Limit Theorems of Integrals with Respect to Vector Random Measures in Complete Paranormed Spaces

This paper studies random integral of the form gdM ∫ , where f is a function taking value in a paranormed vector space X, and M is an independent scattered vector random measure. Random integrals of this type are a natural generalization of random series with paranormed space valued coefficients. Some limit theorems of integrals with respect to vector random measures are proved.


Introduction
The idea of random measures first appeared in Bochner's paper [1], and a variety of discussions followed [8,10,12,15,17,18,19,20]. The aim of this article is to study random integral of the form gdM ∫ , where g is a function taking value in a paranormed vector space X, and M is an independent scattered vector random measure. Random integrals of this type are a generalization of random series with Banach space valued coefficients. It is well known that the asymptotic behavior of such series depends also on some geometric properties of the Banach space X (or a metrizable space) [3,4,6,8,16]. On the other hand, the existence of certain bounded linear operators on appropriate function spaces which we call random integrals depends in general on a geometric structure of X [12,18,19].
This paper defines and studies random integrals without any restriction in the geometry of X, devotes to the study of independently scattered vector random measures with emphasis on their convergence properties, presents convergence theorems of random integrals of the forms

Paranormed Spaces
Let K be the field of real numbers or the field of complex numbers, and X be a vector space over K.
And the paranorm on X can be given by For any sequence{ } n x X ⊂ , and x X ∈ , the following are

Random Integrals
Let (T, Σ) be a measurable space, and ( , , ) P Ω Γ be a probability space. A function is said to be an independently scattered random measure if for every pairwise disjoint sets n A ∈ Σ , random variable ( ) n M A are independent, n =1, 2, …, and In what follows we assume that ν is symmetric. From [3], while m is a symmetric σ -finite measure on 1 (1) and (2) hold. In the sequel let X be a paranormed space, ' X be the topological dual space of X.
A function g: T → X is said to be a simple function if there exist pairwise disjoint measurable sets B j ∈Σ and x j ∈X (j= 1, the µ-Integral of the simple function g is defined as i.e., there exists E ∈ Σ and ; and for every then the symmetry and independence assumptions imply that for every seminorm Q q ∈ , every simple function g: T → X, and every Proof. Without loss of generality, we may assume that Therefore, for any simple function where B j ∈Σ pairwise disjoint measurable sets, and for any seminorm q in Q we have where M a is pure atomic, and M c is atomless.
And by the uniqueness of the limit one concludes that

Limit Theorems
This section devotes to the proofs of some limit theorems of integrals with respect to vector random measures.
and denoted by lim k Definition 4.1 A probability measure µ is said to be a control measure of the random vector measure M, and denoted by µ of random vector measures is said to be weakly converges in probability to a random vector measure M, denoted by lim k Combining (5) and (6)

Conclusion Remark
Let (T, Σ) be a measurable space. A countably additive set function on Σ and taking values in 0 ( , ) L P Ω is said to be a random measure. Vector random measures arise naturally as a "Banach space generalization" of real-valued random measures. Let X be a Banach space, 0 ( , ) X L P Ω the set of all X-valued random variables. A countably additive set function on Σ taking values in 0 ( , ) X L P Ω is said to be an X-valued random measure. Vector random measures can be regarded as a "natural random generalization" of vector non-random measures studied by many authors.
This paper extends the definitions of random measure and random integral to complete paranormed spaces, and devotes to the study of limit theorems of random integrals with respect to vector random measures.