Asymptotic notation and random variables
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 $X,Y\ge1$).
Can we formalize the exact meaning of that notation?
Here is one natural definition.

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 $I_n=[1,n]$.