MRL order, log-concavity and an application to peacocks

We provide an equivalent log-concavity condition to the mean residual life (MRL) ordering for real-valued processes. This result, combined with classical properties of total positivity of order 2, allows to exhibit new families of integrable processes which increase in the MRL order (MRL processes). Note that MRL processes with constant mean are peacocks to which the Az'ema-Yor (Skorokhod embedding) algorithm yields an explicit associated martingale.