Let $X=(X_n)_{n\in \mathbb{N}}$ be a supermartingale bounded in $\mathcal{L}^1$. By Doob's Forward convergence theorem, $\lim_{n\to\infty}X_n = X_\infty$ exist and is finite a.s.
Now, if $Z_n=n^{-1}S_n - X_n$, where $S_n= \sum_{i=0}^n X_i$, we have by the Law of Large numbers that $n^{-1}S_n \to \mu$. Also, intuitively $\mu \to X_\infty$ and therefore $Z_n \to 0$.
I would like to say something like: Consider $\mathbb{E}[Zn]$ and using an argument CLT-like estimate $\mathbb{E}[Z_n] \leq \frac{1}{\sqrt{n}}$ (Since that is the convergence rate of the CTL). Is this possible? or what kind of additional assumptions should I consider?