I have a (fixed, long) string of bytes that I want to compress, $C$. I use a typical (good) lossless compression algorithm on it, to generate a compressed string of bytes, $C^*$. Then I define a random variable $X$ which samples a bit uniformly at random from $C^*$. Should I expect Prob[X = 0] to be close to 1/2?
A good compression algorithm is defined as one that achieves good information-theoretic bounds, such as Shannon coding.
A counter-example would be a compression algorithm that achieves, on average, a non-equal proportion of bits as the length of $C$ goes to infinity, where the contents of $C$ itself are chosen uniformly at random among all strings of that length.
This is to say the following: Let $S(n)$ be a string of length $n$ chosen uniformly at random. Let $C(s)$ be the result of the compression algorithm with input $s$. Let $X(s)$ denote the proportion of '0's in the string $s$. Then the question is whether $X(C(S(n)))$ necessarily converges to $1/2$ as $n$ goes to infinity.