Tree Martingale Theory And Their Applications

The thesis studies the tree martingale theory and their applications,which hasa broadly applied background.The main results of this study are included in thefollowing contents:Firstly,a graph-theoretic decomposition of an index set for tree martingales isconstructed,and this decomposition exhibits the structures on the index set of treemartingales,based on which it is proved that the tree martingale index set is isomor-phic to a directed forest and that in this directed forest,each tree is a directed locallyfinite one.Therefore,the tree martingale index set has a decomposition,each part ofwhich is isomorphic to a directed infinite and locally finite tree.On this basis,thelocally finite tree set is defined,and the locally finite tree martingales are defined interms of the locally finite tree set,and then a tree martingale decomposition theoremis obtained by the locally finite tree martingales.Moreover,in the light of constructingan one-to-one mapping which preserves ordering,it is shown that each locally fmitetree martingale is isomorphic to a Cairoli-Walsh martingale,and in virtue of which,the convergence of locally finite tree martingales is shown by the use of Cairoli-Walshmartingale theory.Based on the tree martingale decomposition theorem and the con-vergence of locally finite tree martingales,a tree martingale convergence theorem isverified.Secondly,based on tree martingale decomposition theory,a kind of simplemethod of verifying tree martingale inequality is developed.With the help of mildconditions,Burkholder-Davis-Gundy's inequality and Doob's inequality of tree mar-tingales are proved by the application of this method.In contrast with F.Schipp and F.Weisz's method,this method is not restricted by either previsiblity or regularity,andthe procession of the proof of these inequalities is simpler;moreover,the accuracycoefficients of the corresponding inequality can be obtained through this method.Thirdly,quasi-normal inequalities of a-variation maximal operator and a-conditional variation maximal operator of scalar predictable tree martingales areidentified by the use of martingale transforms and by the construction of convex or concave function method;on this basis and with the help of previsiblity or regu-larity,Burkholder-Davis-Gundy's inequality of a-variation maximal operator and a-conditional variation maximal operator of scalar predictable tree martingales are iden-tified by the application of Hardy-Lorentz interpolation theory.At the same time,bythe use of G.Pisier's results,under tree martingale valued space X is isomorphic toan a-uniformly convex space (2≤a＜∞),some quasi-normal or normal inequalitiesof a-variation maximal operator and a-conditional variation maximal operator of X-valued predictable tree martingales are identified.In contrast with scalar case,thereis a close connection between 2≤a＜∞and whether these quasi-normal or normalinequalities of X-valued tree martingales hold or not?Fourthly,in the UMD spaces,the tree martingale transforms and their maximaloperators are defined,and then based on the geometrical property of UMD spaces andcombining new interpolation techniques of Hardy-Lorentz spaces,the inequality ofUMD space valued tree martingale transform maximal operators is obtained.Finally,the inequality of maximal operators of tree martingale transformsis applied for obtaining the UMD space valued Vilenkin system and the sum ofVilenkin-Fourier series of UMD space valued Vilenkin system is convergent a.e.or convergent in LP(X)-norm.It is an important application of vector-valued treemartingales in Fourier analysis,an expansion of well-known Carleson's theorem.Vilenkin system,in particular,an example of Vilenkin system-Walsh system hasbeen applied broadly in information science,such as image processing and signalprocessing.Next,a directed stochastic network with weights based on tree martin-gales is introduced,and the convergence of this directed stochastic network withweights is obtained by using tree martingale theory;a detail simple example of thedirected stochastic network with weights is given out,and this example can simulatea kind of intelligence system.At the same time,a stochastic complex dynamicalnetwork model is also introduced,the existence and uniqueness of solution of thisstochastic complex dynamical network is identified;furthermore,this stochasticcomplex dynamical network's synchronization in probability is investigated and twosynchronization theorems of this stochastic complex dynamical network are givenout.numerical simulations verify that the two synchronization theorems are effective.