ABSTRACT
ABSTRACT In this piaper we provide a survey of the theory of tlecoupling, emphasizing its wide rmge of applicatioris. Decouplirig was horn out of n need to exter~ci the kriow~i r~~;trtingale inequalities for real and Hilbert-space valued ra~ldorn variables to variables taking values in rrrore general spaces like U;tii;i.ch spaces. Among its many ilses is that it ci1;tbles one to sy~rmetrize highly depe~ltlent raiidom variables. In fact, sorne of the basic rcslllts in decouplirig theory were rriotiviatcd by s,yrrlrnetriaatiorl tecliiiiqucs for U-p~.ocesscs which corlsist of U-statistics irltlexeti by ;a class of fimctio~is. Other i~pplications include sequentii~l a.n:ilysis, rliartiilgale theory, stochastic intcgrittion ant1 weak convergence. We consider three typcs of tlecoupling. The first type, which we call cornplctc tiecoupling, completely replaces the sumnlailtls in a sum of tlepenclerit rantlorrl varia'r~lcs by indeperlderit ones with the s;trne rn;trgin;al tlistributio~ls. The second type of dcc:oupling is deconplii~g of tangent sequences. In this approiacli, two sequeuces adapted to the same filtri~tion and having the same conditioiial distrilmtioi~s (givcrr t,he past) iarc corr~piircd. Tlie cfrectiveilcss of this approach is realized tlirougli the use of a sequence with a more tra.ctahle dependence structure tliarl the original one. The third type of dccoupling is called "total dccoupling of stopping times". In this approach, problems iilvolving st,oppirlg times are harrdled by esthblislling iriequalities that replace the origiilal process Ijy i ~ i r iiideperder~t copy which is therefore irltlcpendent of the stopping time.
