Probability and Mathematical Statistics: Vol. 18, Fasc. 1

OPERATORS ON MARTINGALES, $P$ -SUMMING OPERATORS, AND THE CONTRACTION PRINCIPLE

Stefan Geiss

Abstract: For the absolutely $P$ -summing operators $T : X --> Y$ between Banach spaces $X$ and $Y$ we consider martingale inequalities of the type

|||| || sum k || |||| |||| ( N sum )1/2|||| || sup || T dl|| Y || < c|| sup |<dk,ai>|2 || , 1<k<N l=1 L2 i=1,2,... k=1 L2

where $(dk)N < LX1 (_O_,F,P) k=0$ is a martingale difference sequence and $(ai) oo i=1$ is a sequence of normalized functionals on $X,$ and we show that these inequalities are useful in different directions. For example, for a Banach space $X, x1,...,xn (- X,$ independent standard Gaussian variables $g1,...,gn,$ and $1 < r < oo$ we deduce that

|||| sum n [ sum ti ] |||| V~ -|||| ti-1 ti|||| |||| sum n |||| || dk xi||LXr < c r|| 1<suip<n S2( f ||Lr|| gixi||LX1 , i=1 k=ti- 1+1 i=1

where $f = (dk)Nk=0$ is a scalar-valued martingale difference sequence such that $(|dk| )Nk=1$ is predictable, $0 = t0 < t1 < ...< tn = N$ is a sequence of stopping times, and

ti- 1 ti ( sum ti 2)1/2 S2( f ) := |dk| . k=ti- 1+1

2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

Key words and phrases: -