Does anyone know of any references for how to evaluate stochastic inequalities? Surprisingly, I can't find any good references for general problems.
For example, suppose we have three random variables, $X,$ $Y$, $Z$. What can we learn about inequalities like $P(X>Y>Z)$ supposing the variables are i) identically distributed but not independent, ii) independent, or iii) simply $iid$?
A first step I thought of was to write expressions like
\begin{eqnarray*} & & \int_{x}\int_{y}\int_{z}1\{X-Y>0,Y-Z>0\}f_{X,Y,Z}(x,y,z)dy dx dz \end{eqnarray*}
If $X,Y,Z$ are independent then we can write expressions like:
\begin{eqnarray*} & & \int_{x}\int_{y}\int_{z}1\{X-Y>0,Y-Z>0\}f_{X}(x)f_{Y}(y)f_{Z}(z)dydxdz \end{eqnarray*}
But I don't know what this gives me. Would a change of variables be of use? Thanks. I hope my question isn't too vague.
Edit: Basically, I want to argue statements like $P(X>Y)$$=P(Y>X)$ if $F_{X,Y}(x,y)$ is symmetric in its arguments.
The distribution of $(X,Y)$ is symmetric when the distributions of $(X,Y)$ and $(Y,X)$ coincide. Expectations depend only on distributions hence this implies $E(H(X,Y))=E(H(Y,X))$ for every measurable function $H$. If $H:(x,y)\mapsto\mathbf 1_{x\lt y}$, one gets $P(X\lt Y)=P(Y\lt X)$.
Likewise, if the distribution of $(X,Y,Z)$ is symmetric, $P(X\gt Y\gt Z)=P(Y\gt Z\gt X)$ which is also equal to 4 other probabilities of similar events.
When there is no tie, that is, when the event $[X=Y]\cup[Y=Z]\cup[Z=X]$ has probability zero, $[X\gt Y\gt Z]$ and the 5 similar other events yield a partition of the probability space hence these considerations yield $P(X\gt Y\gt Z)=\frac16$. Otherwise, one can only be sure that $P(X\gt Y\gt Z)\leqslant\frac16$.
The setting where this applies includes $(X,Y,Z)$ being i.i.d. with no atom but also less restrictive situations such that $(X,Y,Z)$ being exchangeable.