessary and sufficient condition is given in order that CE(Y; H) be the closed, convex hull of its extreme points. For the case of finite F we explicitly calculate the extreme points of CE(Y; H), identify pairs of adjacent extreme points, and characterize extreme points of CE(Y; H) ? CE(Z; G), where G is a second sub-σ-algebra of F and Z ∈ pG. When H = σ(Y) and appropriate topological hypotheses hold, extreme points of CE(Y; H) are shown to be in explicit one-to-one correspondence with certain left inverses of Y. Finally, it is shown how the same approach can be applied to the problem of extremal random measures on $\text{R}$₊ with a prescribed compensator, to deduce that the number of extreme points is zero or one.     

Local times for stochastic processes which are subordinate to Gaussian processes

Let X and Y be random vectors of the same dimension such that Y has a normal distribution with mean vector O and covariance matrix R. Let g(x), x≥0, be a bounded nonincreasing function. X is said to be g-subordinate to Y if |Ee^iu′X| ≤ g(u′Ru) for all real vectors u of the same dimension as X. This is used to define the g-subordination of a real stochastic process X(t), 0 ≤ t ≤ 1, to a Gaussian process Y(t), 0 ≤ t ≤ 1. It is shown that the basic local time properties of a given Gaussian process are shared by all the processes that age g-subordinate to it. It is shown in particular that certain random series, including some random Fourier series, are g-subordinate to Gaussian processes, and so have their local time properties.     

A generalization of a result of Sylvester's

Let v₁,…,v_n be vectors in Zⁿ with D = det(v₁,…,v_n) > 0. Let v_{n + 1} be in the cone generated by v₁,…,v_n and such that v₁,…,v_v, v_{n + 1} generate Zⁿ as a Z-module. There exists a unique “largest“ χ not expressible as a nonnegative integer combination of v₁,…,v_n, v_{n + 1} and χ = Dv_{n + 1} ? (v₁ + … v_n + v_{n + 1}).     

A duality principle for lattices and categories of modules

Let R be a ring with 1, R^op the opposite ring, and R-Mod the category of left unitary R-modules and R-linear maps. A characterization of well-powered abelian categories $\text{A}$ such that there exists an exact embedding functor $\text{A}$→R-Mod is given. Using this characterization and abelian category duality, the following duality principles can be established.Theorem. There exists an exact embedding functor $\text{A}$→R-Mod if and only if there exists an exact embedding functor $\text{A}$^op→R^op-Mod.Corollary. If R-Mod has a specified diagram-chasing property, then R^op-Mod has the dual property.A lattice L is representable by R-modules if it is embeddable in the lattice of submodules of some unitary left R-module; $\text{L}$(R) denotes the quasivariety of all lattices representable by R-modules.Theorem. A lattice L is representable by R-modules if and only if its order dual L¹ is representable by R^op-modules. That is, $\text{L}\text{(}\text{R}{}^{\mathrm{op}}\text{)={L}{}^1\text{:L?}\text{L}\text{(}\text{R}\text{)}}$.If $\text{R}$ is a commutative ring with 1 and a specified diagram-chasing result is satisfied in R-Mod, then the dual result is also satisfied in R-Mod. Furthermore, $\text{L}\text{(}\text{R}\text{)}$ is self-dual: $\text{L}\text{(}\text{R}\text{)= {L}{}^1\text{:L?}\text{L}\text{(}\text{R}\text{)}.}$     

On the existence of free topological groups

Given a Tychonoff space X and classes U and V of topological groups, we say that a topological group G = G(X, U, V) is a free (U,V)-group over X if (a) X is a subspace of G, (b) G ϵ U, and (c) every continuous f: X → H with H ϵ V extends uniquely to a continuous homomorphism f̄: G→H. For certain classes U and V, we consider the question of the existence of free (U,V)- groups. Our principal results are the following. Let PA and CA denote, respectively, the class ofpseudocompact Abelian groups and the class of compact Abelian groups. Then

• 1.(a) there is a free (PA,PA)-group over X iff; X=Ø and
• 2.(b) there is for each X a free (PA,CA)-group over X in which X is closed.

     

Positively regular homeomorphisms of Euclidean spaces

A homeomorphism of Rⁿ onto itself is called positively regular (or EC⁺) iff its family of non-negative iterates is pointwise equicontinuous. For EC⁺ homeomorphism of Rⁿ such that some point of Rⁿ has bounded positive semi-orbit, the nucleus M is defined, and the following theorems are proved.Theorem 1. If such a homeomorphism h:Rⁿ→Rⁿ has compact nucleus M, then M is a fully invariant compact AR. Further, for n≠4,5,h:Rⁿ/M→Rⁿ/M is conjugate to a contraction on Rⁿ.Theorem 2. In Rⁿ,n≠4,5,M compact iff there existsa disk D such that h(D)?IntD.Theorem 3. In R², either M is a disk and h|M is a rotation, or h|M is periodic. The relationship between M and the irregular set of ? is also studied.     

