Asymptotic notation and random variables

Question

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.