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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'. 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 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.

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    $\begingroup$ I upvoted because I think this is a fun question though I do not believe that Terence Tao's article proves that your question is a good one (I personally think that it is reasonably good but my set of reasons has empty intersection with this article). $\endgroup$
    – user141886
    Commented Jun 14, 2019 at 14:44

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See, for example, this paper, 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.

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  • $\begingroup$ Thanks. I find it interesting to note that they came up with a very similar definition. $\endgroup$ Commented Jun 14, 2019 at 12:17

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