von Neumann��s mean ergodic theorem on complete random inner product modules

We first prove two forms of von Neumann’s mean ergodic theorems under the framework of complete random inner product modules. As applications, we obtain two conditional mean ergodic convergence theorems for random isometric operators which are defined on L _ℱ ^p (ℰ, H) and generated by measure-preserving transformations on Ω, where H is a Hilbert space, L ^p (ℰ, H) (1 ⩽ p < ∞) the Banach space of equivalence classes of H-valued p-integrable random variables defined on a probability space (Ω, ℰ, P), F a sub σ-algebra of ℰ, and L _ℱ ^p (ℰ(E,H) the complete random normed module generated by L ^p (ℰ, H).     

Structure of spaces ofC ∞-functions on nuclear spaces

LetE be a real nuclear locally convex space; we prove that the space ℰ_ub(E), of allC ^∞-functions of uniform bounded type onE, coincides with the inductive limit of the spaces ℰ_Nbc(E _v) (introduced by Nachbin-Dineen), whenV ranges over a basis of convex balanced 0-neighbourhoods inE. LetE be a real nuclear bornological vector space; we prove that the space ℰ(E) of allC ^∞-functions onE coincides with the projective limit of the spaces ℰ_Nbc(E _B), whenB is a closed convex balanced bounded subset ofE. As a consequence we obtain some density results and a version of the Paley-Wiener-Schwartz theorem. Research done during the stay of this author at the University of Bordeaux (France) in the academic year 1980–1981.     

Strong orbit realization for minimal homeomorphisms

LetXbe a Cantor set,S a minimal self-homeomorphism ofX, and Μ anS-invariant Borel probability. LetT be an ergodic automorphism of a non-atomic Lebesgue probability space(Y,Ν). Then there is a minimal homeomorphismS′ with the same orbits asS such that (S′, Μ ) is measurably conjugate to (T, Ν). HereS′ can be chosen strongly orbit equivalent toS if and only if the periodic spectrum ofS is contained in the discrete spectrum ofT. Corollaries of these results generalize Dye’s Theorem and the Jewett-Krieger Theorem.     

An Ergodic Decomposition Defined by Transition Probabilities

Our main goal in this paper is to prove that any transition probability P on a locally compact separable metric space (X,d) defines a Kryloff-Bogoliouboff-Beboutoff-Yosida (KBBY) ergodic decomposition of the state space (X,d). Our results extend and strengthen the results of Chap. 5 of Hernández-Lerma and Lasserre (Markov Chains and Invariant Probabilities, [2003]) and extend our KBBY-decomposition for Markov-Feller operators that we have obtained in Chap. 2 of our monograph (Zaharopol in Invariant Probabilities of Markov-Feller Operators and Their Supports, [2005]). In order to deal with the decomposition that we present in this paper, we had to overcome the fact that the Lasota-Yorke lemma (Theorem 1.2.4 in our book (op. cit.)) and two results of Lasota and Myjak (Proposition 1.1.7 and Corollary 1.1.8 of our work (op. cit.)) are no longer true in general in the non-Feller case. In the paper, we also obtain a “formula” for the supports of elementary measures of a fairly general type. The result is new even for Markov-Feller operators. We conclude the paper with an outline of the KBBY decomposition for a fairly large class of transition functions. The results for transition functions and transition probabilities seem to us surprisingly similar. However, as expected, the arguments needed to prove the results for transition functions are significantly more involved and are not presented here. We plan to discuss the KBBY decomposition for transition functions with full details in a small monograph that we are currently trying to write. I am indebted to Sean Meyn for a discussion that we had in November 2004, which helped me to significantly improve the exposition in this paper, and to two anonymous referees for useful recommendations.     

The symplectic Gram-Schmidt theorem and fundamental geometries for A-modules

Like the classical Gram-Schmidt theorem for symplectic vector spaces, the sheaf-theoretic version (in which the coefficient algebra sheaf A is appropriately chosen) shows that symplectic A-morphisms on free A-modules of finite rank, defined on a topological space X, induce canonical bases (Theorem 1.1), called symplectic bases. Moreover (Theorem 2.1), if (ℰ, φ) is an A-module (with respect to a ℂ-algebra sheaf A without zero divisors) equipped with an orthosymmetric A-morphism, we show, like in the classical situation, that “componentwise” φ is either symmetric (the (local) geometry is orthogonal) or skew-symmetric (the (local) geometry is symplectic). Theorem 2.1 reduces to the classical case for any free A-module of finite rank.     

