Let be an irreducible reversible discrete time
Markov chain on a finite state space $\Omega P(X_t^{\mathrm{c}}){t \ge 0} $ whose kernel is given by $H_t:=e^{-t}\sum_k
(tP)^k/k! (X_t^{\mathrm{ave}}){t = 0}^{\infty}$, whose distribution at time is
obtained by replacing by .
A sequence of Markov chains is said to exhibit (total-variation) cutoff if
the convergence to stationarity in total-variation distance is abrupt. Let
be a sequence of irreducible reversible
discrete time Markov chains. In this work we prove that the sequence of
associated continuous-time chains exhibits total-variation cutoff around time
iff the sequence of the associated averaged chains exhibits
total-variation cutoff around time . Moreover, we show that the width of
the cutoff window for the sequence of associated averaged chains is at most
that of the sequence of associated continuous-time chains. In fact, we
establish more precise quantitative relations between the mixing-times of the
continuous-time and the averaged versions of a reversible Markov chain, which
provide an affirmative answer to a problem raised by Aldous and Fill.