  This paper is devoted to the Gaussian fluctuations and deviations of the
traces of tridiagonal random matrix. Under quite general assumptions, we prove
that the traces are approximately normal distributed. Multi-dimensional central
limit theorem is also obtained here. These results have several applications to
various physical models and random matrix models, such as the Anderson model,
the random birth-death Markov kernel, the random birth-death $Q$ matrix and the
$\beta$-Hermite ensemble. Furthermore, under some independent and identically
distributed condition, we also prove the large deviation principle as well as
the moderate deviation principle for the traces.
