tailieunhanh - Báo cáo "Series representation of random mappings and their extension "

In this paper, we introduce a method of extending the domain of a random mapping admitting the series expansion. This method is based on the convergence of certain random series. Some conditions under which a random mapping can be extended to apply to all $X$ - valued random variables will be presented. AMS Subject classification 2000: Primary $60H05$; Secondary: $60B11$, $60G57$, $60K37$, $37L55$. | VNU Journal of Science Mathematics - Physics 25 2009 237-248 Series representation of random mappings and their extension Dang Hung Thang Tran Manh Cuong Department of Mathematics Vietnam National University 334 NguyenTrai Str Hanoi Vietnam Received 28 February 2009 Abstract. In this paper we introduce a method of extending the domain of a random mapping admitting the series expansion. This method is based on the convergence of certain random series. Some conditions under which a random mapping can be extended to apply to all X -valued random variables will be presented. AMS Subject classification 2000 Primary 60LĨ05 Secondary 60B11 60G57 607C37 37L55. Keywords and phrases random operator bounded random operator domain of extension action on random inputs. 1. Series representation of random mappings Let X Y be separable metric spaces. By a random mapping from X into Y we mean a rule 4 that assigns to each element X G X a unique Y - valued random variable . Equivalently it is a mapping I Q X X Y such that for each fixed X G X the map . x Q - Y is measurable. In this point of view two mappings l I íỉ X Y y. 2 Q X X Y define the same random mapping if for each X G X i a cu 2 a cu . Noting that the exceptional set can depend on X. In this case we say that the random mapping f 2 is a modification of the random mapping l . Definition A random mapping l from X into Y is said to admit the series expansion if there exists a sequence of deterministic measurable mappings from X into Y rep. from X into ll and a sequence an of real-valued random variables rep. F-valued . s such that T 2 a-nfnx n l where the series converges in Lq . In the case the sequence an are independent we say that l admits an independent series expansion. Corresponding authors. Tel. E-mail 237 238 . Thang . Cuong VNU Journal of Science Mathematics - Physics 25 2009 237-248 Proposition Let be a random operator from X into Y and suppose that X is