Stationary processes having the blowing-up property

Let X={X_i} be a stationary process, X_i∈C, C finite. q denotes the distribution of X. X, or q, has the blowing-up property if for any ε>0 there are δ>0 and n₀ such that for n⩾n₀ and A⊂Cⁿ q(A)⩾exp(-nδ) implies q([A]_ε )⩾1-ε where [A]_ε is the ε-neighborhood of A in the sense of Hamming distance: [A]_ε={yⁿ∈Cⁿ:d(xⁿ ,yⁿ)⩽ε for some xⁿ∈A}. It is well-known that if q is i.i.d. then it has the blowing-up property. A way to prove this is by an inequality between informational divergence and d-distance of probability measures [Marton 1986]. The present authors generalize this inequality for some processes with memory, including mixing Markov chains. Furthermore, they prove a characterization of the blowing-up property