However, we do not need positive recurrence of the modified Markov Chain to arrive at the conclusion that

for a positive proportion of indices

, where the tilde refers to the state evolution in the new Markov Chain. Indeed, since the old Markov Chain (before removing transition

) had a strictly positive probability to reach state

from any state

within

steps (

= number of states in the Markov Chain), the same applies to the new Markov Chain (this can be verified easily). It follows that

for a positive proportion of indices

.