Centering Sequences with Bounded Differences

Inequalities for martingales with bounded differences have recently proved to be very useful in combinatorics and in the mathematics of operational research and computer science. We see here that these inequalities extend in a natural way to ’centering sequences’ with bounded differences, and thus include for example better inequalities for sequences related to sampling without replacement. 1 Centering sequences Given a sequence X = (X1, X2, . . .) of (integrable) random variables the corresponding difference sequence is Y = (Y1, Y2, . . .) where Yk = Xk −Xk−1 (and where we always set X0 ≡ 0). Let μk(x) = E(Yk|Xk−1 = x), that is μk(Xk−1) is a version of E(Yk|Xk−1). We call the sequence X centering if for each k = 2, 3, . . . we may take μk(x) to be a non-increasing function of x. If we interpret Xk as your capital at time k, then this corresponds to the worthy maxim ”the less you have the more you get on average”. Observe that a martingale is trivially centering, since we may take each μk(x) ≡ 0. Also various time series (economic or other) that are controlled towards some target should be centering. Inequalities for martingales with bounded differences due to Hoeffding [11] and others and presented in [5, 18] have proved to be very useful in combinatorics and in the mathematics of operational research and theoretical