The entropy rate of the binary symmetric channel in the rare transitions regime

Consider a stationary ergodic sequence of random variables X = (Xn)n≥1 with values in a finite alphabet X = {0, ..., dx − 1} of size dx ≥ 2. Denote by pn(x), x = (x1, ..., xn) ∈ X the function pn(x) = P(X1 = x1, ..., Xn = xn) and let X be the vector with the entries X1, ..., Xn. As is well known, the entropy of X : H(X) = −E log pn(X) is the minimal number of bits on average, needed to encode a realization of X in a uniquely decodable way (see e.g. [6]). By the Shannon-McMillan-Breiman theorem the limits