We study three quantities that can each be viewed as the time
needed for a finite irreducible
Markov chain to ‘forget’ where it started. One of these is
mixing time, the minimum
mean length of a stopping rule that yields the stationary distribution
from the worst starting
state. A second is the forget time, the minimum mean length
of any stopping rule that yields
the same distribution from any starting state. The third is the
reset time, the minimum
expected time between independent samples from the stationary distribution.
Our main results state that the mixing time of a chain is equal to the
mixing time of the
time-reversed chain, while the forget time of a chain is equal
to the reset time of the reverse
chain. In particular, the forget time and the reset time of a time-reversible
chain are equal.
Moreover, the mixing time lies between absolute constant multiples of the
sum of the forget
time and the reset time.
We also derive an explicit formula for the forget time, in terms
of the 'access times' introduced in . This enables us to relate the forget and
reset times to other mixing measures of the chain.