Ratio prophet inequalities for convex functions of partial sums

Abstract

Let x,, x,,... be independent mean zero random variables and set S, =X, + . * * +X,, and S,* = maxi ~ k ~ ,,S: . We are interested in comparing Sz and S,* for large values of their ranges. Since we cannot compare their tail probabilities directly, we compare ENS,*) with E@(Sz), where @ is any non-decreasing non negative convex function on [O, m). For the convex functions Q,(x) = I x I ‘, square function inequalities are known to become proportionately worse and worse as P + 00. (See Hitczenko (1990) for a summary of what is known.) It may therefore seem surprising that for all @, n and Xi,. . . , X,,, E@(S,*) , Y) (3) for all y sufficiently large (perhaps depending on IZ and Xi, . . . , X,). This inequality is trivial if the variates are symmetric. But even in the i.i.d. case it is essentially false, as shown by B. Davis (for n = 2). To see this, let X, and X, be i.i.d. mean zero random variables with P(X, 'Y) = & and P(X,= -:)=A.