ABSTRACT
Let {d,. .F,, I > 1) be a mean-zero inastingale difference sequence. Let {c,} be C1 (conditionnlly independent) given G m d such that {cl,) ancl
Theory ancl Applications of Decoupling l29
for some universal constant C. Since the e, are independent zero-mean ra~idorri variables given G, it then follows from Rosenthal's inequality that
Hence
where S: = lS(d:l.E-l) -- E(e:/F,-1) = E;'E(~:~G) arid e; = supksn ek 1 . Moreover, lie;? i p 5 2 " ~ lid, l p by (7.3.2). Hcnce. dccoupling inequalities for tangent sequences and Rosenthal's inequality for sums of independent zeso-mean random variables yield the martingalc inequality
which was derived by Burkholder (1973) using distribution furiction inequalities. Hitczenko (1990) used this approach to derive the "best possible" constant B,, tleperiding only on p, with the slowest growth rate p/ logp as p -1 CO. In the case p 2 2, Hitczenko (1994) also derived by a similar dccoupling argument the following variant of (7.4.2):
thereof (such as Theorem 7.3.4 and Corollary 7.3.5) are particularly useful for developing extensions of typical exponential inequalities for sums
of zeio-mean independent random variables to the case of the ratio of a mar tingale over its conditional variance. Let {d, , F,, z > l} be a mal tingale difference sequence. Let Mn = d, and V: - C:" ,E ( d $ IF7 ). Consider the problern of hounding the tail probability of P(M,/y/;B 2 X). Following the standard approach to such prok)lerns, one obtains
d,) 2 e x p ( k V,;)
< exp ( ( z a r c iirlii E) . 2~ 2
In this section we let XI , X 2 , . . . be independent random variables talcing values in sorne measurable space S. Motivated in part by the tlesire to extend the theory of multiple stochastic integration, the work of
McCorinell arid T;~qqu (1986) provided dccoupling iiiequalitics for rrlultiliriear forms of indepcndent symmetric varial~les. After their paper, a large amount of work has bcen tioric, providing extensions of this result in rrlany directions, includirlg Kwapien (1987), Bourgain and Tzafriri (1987), tic Acosta (1987), Krakowiak and Szulga (1988), Kwapieri arid Woyczynski (1992): de la Pelia (1992), de la Peria, Morltgornery-Smith, and Szulga. (1994), de 1a Peiia and Mor~tgomery-Smith (1994, 1995), and Szulga (1998). In particular, de la Pelia (1992) provided an exterisiori to a general class of statistics that include both U-~t~atistics and midtilinear forms, while de la Peiia, and Moritgorr1er.y-Smitki (1993, 1995) established the following tail probability conlpa.rison for s l ~ h statistics.
