Set-valued stationary processes
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\}, \(( \Omega, {\mathcal F}, P)\) is a complete probability space, \(F_0\in M[\Omega;X]\) \((M[\Omega;X]\) is the collection of all \({\mathcal F}\)-measurable random sets). Then the main result of this section is the following theorem: Suppose that \(F= \{F_t: t\geq 0\}\) is an SVSP, \(F(\cdot, \cdot): (\Omega \times \mathbb{R}_+, {\mathcal F} \times {\mathcal B} (\mathbb{R}_+)) \to(P_f(X), \sigma(I_*))\) is measurable w.r.t. \({\mathcal F} \times{\mathcal B} (\mathbb{R}_+)\); \({\mathcal U}\) is the invariant \(\sigma\)-field of \(\{F_t: t\geq 0\}\). If one of the following conditions holds: (i) \(X^*\) is separable, (ii) for almost all \(\omega \in \Omega\), for any \(t\geq 0\), \(F_t (\omega)\in G(\omega)\), \(G(\omega) \in P_{\text{wkc}}(X)\), then we have \[ {1\over T} \int^T_0 F_t(\omega) dt@>w>> \overline {\text{co}} E(F_0| {\mathcal U}) \quad \text{a.s.};\;T\to \infty. \]
scientific article; zbMATH DE number 1085314
scientific article