On Stochastic Orders and Total Positivity

The usual stochastic order and the likelihood ratio order between probability distributions on the real line are reviewed in full generality. In addition, for the distribution of a random pair $(X,Y)$, it is shown that the conditional distributions of $Y$, given $X = x$, are increasing in $x$ with respect to the likelihood ratio order if and only if the joint distribution of $(X,Y)$ is totally positive of order two (TP2) in a certain sense. It is also shown that these three types of constraints are stable under weak convergence, and that weak convergence of TP2 distributions implies convergence of the conditional distributions just mentioned.