### Motivation:


At the present time it really isn't clear to me why this question might be inappropriate for the MathOverflow. However, it appears that some people are down-voting this question even if the notation we use is a key part of doing mathematics. Mathematicians of the contrary opinion may read Terrence Tao's article on notation: ['Use good notation'](https://terrytao.wordpress.com/advice-on-writing-papers/use-good-notation/). I'd like to add that the relationship in question appears to lack a reliable definition. 


### Question:


Let's suppose we have two probability distributions $P$ and $Q$ defined $\forall x \in \mathbb{R}^n$ such that: 


\begin{equation}
\forall x, \Delta x \in \mathbb{R}^n, (P(x+\Delta x)-P(x)) \cdot (Q(x+\Delta x)-Q(x)) \geq 0 \tag{1}
\end{equation}


At present, for lack of a better symbol, I express this relationship as:


\begin{equation}
P \propto Q \tag{2}
\end{equation}


but this symbol is typically used to mean 'proportional to' which isn't exactly what I mean. 


Might there be a specific symbol for a monotone relationship between two probability distributions? I have actually looked through the [list of latex symbols for relation operators](https://oeis.org/wiki/List_of_LaTeX_mathematical_symbols#Relation_operators) and couldn't find a symbol for 'monotone relationship'.


### Addendum:


Equation (1) and therefore the answer provided by @kodlu appears to be inappropriate due to the possibility that $P$ or $Q$ might correspond to the uniform distribution in which case: 


\begin{equation}
\forall x, \Delta x \in \mathbb{R}^n, (P(x+\Delta x)-P(x)) \cdot (Q(x+\Delta x)-Q(x)) =0 \tag{\*}
\end{equation}


In order to address this case, where $P$ and $Q$ aren't necessarily *similarly-ordered* or *monotone*, a more subtle definition is required. I attempt to provide such a definition in my answer below using level sets. 



Symbol for monotone relationship between two probability distributions
See, for example, [this paper](http://www.ams.org/journals/proc/1998-126-05/S0002-9939-98-04702-9/S0002-9939-98-04702-9.pdf), and apply the definition to $X=\mathbb{R}^n$:


**Definition 1**. Two functions $f : X → \mathbb{R},$ and $g : X → \mathbb{R},$ are said to be *similarly
ordered*, in short $f$ s.o. $g,$ if
$$(f(x) − f(y))(g(x) − g(y)) ≥ 0$$ for every $x, y ∈ X.$ A specific symbol does not exist, as far as I am aware.