On the asymptotic convergence factor of the total step method in interval computation

For a special class of n×n interval matrices A we derive a necessary and sufficient condition for the asymptotic convergence factor α of the total step method x^(m+1)=Ax^(m)+b to be less than the spectral radius ϱ(|A|) of the absolute value |A| of A.     

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.     

A topological property enjoyed by near points but not by large points

Let H denote the halfline [0,∞). A point p?βH?H is called a near point if p is in the closure of some countable discrete closed subspace of H. In addition, a point p?βH?H is called a large point if p is not in the closure of a closed subset of H of finite Lebesgue measure. We will show that for every autohomeomorphism ? of βH?H and for each near point p we have that ?(p) is not large. In addition, we establish, under CH, the existence of a point x?βH?H such that for each autohomeomorphism ? of βH?H the point ?(x) is neither large nor near.     

Factorizing multivariate time series operators

Motivated by problems occurring in the empirical identification and modelling of a n-dimensional ARMA time series X(t) we study the possibility of obtaining a factorization (I + a₁B + … + a_pB^p) X(t) = [Π_i=1^p (I ? α_iB)] X(t), where B is the backward shift operator. Using a result in [3] we conclude that as in the univariate case such a factorization always exists, but unlike the univariate case in general the factorization is not unique for given a₁, a₂,…, a_p. In fact the number of possibilities is limited upwards by $\text{(np)!}\text{(n!)}{}^{\mathrm{p}}$, there being cases, however, where this maximum is not reached. Implications for the existence and possible use of transformations which removes nonstationarity (or almost nonstationarity) of X(t) are mentioned.     

A unique exchange property for bases

We define the concept of unique exchange on a sequence (X₁,…, X_m) of bases of a matroid M as an exchange of x ? X_i for y ? X_j such that y is the unique element of X_j which may be exchanged for x so that (X_i ? {x}) ∪ {y} and (X_j ? {y}) ∪ {x} are both bases. Two sequences X and Y are compatible if they are on the same multiset. Let UE(1) [UE(2)] denote the class of matroids such that every pair of compatible basis sequences X and Y are related by a sequence of unique exchanges [unique exchanges and permutations in the order of the bases]. We similarly define UE(3) by allowing unique subset exchanges. Then UE(1),UE(2), and UE(3) are hereditary classes (closed under minors) and are self-dual (closed under orthogonality). UE(1) equals the class of series-parallel networks, and UE(2) and UE(3) are contained in the class of binary matroids. We conjecture that UE(2) contains the class of unimodular matroids, and prove a related partial result for graphic matroids. We also study related classes of matroids satisfying transitive exchange, in order to gain information about excluded minors of UE(2) and UE(3). A number of unsolved problems are mentioned.     

Dimension and embedding theorems for geometric lattices

Let G be an n-dimensional geometric lattice. Suppose that 1 ? e, f ? n ? 1, e + f ? n, but e and f are not both n ? 1. Then, in general, there are E, F? G with dim E = e, dim F = f, E ? F = 1, and dim E ∧ F = e + f ? n ? 1; any exception can be embedded in an n-dimensional modular geometric lattice M in such a way that joins and dimensions agree in G and M, as do intersections of modular pairs, while each point and line of M is the intersection (in M) of the elements of G containing it.     

A Class of Nonlinear Averaging Integral Operators

LetHbe the class of analytic functions defined in the unit discU, and let coEdenote the convex hull of a setEinC. IfK⊂H, then an operatorI:K→His an averaging operator ifI[f](0) =f(0) andI[f](U) ⊂ cof(U), for allf∈K. The authors show that the operatorI_β,γ[f](z) ≡ [γz^−γ∫^z₀f^β(t)t^γ−1dt]^1/βis an averaging operator on certain subsets ofH.     

Closure operators I

Closure operators in an (E, M)-category X are introduced as concrete endofunctors of the comma category whose objects are the elements of M. Various kinds of closure operators are studied. There is a Galois equivalence between the conglomerate of idempotent and weakly hereditary closure operators of X and the conglomerate of subclasses of M which are part of a factorization system. There is a one-to-one correspondence between the class of regular closure operators and the class of strongly epireflective subcategories of X. Every closure operators admits an idempotent hull and a weakly hereditary core.Various examples of additive closure operators in Top are given. For abelian categories standard closure operators are considered. It is shown that there is a one-to-one correspondence between the class of standard closure operators and the class of preradicals. Idempotent, weakly hereditary, standard closure operators correspond to idempotent radicals (= torsion theories).     

