I have a random variable $X_n$ that arises from the following random walk on $\mathbb{Z}^n$: let $X_0 = 0$ and $\mathbf{P}(X_{n+1} = X_n + 1 \mid X_n) = p$ and $\mathbf{P}(X_{n+1} = X_n - 1 \mid X_n) = 1-p$.
Suppose I fix $p = \frac{2}{3}$. I am asked to find the limiting distribution of $\frac{X_n}{n}$.
We can remove casework and just write $X_n = X_{n-1} + Z_n$ where $Z_n$ are i.i.d. Bernoulli R.V taking values $\pm 1$ with probability $\frac{2}{3}, \frac{1}{3}$, respectively. Using this recursive formulation, I re-wrote $$X_n = \sum_{i=1}^n Z_n$$ in which case $$ \frac{X_n}{n} = \overline{Z_n}. $$
The weak law of large numbers says that $$ \overline{Z_n} \to \mu $$ in probability. By this, I think I can conclude that $\frac{X_n}{n} \to \frac{1}{3} = \frac{2}{3} \cdot 1 + \frac{1}{3} \cdot -1.$
However, I wanted to verify that this work was correct and ask if there's a stronger claim on the rate of convergence I can make.