  Here we develop an effective approach to simplify two-time-scale Markov
chains with infinite state spaces by removal of states with fast leaving rates,
which improves the simplification method of finite Markov chains. We introduce
the concept of fast transition paths and show that the effective transitions of
the reduced chain can be represented as the superposition of the direct
transitions and the indirect transitions via all the fast transition paths.
Furthermore, we apply our simplification approach to the standard Markov model
of single-cell stochastic gene expression and provide a mathematical theory of
random gene expression bursts. We give the precise mathematical conditions for
the bursting kinetics of both mRNAs and proteins. It turns out that random
bursts exactly correspond to the fast transition paths of the Markov model.
This helps us gain a better understanding of the physics behind the bursting
kinetics as an emergent behavior from the fundamental multi-scale biochemical
reaction kinetics of stochastic gene expression.