Properties of Euclidean and non-Euclidean distance matrices

A distance matrix D of order n is symmetric with elements $\text{?}\text{1}\text{2}\text{d}{}_{\mathrm{ij}}{}^2$, where d_ii=0. D is Euclidean when the $\text{1}\text{2}\text{n(n?1)}$ quantities d_ij can be generated as the distances between a set of n points, X (n×p), in a Euclidean space of dimension p. The dimensionality of D is defined as the least value of p=rank(X) of any generating X; in general p+1 and p+2 are also acceptable but may include imaginary coordinates, even when D is Euclidean. Basic properties of Euclidean distance matrices are established; in particular, when ρ=rank(D) it is shown that, depending on whether e^TD^?e is not or is zero, the generating points lie in either p=ρ?1 dimensions, in which case they lie on a hypersphere, or in p=ρ?2 dimensions, in which case they do not. (The notation e is used for a vector all of whose values are one.) When D is non-Euclidean its dimensionality p=r+s will comprise r real and s imaginary columns of X, and (r, s) are invariant for all generating X of minimal rank. Higher-ranking representations can arise only from p+1=(r+1)+s or p+1=r+ (s+1) or p+2=(r+1)+(s+1), so that not only are r, s invariant, but they are both minimal for all admissible representations X.     

A characterization of the multivariate normal distribution

Let X_j (j = 1,…,n) be i.i.d. random variables, and let Y′ = (Y₁,…,Y_m) and X′ = (X₁,…,X_n) be independently distributed, and A = (a_jk) be an n × n random coefficient matrix with a_jk = a_jk(Y) for j, k = 1,…,n. Consider the equation U = AX, Kingman and Graybill [Ann. Math. Statist.41 (1970)] have shown U ～ N(O,I) if and only if X ～ N(O,I). provided that certain conditions defined in terms of the a_jk are satisfied. The task of this paper is to delete the identical assumption on X₁,…,X_n and then generalize the results to the vector case. Furthermore, the condition of independence on the random components within each vector is relaxed, and also the question raised by the above authors is answered.     

An analysis of (h,k, 1)-Shellsort

One classical sorting algorithm, whose performance in many cases remains unanalyzed, is Shellsort. Let h be a t-component vector of positive integers. An h-Shellsort will sort any given n elements in t passes, by means of comparisons and exchanges of elements. Let S>_j(h; n) denote the average number of element exchanges in the jth pass, assuming that all the n! initial orderings are equally likely. In this paper we derive asymptotic formulas of S_j(h; n) for any fixed h = (h, k, l), making use of a new combinatorial interpretation of S₃. For the special case h = (3, 2, 1), the analysis is further sharpened to yield exact expressions.     

The concept lattice functors

This paper is concerned with the relationship between contexts, closure spaces, and complete lattices. It is shown that, for a unital quantale L, both formal concept lattices and property oriented concept lattices are functorial from the category L-Ctx of L-contexts and infomorphisms to the category L-Sup of complete L-lattices and suprema-preserving maps. Moreover, the formal concept lattice functor can be written as the composition of a right adjoint functor from L-Ctx to the category L-Cls of L-closure spaces and continuous functions and a left adjoint functor from L-Cls to L-Sup.     

Maximizing the probability of correctly ordering random variables using linear predictors

Let (T₁, x₁), (T₂, x₂), …, (T_n, x_n) be a sample from a multivariate normal distribution where T_i are (unobservable) random variables and x_i are random vectors in R^k. If the sample is either independent and identically distributed or satisfies a multivariate components of variance model, then the probability of correctly ordering {T_i} is maximized by ranking according to the order of the best linear predictors {E(T_i|x_i)}. Furthermore, it orderings are chosen according to linear functions {b′x_i} then the conditional probability of correct order given (T_i = t₁; i = 1, …, n) is maximized when b′x_i is the best linear predictor. Examples are given to show that linear predictors may not be optimal and that using a linear combination other that the best linear predictor may give a greater probability of correctly ordering {T_i} if {(T_i, x_i)} are independent but not identically distributed, or if the distributions are not normal.     

Antichains in the set of subsets of a multiset

A set F of distinct subsets x of a finite multiset M (that is, a set with several different kinds of elements) is a c-antichain if for no c + 1 elements x₀, x₁, …, x_c of F does x₀ ? x₁ ? ··· ? x_c hold. The weight of F, wF, is the total number of elements of M in the various elements x of F. For given integers f and c, we find min wF, where the minimum is taken over all f-element c-antichains F. Daykin [9, 10] has solved this problem for ordinary sets and Clements [3] has solved it for multisets, but only for c = 1.