A new hybrid Ishikawa iteration scheme is presents and the strong convergence theorems of the fixed points for nonexpansive mappings in Hilbert spaces is proved.

A strong convergence theorem for a family of nonexpansive non-self mappings;

The convergence theorem for(H) integral of vector valued function on infinite interval;

LSRS Convergence Theorem of Mcshane Integral;

Convergence Theorems of the Choquet Integral;

The characteristics and convergent theorems of F valued (K) integral;

In this paper,some properties of BSDEs on infinite horizon and g-expectations are studied,and then the Levi、Fatou and Lebesgue convergent theorems of the generalized g-expectations are verified.

Osilike has proved the convergence theorems and some results of the implicit iteration process for common fixed points of a finite family of strictly pseudocontractive mapings which improved the results of Xu and Ori,about the nonexpansive mappings.

Levi,Fatou and Lebesgue convergence theorems for g-expectation are put forward and proved by constrasting with classical mathematical expectation.

In this paper,we introduce the martingale convergence theorem and martingale hyperconvergence theorem for analyzing performances of identification methods and states their application ranges.

Martingale method is employed to investigate a local convergence theorem for countable Mar-kov chains indexed by a generalized Bethe tree by using a martingale and Doob martingale convergence theorem and some special inequalities.

Under the condition that the moment of Markov chains in bi-infinite environments satisfies some conditions,this paper proves a class of limit theorems for functionals of Markov chains in bi-infinite environments by stopping time and martingale convergence theorem.