An extremal property of independent random variables

Abstract:

In a previous paper the first author proved $Ef(\smallint _0^t {e db) \leqq Ef(M{b_t})}$, where e is a Brownian functional $\leqq M$ in absolute value and f is a convex function such that the right side is finite. We now prove a discrete analog of this inequality in which the integral is replaced by a martingale transform: $Ef(\sum \nolimits _1^n {{d_k}{y_k}) \leqq Ef(M\sum \nolimits _1^n {{y_k})} }$. (The ${y_j}$’s are independent variables with mean zero, $j \to {d_1}{y_1} + \cdots + {d_j}{y_j}$ is a martingale, and $0 \leqq {d_j} \leqq M$.) We further show that these inequalities are false if t or n is a stopping time, or if ${d_j} \ngtr 0$.