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.

翻译：全面回顾了实数线上概率分布之间的通常随机序和似然比序。此外，对于随机对$(X,Y)$的分布，证明了当且仅当$(X,Y)$的联合分布在某种意义上是二阶全正（TP2）时，给定$X=x$下$Y$的条件分布随$x$关于似然比序单调递增。还表明这三类约束在弱收敛下是稳定的，并且TP2分布的弱收敛蕴含了上述条件分布的收敛。