# Symbol for monotone relationship between two probability distributions

### 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:

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

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

$$$$P \propto Q \tag{2}$$$$

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'.

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:

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

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.

• 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).
– user141886
Commented Jun 14, 2019 at 14:44

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.