Stochastic Processes

This chapter begins with some fundamental ideas concerning random sequences, and related convergence concepts. It discusses the underlying probability model, and develops the idea of infinite dimensional Euclidean space and the associated Borel field, leading on to the Kolmogorov consistency theorem. The chapter concludes with consideration of uniform and limiting properties, including uniform boundedness and uniform integrability.

Lacunary distributional convergence in topological spaces

Most of the summability methods cannot be defined in an arbitrary Hausdorff topological space unless one introduces a linear or a group structure. In the present paper, using distribution functions over the Borel ?-field of the topology and lacunary sequences we define a new type of convergencemethod in an arbitrary Hausdorff topological space and we study some inclusion theorems with respect to the resulting summability method. We also investigate the inclusion relation between lacunary sequence and lacunary refinement of it.

CHARACTERIZATIONS OF THE BOREL -FIELDS OF THE FUZZY NUMBER SPACE

In this paper, we characterize Borel $\unicode[STIX]{x1D70E}$-fields of the set of all fuzzy numbers endowed with different metrics. The main result is that the Borel $\unicode[STIX]{x1D70E}$-fields with respect to all known separable metrics are identical. This Borel field is the Borel $\unicode[STIX]{x1D70E}$-field making all level cut functions of fuzzy mappings from any measurable space to the fuzzy number space measurable with respect to the Hausdorff metric on the cut sets. The relation between the Borel $\unicode[STIX]{x1D70E}$-field with respect to the supremum metric $d_{\infty }$ is also demonstrated. We prove that the Borel field is induced by a separable and complete metric. A global characterization of measurability of fuzzy-valued functions is given via the main result. Applications to fuzzy-valued integrals are given, and an approximation method is presented for integrals of fuzzy-valued functions. Finally, an example is given to illustrate the applications of these results in economics. This example shows that the results in this paper are basic to the theory of fuzzy-valued functions, such as the fuzzy version of Lebesgue-like integrals of fuzzy-valued functions, and are useful in applied fields.

Actions of amenable equivalence relations on CAT(0) fields

We present the general notion of Borel fields of metric spaces and show some properties of such fields. Then we make the study specific to the Borel fields of proper CAT(0) spaces and we show that the standard tools we need behave in a Borel way. We also introduce the notion of the action of an equivalence relation on Borel fields of metric spaces and we obtain a rigidity result for the action of an amenable equivalence relation on a Borel field of proper finite dimensional CAT(0) spaces. This main theorem is inspired by the result obtained by Adams and Ballmann regarding the action of an amenable group on a proper CAT(0) space.

A characterization and moving average representation for stable harmonizable processes

In this paper we provide a characterization for symmetric α-stable harmonizable processes for 1<α≤2. We also deal with the problem of obtaining a moving average representation for stable harmonizable processes discussed by Cambanis and Soltani [3], Makegan and Mandrekar [9], and Cambanis and Houdre [2]. More precisely, we prove that if Z is an independently scattered countable additive set function on the Borel field with values in a Banach space of jointly symmetric α-stable random variables, 1<α≤2, then there is a function k∈L2(λ) (λ is the Lebesgue measure) and a certain symmetric-α-stable random measure Y for which ∫−∞∞eitxdZ(x)=∫−∞∞k(t−s)dY(s),t∈R, if and only if Z(A)=0 whenever λ(A)=0. Our method is to view SαS processes with parameter space R as SαS processes whose parameter spaces are certain Lβ spaces.

A Class of Singular Functions

Kakutani (2) has proved a very general theorem, giving necessary and sufficient conditions for two infinite product measures to be mutually absolutely continuous. To formulate Kakutani's result, let us first recall that a measurable space is a pair (E, B), where B denotes a Borel field (also called σ-ring) of subsets of E, and a measure m on this space is a countably additive set function on B (see Halmos (1)).

On the Hausdorff measure of Brownian paths in the plane

Ω will denote the space of all plane paths ω, so that ω is a short way of denoting the curve . we assume that there is a probability measure μ defined on a Borel field of (measurable) subsets of Ω, so that the system (Ω, , μ) forms a mathematical model for Brownian paths in the plane. [For details of the definition of μ, see for example (9).] let L(a, b; μ) be the plane set of points z(t, ω) for a≤t≤b. Then with probability 1, L(a, b; μ) is a continuous curve in the plane. The object of the present note is to consider the measure of this point set L(a, b; ω).

The Equivalence of Two Extremum Problems

Let f1 , …, fk be linearly independent real functions on a space X, such that the range R of (f1, …, fk) is a compact set in k dimensional Euclidean space. (This will happen, for example, if the fi are continuous and X is a compact topological space.) Let S be any Borel field of subsets of X which includes X and all sets which consist of a finite number of points, and let C = {ε} be any class of probability measures on S which includes all probability measures with finite support (that is, which assign probability one to a set consisting of a finite number of points), and which are such thatis defined. In all that follows we consider only probability measures ε which are in C.