In probability theory, the mixing time of a Markov chain is the time until the Markov chain is "close" to its steady state distribution. Lets fix some $\varepsilon\in (0,1]$ as our closeness parameter.
Now suppose that we have a Markov Chain M with mixing time T, and suppose that we want to draw 10 independent samples from the stationary distribution of M. What is the correct way to do that?
- Do we have to apply the Markov chain for $10\cdot T$ steps, and pick the resulting states at time $T, 2T, ...,10T$ as my samples, or...
- is it enough to apply the Markov chain for T+10 steps, and pick the resulting states at time $T+1,T+2,...,T+10$ as my samples?
The justification for the first alternative is that once we have drawn a sample state $s$ at time $T$, then the immediate samples following $s$ will be correlated with $s$ according to the transition matrix of the chain.
On the other hand, maybe it is possible to use some property about Markov chains to show that alternative 2 is enough? For instance, by arguing that the distribution at time T+i is close to uniform with respect to the initial state anyways. This leads to question 3 below...
3) Is there some class of Markov chains for which alternative 2 is enough?