Strong consistency of kernel estimators for Markov transition densities

Let P(x, dy) = t (x, y)ν(d y) be the transition kernel of a Markov chain, where t (x, y) is a density with respect to a σ-finite measure ν on (E,ℰ), with E ⊂ R ^d . In this note, we propose a general class of estimates for t (x, y) that are strongly consistent and that extend the classical results for continuous densities on R ^d . Received: 2 June 2002     

Graph factors

This exposition is concerned with the main theorems of graph-factor theory, Hall’s and Ore’s Theorems in the bipartite case, and in the general case Petersen’s Theorem, the 1-Factor Theorem and thef-Factor Theorem. Some published extensions of these theorems are discussed and are shown to be consequences rather than generalizations of thef-Factor Theorem. The bipartite case is dealt with in Section 2. For the proper presentation of the general case a preliminary theory of “G-triples” and “f-barriers” is needed, and this is set out in the next three Sections. Thef-Factor Theorem is then proved by an argument of T. Gallai in a generalized form. Gallai’s original proof derives the 1-Factor Theorem from Hall’s Theorem. The generalization proceeds analogously from Ore’s Theorem to thef-Factor Theorem.     

A 10-point circle is associated with any general point of the ellipse. New properties of Fagnano’s point

Let H be an ellipse with semiaxes a and b (a > b). Two circles concentric with H, and with radii a − b and a + b, are described, each of them being the locus of the intersections between couples of noteworthy H-related lines (Theorems 1 and 2). Tight, as well as unexpected links among such circles and Monge’s circle are shown (Theorems 4, 5, and 6). A surprising pythagorean relationship involving segments related to the ellipse is shown (Theorem 3). A set of 10 concyclic points is associated with any general point of H (Theorem 9). New properties of Fagnano’s point are described (Theorems 10 through 13). Only elementary facts from trigonometry and analytic geometry are used.      

The conjugate gradient method for computing all the extremal stationary probability vectors of a stochastic matrix

Summary The conjugate gradient method is developed for computing stationary probability vectors of a large sparse stochastic matrixP, which often arises in the analysis of queueing system. When unit vectors are chosen as the initial vectors, the iterative method generates all the extremal probability vectors of the convex set formed by all the stationary probability vectors ofP, which are expressed in terms of the Moore-Penrose inverse of the matrix (P−I). A numerical method is given also for classifying the states of the Markov chain defined byP. One particular advantage of this method is to handle a very large scale problem without resorting to any special form ofP. The Institute of Statistical Mathematics     

On Fefferman and Burkholder–Davis–Gundy inequalities for ℰ-martingales

In a previous paper we introduced a new concept, the notion of ℰ-martingales and we extended the well-known Doob inequality (for 1 < p < + ∞) and the Burkholder–Davis–Gundy inequalities (for p = 2) to ℰ-martingales. After showing new Fefferman-type inequalities that involve sharp brackets as well as the space bmo _q , we extend the Burkholder–Davis–Gundy inequalities (for 1 < p < + ∞) to ℰ-martingales. By means of these inequalities we give sufficient conditions for the closedness in L ^p of a space of stochastic integrals with respect to a fixed ℝ^d-valued semimartingale, a question which arises naturally in the applications to financial mathematics. Finally we investigate the relation between uniform convergence in probability and semimartingale topology. Received: 22 July 1997 / Revised version: 3 July 1998     

Statistical maps: A categorical approach

In probability theory, each random variable f can be viewed as channel through which the probability p of the original probability space is transported to the distribution p _f , a probability measure on the real Borel sets. In the realm of fuzzy probability theory, fuzzy probability measures (equivalently states) are transported via statistical maps (equivalently, fuzzy random variables, operational random variables, Markov kernels, observables). We deal with categorical aspects of the transportation of (fuzzy) probability measures on one measurable space into probability measures on another measurable spaces. A key role is played by D-posets (equivalently effect algebras) of fuzzy sets. Supported by VEGA 1/2002/06.     

Potential Analysis on Carnot Groups，Part II：Relationship between Hausdorff Measures and Capacities

Abstract In this paper, we establish the relationship between Hausdorff measures and Bessel capacities on any nilpotent stratified Lie group (i. e., Carnot group). In particular, as a special corollary of our much more general results, we have the following theorem (see Theorems A and E in Section 1): Let Q be the homogeneous dimension of . Given any set E ⊂ , B _α,p (E) = 0 implies ℋ ^{Q−αp+ ε}(E) = 0 for all ε > 0. On the other hand, ℋ ^Q−αp (E) < ∞ implies B _α,p (E) = 0. Consequently, given any set E ⊂ of Hausdorff dimension Q − d, where 0 < d < Q, B _α,p (E) = 0 holds if and only if αp ≤ d. A version of O. Frostman’s theorem concerning Hausdorff measures on any homogeneous space is also established using the dyadic decomposition on such a space (see Theorem 4.4 in Section 4). Research supported partly by the U. S. National Science Foundation Grant No. DMS99–70352     

