A theorem on the entropy of certain binary sequences and applications--I

In this, the first part of a two-part paper, we establish a theorem concerning the entropy of a certain sequence of binary random variables. In the sequel we will apply this result to the solution of three problems in multi-user communication, two of which have been open for some time. Specifically we show the following. Let
X
and
Y
be binary random
n
-vectors, which are the input and output, respectively, of a binary symmetric channel with "crossover" probability
p_0
. Let
H\{X\}
and
H\{ Y\}
be the entropies of
X
and
Y
, respectively. Then \begin{equation} \begin{split} \frac{1}{n} H\{X\} \geq h(\alpha_0), \qquad 0 \leq \alpha_0 &\leq 1, \Rightarrow \\ \qquad \qquad \&\qquad \frac{1}{n}H\{Y\} \geq h(\alpha_0(1 - p_0) + (1 - \alpha_0)p_0) \end{split} \end{equation} where
h(\lambda) = -\lambda \log \lambda - (1 - \lambda) \log(l - \lambda), 0 \leq \lambda \leq 1
.