Let $S$ be a Markov Chain on the non-negative integers such that $S_0 = s_0 \in \mathbb{Z}$ and $S_{k+1} = S_k +1$ with probability $p$ and $S_{k+1} = S_k -1$ with probability $1-p$.
For $n \in \mathbb{N}$, define $$\bar{S}_n(t) = \frac{1}{\sqrt{n}}S_{\lfloor nt \rfloor}$$
Fix $t \geq 0$. I want to determine $\sup_n |\bar{S}_n(t)|$. I am guessing an additional condition is needed for this supremum to be finite.
So far, I have
\begin{align*} |\bar{S}_n(t)| &= \left| \frac{S_{\lfloor nt \rfloor}}{\sqrt{n}} \right|\\ &\leq \frac{|s_0| + \lfloor nt\rfloor}{\sqrt{n}}\\ &\leq |\bar{S}_n(0)|+ \frac{nt}{\sqrt{n}} = |\bar{S}_n(0)| + \sqrt{n}t \end{align*}
Can we improve this upper bound that would be independent of $n$? Or what additional condition can we have?