# Asymptotic notation and random variables

I have two random variables $$X$$ and $$Y$$ and I want to bound the value of one in terms of the other (for now, I don't care about the actual distribution of their values).

Suppose that the two variables can have different distributions with values chosen from $$[1, n]$$. But $$X$$ is always upper bounded by $$Y \cdot c\log{n}$$ for some constant $$c$$. Can I write this as $$X = O(Y\log{n})$$ (if I care about the behavior for large $$n$$). I'm not sure what is the convention wrt to random variables and asymptotic notation.

• "$X$ is always upper bounded by $Y⋅\log n$." You write that as $X\le Y\log n$. What is the problem with that? Do you have a different relation between $X$ and $Y$? – John L. Jul 7 '19 at 8:05
• sorry, I forgot to add the constant for the upper bound. There exists a constant $c$ for large $n$ such that the bound holds, so I want to write this in asymptotic notation. – user1246462 Jul 7 '19 at 15:00

On first look, the notation $$X = O(Y\,\log{n})$$ seems intuitively reasonable since $$X$$ is always upper bounded by $$c (Y\,\log n)$$ for some constant $$c$$. ($$c$$ must be positive since $$X,Y\ge1$$).
Let $$(\Omega,S,T)$$ represents a set of 2-dimensional random vectors on $$\Omega$$ parameterized by $$n$$, i.e., for all $$n$$, $$(S_n, T_n)$$ is a random vector from $$\Omega_n$$ (which is a set of possible outcomes) to $${\Bbb R}^2$$. We write
$$S=O(T){\text{ as }}n\to \infty$$ if and only if there is a positive constant $$c$$ such that for all sufficiently large values of $$n$$, for all $$\omega\in\Omega_n$$, the absolute value of $$S_n(w)$$ is at most $$c\,T_n(w)$$.
We can apply the above definition to $$(I, X, Y\,\log n)$$ where $$I_n=[1,n]$$.