The overall performance of the Shannon code

Abstract
It is well-known that the redundancy of the Shannon code lies in the interval [0, 1). In order to study the overall performance of the Shannon code, we consider its redundancy as a random variable on the set of sources with n symbols, i.e., $R^{sh}(n)$, and examine its statistical parameters. It is shown that the mean of $R^{sh}(n)$ gets close to $0.5$ for sources with large alphabet size $n$. Moreover, we observe that its variance tends to zero as $n$ increases. Briefly speaking, for almost all sources with a large alphabet size, the redundancy of
the Shannon code is almost $0.5$ bits.