We study the class of Dunford-Pettis sets in Banach lattices. In particular, we establish some sufficient conditions for which a Dunford-Pettis set is relatively weakly compact (resp. relatively compact).

1 Last two lectures ¯ probability spaces ¯ probability measure ¯ random variables and stochastic processes ¯ distribution functions ¯ independence ¯ conditional probability ¯ memoriless property of geometric and exponential distributions ¯ expectation ¯ conditional expectation (double expectation) ¯ mean-square estimation 1

In this paper, we introduce the Choquet-Pettis integral of set-valued mappings and investigate some properties and convergence theorems for the set-valued Choquet-Pettis integrals.

Because of a lack of a priori information, the minimum mean-squared error predictor, the conditional expectation, is often not known for a non-Gaussian time series. We show that the nonparametric kernel regression estimator of the conditional expectation is mean-squared consistent for a time series: When used as a predictor, the estimator asymptotically matches the mean-squared error produced b...

1 Conditional expectation 3 1.1 Discrete case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Product measure and Fubini’s theorem . . . . . . . . . . . . . . . . . . . . . 11 1.4 Examples of conditional expectation . . . . . . . . . . . . . . . . . . . . . . . 11 1.4.1 Gaussian ...

We investigate the sufficient condition under which each positive b-weakly compact operator is Dunford-Pettis. We also investigate the necessary condition on which each positive b-weakly compact operator is Dunford-Pettis. Necessary condition on which each positive b-weakly compact operator is weakly compact is also considered. We give the operator that is semi-compact, but it is not bweakly. W...

Two conditional expectations in unbounded operator algebras (O∗-algebras) are discussed. One is a vector conditional expectation defined by a linear map of an O∗-algebra into the Hilbert space on which the O∗-algebra acts. This has the usual properties of conditional expectations. This was defined by Gudder and Hudson. Another is an unbounded conditional expectation which is a positive linear m...