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Imagine I have some Markov process with stationary distribution $\pi$ and a mixing time of $\tau$ after which $|Prob[x=s_i] - \pi(s_i)| \leq \epsilon$.

Can I assume the following:

A state $(x=s_i)$ will be sampled with probability at least $\pi(s_i)$ in time $\Theta(\tau)$? Note that I'm talking about the average case, and saying nothing about variance.

If this assumption is false, how do we rectify this with the notion that we converge on $\pi$ after some appropriate amount of mixing time $\tau$?

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Intuitively not: in time $\Theta(g(\tau))$ the probability the particle is in state $i$ may be less than $\pi_i$ –  Alex Jan 9 '13 at 23:07
@Alex But if the stationary distribution is what is approached after sufficient mixing, shouldn't we treat it as a probability distribution for where our walker will be after mixing? –  PlacidLake Jan 9 '13 at 23:11
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