  The Good-Turing (GT) estimator for the missing mass (i.e., total probability
of missing symbols) in $n$ samples is the number of symbols that appeared
exactly once divided by $n$. For i.i.d. samples, the bias and squared-error
risk of the GT estimator can be shown to fall as $1/n$ by bounding the expected
error uniformly over all symbols. In this work, we study convergence of the GT
estimator for missing stationary mass (i.e., total stationary probability of
missing symbols) of Markov samples on an alphabet $\mathcal{X}$ with stationary
distribution $[\pi_x:x \in \mathcal{X}]$ and transition probability matrix
(t.p.m.) $P$. This is an important and interesting problem because GT is widely
used in applications with temporal dependencies such as language models
assigning probabilities to word sequences, which are modelled as Markov. We
show that convergence of GT depends on convergence of $(P^{\sim x})^n$, where
$P^{\sim x}$ is $P$ with the $x$-th column zeroed out. This, in turn, depends
on the Perron eigenvalue $\lambda^{\sim x}$ of $P^{\sim x}$ and its
relationship with $\pi_x$ uniformly over $x$. For randomly generated t.p.ms and
t.p.ms derived from New York Times and Charles Dickens corpora, we numerically
exhibit such uniform-over-$x$ relationships between $\lambda^{\sim x}$ and
$\pi_x$. This supports the observed success of GT in language models and
practical text data scenarios. For Markov chains with rank-2, diagonalizable
t.p.ms having spectral gap $\beta$, we show minimax rate upper and lower bounds
of $1/(n\beta^5)$ and $1/(n\beta)$, respectively, for the estimation of
stationary missing mass. This theoretical result extends the $1/n$ minimax rate
for i.i.d. or rank-1 t.p.ms to rank-2 Markov, and is a first such minimax rate
result for missing mass of Markov samples.
