The mixing time $t_{mix}$ of a $\beta$ biased random walk on a line segment $\{0,1,\cdots n\}$ is given by $\beta^{-1}n+O(\sqrt{n})$. I am not interested in the proof but an understanding of the statement I was told: "If we know that the average time it takes to reach $n$ from $0$ is $\beta^{-1}n$ (inverse of speed times distance) and if we know that the chain mixes when the walk reaches $n$, then $O(\sqrt{n})$ term follows from the central limit theorem (CLT)". May I know how this follows from the CLT?
I know by CLT, if $X_i$'s are iid with finite variance and $S_n:=\sum_{i=1}^n X_i$ , then $S_n = ES_n + O(\sqrt{n})$ but am not sure how that is analogous to $t_{mix} = \beta^{-1}n + O(\sqrt{n})$? Is $t_{mix}$ a sum of $n$ iid random variables?