Set-valued stationary processes
type: Item
instance of: scholarly article
title: Set-valued stationary processes
zbMATH Open document ID: 0898.60017
author: Zhenpeng Wang
author: Rongming Wang
published in: Journal of Multivariate Analysis
publication date: 1998-11-01T00:00:00Z
review text: This paper deals with set-valued processes \(F=\{F_t: F_t\) is a random set \( \forall t\in \mathbb{R}_+\}\) in a real separable Banach space \((X,\| \cdot\|)\) with the dual space \(X^*\). The organization of the paper is as follows. In Section 1 the authors present many of the new results (expectation, multivalued conditional expectation), ideas and developments in the theory of random sets. The definition of the set-valued stationary processes (SVSP) is proposed: a set-valued process \(F= \{F_t:t \geq 0\}\) is a stationary process if \[ P\{\omega: F_{t_k} (\omega)\in U_k,\;k=1, \dots,i\} =P\{\omega: F_{t_k+t} (\omega)\in U_k,\;k=1, \dots,i\} \] for any \(i\geq 1\), \((t_1, \dots,t_i) \in\mathbb{R}_+\), \(t\in \mathbb{R}_+\). The main result of this section characterizes the relationship between an SVSP and a semigroup \(\{T_t: t\geq 0\}\) of measure-preserving set transformations. In Section 2 the following selection and representation theorem of SVSP is proved: Suppose that \(F=\{F_t: t\geq 0\}\) is a set-valued stochastic process. Then the following statements are equivalent: (1) \(F\) is an SVSP. (2) There exists \(\{(f_t^{(k)}, t \geq 0),\;k\geq 1\} \subset\text{SS}(F)\), such that (2.1) for each \(t\geq 0\), \(F_t= \text{cl} \{f_t^{(k)}: k\geq 1\}\), (2.2) for each \(n\geq 1\), \(\{(f_t^{(1)}, \dots, f_t^{(n)})\), \(t\geq 0\}\) is an \(X^n\) valued stationary stochastic process, where SS\((F)\) is the set of all stationary selections of \(F\) and \(\text{cl} \{\cdot\}\) denotes the norm-closure of \(\{\cdot\}\). The laws of large numbers and ergodicities of set-valued stationary processes are studied in Section 3. Denote \(\overline {\text{co}}A\) is the closed convex hull of \(A\), \(P_f(X)= \{A\subset X:A\) is nonempty and closed\}, \(\sigma (I_*)\) is a \(\sigma\)-field generated by \(I_*\) \((I_*\) is a subbase for a topology on \(P_f(X))\), \(P_{\text{wkc}}(X)= \{A\in P_f(X) :A\) is (bounded) convex and weakly compact\}, \(( \Omega, {\mathcal F}, P)\) i…
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 52A22
zbMATH DE Number: 1085314
zbMATH Keywords: set-valued stationary processes
zbMATH Keywords: selection and representation
zbMATH Keywords: ergodicity
zbMATH Keywords: multivalued conditional expectation
MaRDI profile type: Publication
full work available at URL: jmva.1997.1702
OpenAlex ID: W2091202404
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DOI: 10.1006/JMVA.1997.1702
description: scientific article; zbMATH DE number 1085314