  In this paper we continue our work on adiabatic time of time-inhomogeneous
Markov chains first introduced in Kovchegov (2010) and Bradford and Kovchegov
(2011). Our study is an analog to the well-known Quantum Adiabatic (QA) theorem
which characterizes the quantum adiabatic time for the evolution of a quantum
system as a result of applying of a series of Hamilton operators, each is a
linear combination of two given initial and final Hamilton operators, i.e.
$\mathbf{H}(s) = (1-s)\mathbf{H_0} + s\mathbf{H_1}$. Informally, the quantum
adiabatic time of a quantum system specifies the speed at which the Hamiltonian
operators changes so that the ground state of the system at any time $s$ will
always remain $\epsilon$-close to that induced by the Hamilton operator
$\mathbf{H}(s)$ at time $s$. Analogously, we derive a sufficient condition for
the stable adiabatic time of a time-inhomogeneous Markov evolution specified by
applying a series of transition probability matrices, each is a linear
combination of two given irreducible and aperiodic transition probability
matrices, i.e., $\mathbf{P_{t}} = (1-t)\mathbf{P_{0}} + t\mathbf{P_{1}}$. In
particular we show that the stable adiabatic time $t_{sad}(\mathbf{P_{0}},
\mathbf{P_{1}}, \epsilon) = O (t_{mix}^{4}(\epsilon \slash 2) \slash
\epsilon^{3}), $ where $t_{mix}$ denotes the maximum mixing time over all
$\mathbf{P_{t}}$ for $0 \leq t \leq 1$.
