  In this note, we are concerned with the subgeometric rate of convergence of a
Markov chain with discrete time parameter to its invariant measure in the
$f$-norm. We clarify how three typical subgeometric rates of convergence are
inherited under a discrete time version of Bochner's subordination. The crucial
point is to establish the corresponding moment estimates for discrete time
subordinators under some reasonable conditions on the underlying Bernstein
function.
