ON A DOMINATION OF SUMS OF RANDOM-VARIABLES BY SUMS OF CONDITIONALLY INDEPENDENT ONES

It is known that if (X(n)) and (Y(n)) are two (F(n))-adapted sequences of random variables such that for each k greater-than-or-equal-to 1 t he conditional distributions of X(k) and Y(k), given F(k-1), coincide a.s., then the following is true: \\SIGMAX(k)\\p less-than-or-equal-to B(p)\\SIGMA Y(k)\\p, 1 less-than-or-equal-to p < infinity, for some c onstant B(p) depending only on p. The aim of this paper is to show tha t if a sequence (Y(n)) is conditionally independent, then the constant B(p) may actually be chosen to be independent of p. This significantl y improves all hitherto known estimates on B(p) and extends an earlier result of Mass on randomly stopped sums of independent random variabl es as well as our recent result dealing with martingale transforms of Rademacher sequences.