I have difficulty understanding what is sup$X_n$ and what is limsup $X_n$, for $X_n$ as s sequence of random variables.
Does sup$X_n$ mean for all $\omega$ in the domain of $X_n$, sup $X_n(\omega)$. So sup$X_n$ is a different number for each $X_n$. And $\limsup X_n$ means for each $\omega$, we find sup$X_n(\omega)$ for each n, and set n to infinity.
Then what does limsup$\frac{X_n}{logn}=1$ mean?
Best Answer
No, the supremum is over $n$. For each $\omega \in \Omega$, $$(\sup_{n \in \mathbb{N}} X_n)(\omega) := \sup\{X_n(\omega):n \in \mathbb{N}\}$$ Note that $\sup_n X_n$ is still a random variable!
Similarly, $\limsup X_n$ is a random variable. For each $\omega \in \Omega$, $$(\limsup_{n \to \infty} X_n)(\omega) := \limsup_{n \to \infty} X_n(\omega).$$ That is, for a fixed $\omega$, you have a sequence of real numbers $\{X_1(\omega), X_2(\omega),\ldots\}$, and I assume you know the definition of limit supremum of a sequence of real numbers.
$X_n / \log n$ is again a random variable, so you can apply the reasoning above to talk about $\limsup_{n \to \infty} \frac{X_n}{\log n}$.