Supoose $X_i, i = 1,2,..., T$ are i.i.d. nonnegative random variables. Given any constant $c > 0$, can I prove the following?
$$\mathbb{E}[\sum_{t=1}^T(Tc - \sum_{k=1}^t X_i)^+] - \sum_{t=1}^T(Tc - \sum_{k=1}^t\mathbb{E}[X_i])^+$$
is of $O(\sqrt{T})$, where $(b)^+ = \max\{b, 0\}$.