  We focus on a data sequence produced by repetitive quantum measurement on an
internal hidden quantum system, and call it a hidden Markovian process. Using a
quantum version of the Perron-Frobenius theorem, we derive novel upper and
lower bounds for the cumulant generating function of the sample mean of the
data. Using these bounds, we derive the central limit theorem and large and
moderate deviations for the tail probability. Then, we give the asymptotic
variance is given by using the second derivative of the cumulant generating
function. We also derive another expression for the asymptotic variance by
considering the quantum version of the fundamental matrix. Further, we explain
how to extend our results to a general probabilistic system.
