ABSTRACT
Let l?,, be the conditional distribution of e,, given G. In view of the right continuity of distribution functions, it follows from (7.6.5) that with probability 1, 1-Fn(2) 5 l-G,(x/b) for all n > 1 and rc > 11. Therefore, letting P 1 ( u ) = sup{z : F ( z ) 5 U , ) for 0 5 U < 1 and 0 = bC;l(rl), we have F l l ( u ) < bG;'(u) + Q for all n > 1 and 0 < u < 1. Replacing u in the preceding inequality by i.i.d. uniform random variables U l , U2, . . . on [0, 1) and noting that conditional on G (with respect to which T is rneasurable) the e, are independent random variables with distribution functions F,, we obtain that
Noting that 7' 5 T'/" and that E Y ~ " < m since n = q/2 > p/2, the desired conclusion follows from (7.6.4), (7.6.7) and Burkholder's inequality
Theory and Applications of' Detorrplii~g 137
de la Pefia and Lai (199710) proved the following dccoupling inequality for randomly stopped de-nornialiaed U-statistics in the case p > 2:
Moreover, for the case l < p < 2, assuming that f ( ~ , ( ~ ) , . . . , x , ( ~ ) ) = f (X,, . . . , x k ) for any permutation of ( l , . . . . k), dc la Pefia and Lai (1997b) also proved the decoupling inequality
Another direction of extending inequality (7.6.2) for total decoupling of a stopping time deals with continuous-time processes with independent increments. In particular, the following extension of Theorems 7.6.1
138 Victor H. de 1a Pcria and Tze Leurrg Lai
and 7.6.2 is given by de la P&L imd Eisenbiturri (1997) [see also de la Pena and Girld (l999)j.
