  In this article, we consider products of ergodic Markov chains and discuss
their cutoffs in the total variation. Through a new inequality relating the
total variation and the Hellinger distance, we may identify the total variation
cutoffs with cutoffs in the Hellinger distance. This provides a new scheme to
study the total variation mixing of Markov chains, in particular, product
chains. In the theoretical framework, a series of criteria are introduced to
examine cutoffs and a comparison of mixing between the product chain and its
coordinate chains is made in detail. For illustration, we consider products of
two-state chains, cycles and other typical examples.