Extended F4-buildings and the Baby Monster

Let Τ be the Baby Monster graph which is the graph on the set of {3,4}-transpositions in the Baby Monster group B in which two such transpositions are adjacent if their product is a central involution in B. Then Τ is locally the commuting graph of central (root) involutions in ² E ₆(2). The graph Τ contains a family of cliques of size 120. With respect to the incidence relation defined via inclusion these cliques and the non-empty intersections of two or more of them form a geometry ℰ(B) with diagram for t=4 and the action of B on ℰ(B) is flag-transitive. We show that ℰ(B) contains subgeometries ℰ(² E ₆(2)) and ℰ(Fi ₂₂) with diagrams c.F ₄(2) and c.F ₄(1). The stabilizers in B of these subgeometries induce on them flag-transitive actions of ² E ₆(2):2 and Fi ₂₂:2, respectively. The geometries ℰ(B), ℰ(² E ₆(2)) and ℰ(Fi ₂₂) possess the following properties: (a) any two elements of type 1 are incident to at most one common element of type 2 and (b) three elements of type 1 are pairwise incident to common elements of type 2 if and only if they are incident to a common element of type 5. The paper addresses the classification problem of c.F ₄(t)-geometries satisfying (a) and (b). We construct three further examples for t=2 with flag-transitive automorphism groups isomorphic to 3⋅²E₂:2, E₆(2):2 and 2²⁶ .F₄(2) and one for t=1 with flag-transitive automorphism group 3⋅Fi ₂₂:2. We also study the graph of an arbitrary (non-necessary flag-transitive) c.F ₄(t)-geometry satisfying (a) and (b) and obtain a complete list of possibilities for the isomorphism type of subgraph induced by the common neighbours of a pair of vertices at distance 2. Finally, we prove that ℰ(B) is the only c.F ₄(4)-geometry, satisfying (a) and (b). Oblatum 20-X-1999 & 2-I-2001?Published online: 5 March 2001     

The Banach-Saks property is notL 2-hereditary

We construct a Banach spaceE, which has the Banach-Saks property and such thatL ²(E) does not have the Banach-Skas property. The construction is a somewhat tree-like modification of Baernstein’s space.     

Graph-Theoretic Approach to Stochastic Integrals with Clifford Algebras

Given a fixed probability space (Ω,ℱ,ℙ) and m≥1, let X(t) be an L²(Ω) process satisfying necessary regularity conditions for existence of the mth iterated stochastic integral. For real-valued processes, these existence conditions are known from the work of D. Engel. Engel’s work is extended here to L²(Ω) processes defined on Clifford algebras of arbitrary signature (p,q), which reduce to the real case when p=q=0. These include as special cases processes on the complex numbers, quaternion algebra, finite fermion algebras, fermion Fock spaces, space-time algebra, the algebra of physical space, and the hypercube. Next, a graph-theoretic approach to stochastic integrals is developed in which the mth iterated stochastic integral corresponds to the limit in mean of a collection of weighted closed m-step walks on a growing sequence of graphs. Combinatorial properties of the Clifford geometric product are then used to create adjacency matrices for these graphs in which the appropriate weighted walks are recovered naturally from traces of matrix powers. Given real-valued L²(Ω) processes, Hermite and Poisson-Charlier polynomials are recovered in this manner.     

Optimal Domains for <Emphasis Type="Italic">L</Emphasis><Superscript>0</Superscript>-valued Operators Via Stochastic Measures

We study extension of operators T: E→ L⁰([0, 1]), where E is an F–function space and L⁰([0, 1]) the space of measurable functions with the topology of convergence in measure, to domains larger than E, and we study the properties of such domains. The main tool is the integration of scalar functions with respect to stochastic measures and the corresponding spaces of integrable functions. Partially supported by D.G.I. #MTM2006-13000-C03-01 (Spain).     

Trinomial random walk with one or two imperfect absorbing barriers

Trinomial random walk, with one or two barriers, on the non-negative integers is discussed. At the barriers, the particle is either annihilated or reflects back to the system with respective probabilities 1 − ρ, ρ at the origin and 1 − ω, ω at L, 0 ≤ ρ,ω ≤ 1. Theoretical formulae are given for the probability distribution, its generating function as well as the mean of the time taken before absorption. In the one-boundary case, very qualitatively different asymptotic forms for the result, depending on the relationship between transition probabilities and the annihilation probability, are obtained.      

Two constructions of oriented matroids with disconnected extension space

The extension space ℰ(ℳ) of an oriented matroid ℳ is the poset of all one-element extensions of ℳ, considered as a simplicial complex. We present two different constructions leading to rank 4 oriented matroids with disconnected extension space. We prove especially that if an elementf is not contained in any mutation of a rank 4 oriented matroid ℳ, then ℰ(ℳ\f) contains an isolated point. A uniform nonrealizable arrangement of pseudoplanes with this property is presented. The examples described contrast results of Sturmfels and Ziegler [12] who proved that for rank 3 oriented matroids the extension space has the homotopy type of the 2-sphere.     

A Note on the Browder＇s and Weyl＇s Theorem

Let T be a Banach space operator, E（T） be the set of all isolated eigenvalues of T and π（T） be the set of all poles of T. In this work, we show that Browder＇s theorem for T is equivalent to the localized single-valued extension property at all complex numbers λ in the complement of the Weyl spectrum of T, and we give some characterization of Weyl＇s theorem for operator satisfying E（T） = π（T）. An application is also given.